CALCULUS
with the
Casio FX-9750G Plus
CALCULUS with the Casio FX-9750G Plus
Limits
Derivatives
Continuity
Slope
Linear Functions
Differentiability
Polynomials
Trigonometric Functions
Graphing Models
Slope Fields
Anti Derivatives
Integration
Riemann Sums
Activities for the Classroom
9750-CALC
Casio, Inc.
CALCULUS with the Casio FX-9750G Plus
Activities for the Classroom
All activities in this resource
are also compatible with the
Casio CFX-9850G Series.
CALCULUS
with the
Casio FX-9750G Plus
Kevin Fitzpatrick
® 2005 by CASIO, Inc.
570 Mt. Pleasant Avenue
Dover, NJ 07801
www.casio.com
9750-CALC
The contents of this book can be used by the classroom teacher to make reproductions for
student use. All rights reserved. No part of this publication may be reproduced or utilized
in any form by any means, electronic or mechanical, including photocopying, recording,
or by any information storage or retrieval system without permission in writing from CASIO.
Printed in the United States of America.
Design, production, and editing by Pencil Point Studio
Contents
Activity 1: Looking at Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Teaching Notes
Student Activity
Calculator Notes and Answers
Activity 2: Do Limits Take Sides? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Teaching Notes
Student Activity
Calculator Notes and Answers
Activity 3: A Graphical Look at Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Teaching Notes
Student Activity
Calculator Notes and Answers
Activity 4: Introduction to Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Teaching Notes
Student Activity
Calculator Notes and Answers
Activity 5: Being Locally Linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Teaching Notes
Student Activity
Calculator Notes and Answers
Activity 6: Continuity Meets Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Teaching Notes
Student Activity
Calculator Notes and Answers
Activity 7: Derivative Behavior of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Teaching Notes
Student Activity
Calculator Notes and Answers
Activity 8: Derivative Behavior of Common
Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Teaching Notes
Student Activity
Calculator Notes and Answers
Copyright © Casio, Inc.
Calculus with the Casio fx-9750G Plus
iii
Activity 9: Looking at Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Teaching Notes
Student Activity
Calculator Notes and Answers
Activity 10: Looking at Slope Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Teaching Notes
Student Activity
Calculator Notes and Answers
Activity 11: Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Teaching Notes
Student Activity
Calculator Notes and Answers
Appendix:
Overview of the Calculator Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
iv
Calculus with the Casio fx-9750G Plus
Copyright © Casio, Inc.
Activity 1
Looking at Limits
Teaching Notes
Topic Area: Limits
Class Time: one 45-50 minute class period
Overview
This activity will encourage students to use graphical and numerical representations
to examine the behavior of a function as it approaches a particular input value.
A limit is one of the foundation concepts in any calculus course. The idea behind this
activity is to have the student investigate both numerically and graphically the behavior of the output of a function as its input moves closer and closer to some point of
interest. The emphasis will be on examining the behavior of the function as its gets
near a particular input value. Even though the function may reach that input value,
the activity will be centered more on what happens as the input gets closer and
closer to the value of interest.
Objectives
•
To develop an understanding of meaning of a "limit”
•
To be able to estimate the value of a limit using a numerical view from a
table and a graphical view
Getting Started
Using the Casio fx-9750G Plus, have students work in pairs or small groups
arranged prior to beginning the activity to maximize student involvement and
ownership of the results.
Prior to using this activity:
•
Students should be able to produce and manipulate graphs and tables of
values manually and with the graphing utility.
•
Students should have a basic understanding of the language of functions.
•
Students should be able to identify rational and exponential functions.
Ways students can provide evidence of learning:
•
If given a function, the student can state and explain what the limit is at a
particular value.
•
If given a graphical representation of a function, the student can state and
explain what the limit is at a particular value.
•
If given a tabular representation of a function, the student can state and
explain what the limit is at a particular value.
Common mistakes to be on the lookout for:
•
Students may use viewing windows that appear to show functions being
defined when they are not.
•
Students may use an input or table value with an increment so small that the
calculator will display a rounded value that does not actually exist.
•
Students may use an input or table value with an increment so small that the
calculator will return an error message regarding memory overflow.
Copyright © Casio, Inc.
Activity 1 • Calculus with the Casio fx-9750G Plus
1
Name _____________________________________________ Class ________ Date ________________
Activity 1 • Looking at Limits
Introduction
This activity will encourage you to use graphical and numerical representations to
examine the behavior of a function as it approaches a particular input value.
Using the Casio fx-9750G Plus you will be working in pairs or small groups.
Problems and Questions
2
Examine the value of the function f(x)= x – 1 as the value of x gets close to 1.
x–1
1. Go to the MENU and choose the TABLE option.
2. Enter the function in Y1.
3. Set up the table as shown below.
4. Display the table and record the function values when x = {0,1,2}.
x
y
0
1
2
5. Explain why the values you recorded either did or did not match up with your
expectations.
_____________________________________________________________________________
_____________________________________________________________________________
6. Now have the table start at .5, and change the pitch to .5 as well.
7. Record the values you get for x = {.5, 1, 1.5}
x
y
.5
1
1.5
2
Calculus with the Casio fx-9750G Plus • Activity 1
Copyright © Casio, Inc.
Name _____________________________________________ Class ________ Date ________________
Activity 1 • Looking at Limits
8. Repeat the process, this time starting the table at .75 and changing the pitch to
.25, then record the function values for x = {.75, 1, 1.25}.
x
y
.75
1
1.25
9. Repeat the process twice more.
•
the first time starting at .9 with a pitch of .1 Record the values for
x = {.9, 1, 1.1}
x
y
.9
1
1.1
• the second time starting at .99 with a pitch of .01. Record the values for
x = {.99, 1, 1.01}
x
y
.99
1
1.01
10. What would you expect to see if the pitch was changed to .001, to .0001?
_____________________________________________________________________________
11. What function value does it appear to close in on?
_____________________________________________________________________________
Now examine the graph of the same function to see the behavior.
12. Choose GRAPH from the Menu and set the INIT viewing window as shown
below.
Copyright © Casio, Inc.
Activity 1 • Calculus with the Casio fx-9750G Plus
3
Name _____________________________________________ Class ________ Date ________________
Activity 1 • Looking at Limits
Then sketch the graph on the axis below.
13. Go to ZOOM and press F2 (zoom factors), set the zoom factors as follows:
•
Xfact: 4
•
Yfact: 2
14. Graph the function again, Trace to the point (.9, 1.9) and Zoom-In. Write a
description of what you see and include a sketch to support your statements.
_____________________________________________________________________________
_____________________________________________________________________________
15. Trace to the point (1.025, 2.025) and Zoom-In again. Write a description of what
you see and include a sketch to support your statements.
_____________________________________________________________________________
_____________________________________________________________________________
16. Continue to repeat the process, tracing closer and closer to the value x = 1, from
values both above and below x = 1, each time Zoom-In, until you are
comfortable drawing a conclusion.
17. If the values of a function come closer and closer to a single value, that value is
called the limit of the function and is expressed as "as x approaches some value
(c), f(x) has a limit of L" Rewrite your conclusion to these examinations using
the phrasing shown here.
_____________________________________________________________________________
_____________________________________________________________________________
x2 – 1
18. Examine the function: f(x)= x – 2 around the value x = 2 using a table set up
starting at x = 1, ending at x = 3, and having a pitch of 1, record the values for
x = {1,2,3}.
x
y
1
2
3
4
Calculus with the Casio fx-9750G Plus • Activity 1
Copyright © Casio, Inc.
Name _____________________________________________ Class ________ Date ________________
Activity 1 • Looking at Limits
19. Change the pitch of the table (table increments) as before, first to .5, then to .25,
then to .1 and finally to .01. Each time, recording the values directly above and
below x = 2 in each case.
x
2
y
x
2
y
x
2
y
x
y
2
20. Now use the same graphical analysis process with this function and write a
conjecture based upon the numerical and graphical evidence.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
21. When an input approaches a single value and the output also approaches a
single value the function is said to have a limit, however when the output
does not approach a single value, the function is said to have no limit.
Using the phrasing from Question 17, express your conclusion using the
proper phrasing.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
Further Exploration
Find the limit, if it exists, for each of the following. If it does not exist, explain why.
22.
23.
24.
2
lim 2xx – 50
3 + 15
______________________________________________________________
x
lim 3 – 1
x→0
x
______________________________________________________________
lim x + 2
x+4
______________________________________________________________
x→5
x→0
25. xlim
→-3
Copyright © Casio, Inc.
3x + 12
4–x
______________________________________________________________
Activity 1 • Calculus with the Casio fx-9750G Plus
5
Calculator Notes and Answers for Activity 1
To Get to the TABLE screen:
•
From the Main MENU either press 7, or use the arrow keys to highlight TABLE
and press EXE.
To get to the TABLE SET UP:
•
While in the Table Function, press F5 (RANGE) key.
To get to the Zoom Factors screen:
•
After graphing press SHIFT F2 (Zoom).
•
Press F2 (FACT) key.
Answers:
4.
5. Answers will vary, however, most students should recognize that at x = 1 there
is division by zero and that is creating the error being displayed.
6. n/a
7.
8.
9.
10. Answers will vary but a good answer should contain the fact that they value is
closing in on 2 as x approaches 1.
6
Calculus with the Casio fx-9750G Plus • Activity 1
Copyright © Casio, Inc.
Calculator Notes and Answers for Activity 1
11. Here the answers should not vary, a value of 2 is the correct answer.
12.
A good sketch will show the hole in the graph and look something like this:
Note: The reason for setting the particular viewing window in this activity is to make
sure the hole is visible. The calculator will only show the gap if it is a specific pixel it is
asked to light up and that pixel does not exist at that point. In a many other viewing
windows the point (1,2) would not be one that the 9750 would try to graph, thus in connected mode the hole would not appear and the graph would appear to be continuous.
13. n/a
14. Answers will vary but should contain a statement about the maintenance of
the discontinuity (hole) in the graph.
15. The description should include mention of the hole and a better description
would include a statement about the value closing in on 2, while still not
existing at x = 2.
16. A good conclusion would center around the value getting infinitely close to 2
as x gets closer and closer to 1.
17. "as x approaches 1, f(x) has a limit of 2"
18.
19.
Copyright © Casio, Inc.
Activity 1 • Calculus with the Casio fx-9750G Plus
7
Calculator Notes and Answers for Activity 1
20. Students should produce some graphs showing the following sketch, and the
idea of asymptotes should be mentioned.
Note: This is a good time to discuss the window again, here there is not missing pixel
but care needs to be taken to show both branches of the graph. If the proper vertical window is not set, only one branch will be found leading to an incorrect answer.
21. "as x approaches 2 f(x) has a no limit"
22. 0
23. 1.099 approximately
24. 1
25. .247
Note: Some students may realize that 24, 25 can be done by direct substitution, this
should cement discussion regarding the fact that while it is not necessary for a limit
to actually be a value of the function, it certainly can be. This also can be used to foreshadow a discussion of continuity.
8
Calculus with the Casio fx-9750G Plus • Activity 1
Copyright © Casio, Inc.
Activity 2
Do Limits Take Sides?
Teaching Notes
Topic Area: Limits
Class Time: one 45-50 minute class period
Overview
This activity will encourage students to use graphical and numerical representations
to examine the idea of a limit needing to be the same from both directions of
approach.
The concept of a limit creates the framework for discussing continuity. Using splitdefined functions, the goal of this activity is to put a face on the idea of one-sided
limits.
Objectives
•
To develop an understanding of meaning of one sided limits
•
To be able to understand and communicate the idea that for a function to have
a limit at a point, it must approach the same output value from either direction.
Class Time: This activity is designed to be used in one 45-50 minute class period.
Getting Started
Using the Casio fx-9750G Plus, have students work in pairs or small groups
arranged prior to beginning the activity to maximize student involvement and
ownership of the results.
Prior to using this activity:
• Students should be able to produce and manipulate graphs and tables of
values manually and with the graphing utility.
•
Students should be able to produce split defined (or piecewise) functions.
•
Students should have a basic understanding of the language of functions.
•
Students should be able to identify rational and exponential functions.
Ways students can provide evidence of learning:
• If given a split defined function, the student can produce a picture of the
function using the calculator.
•
If given a graphical representation of a function, the student can state and
explain what the limit is as it approaches an input value from the left side or
the right side.
Common mistakes to be on the lookout for:
• Students may produce a graph on the calculator and not be able to
communicate the concept of a split-defined function as window chosen may
produce the appearance of single formula.
Copyright © Casio, Inc.
Activity 2 • Calculus with the Casio fx-9750G Plus
9
Name _____________________________________________ Class ________ Date ________________
Activity 2 • Do Limits Take Sides?
Introduction
This activity will have you use graphical and numerical representations to examine
the idea of a limit needing to be the same from both directions of approach.
Using the Casio fx-9750G Plus you will be working in pairs or small groups.
Problems and Questions
{
Examine the behavior of the function: f(x)= x – 4, x < 2 as the value of x
x – 1, x > 2
approaches 2:
1. Choose GRAPH from the MENU, enter the function.
2. Set the initial viewing window to Standard by pressing F3 (STD).
3. Copy the graph on the axis shown and describe what you see:
_____________________________________________________________________________
_____________________________________________________________________________
4. Using the trace function, record your observations as to what happens as you
trace along the function moving closer and closer to the value x = 2.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
5. Using any zoom technique you prefer, keep both branches visible and keeping
x = 2 toward the center of the window redraw the graph getting a closer and
closer look at the output of the function. Explain what you see.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
6. From your knowledge of limits, and based upon what you see in this case, what
is the xlim2 f(x) ? Explain your answer.
→
_____________________________________________________________________________
_____________________________________________________________________________
10
_____________________________________________________________________________
Calculus with the Casio fx-9750G Plus • Activity 2
Copyright © Casio, Inc.
Name _____________________________________________ Class ________ Date ________________
Activity 2 • Do Limits Take Sides?
7. The symbolic notation: lim+ f(x) means to investigate the limit of the function,
x→c
f(x), as x approaches some value c through values that are greater than c
(frequently called "from the right"). In this case, using your trace cursor, copy the
graph and show what that means.
8. Describe your results using some ordered pairs to show the respective input and
output relationships.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
9. How would you now answer the question: Find lim+ f(x) ?
x→2
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
10. Based upon this investigation so far, how would you describe the notation:
lim f(x) ?
x 2–
→
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
11. How would you answer the question: Find lim– f(x)? Why?
x→2
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
12. How would you now answer the question: Find xlim2 f(x) ? Why?
→
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
Copyright © Casio, Inc.
Activity 2 • Calculus with the Casio fx-9750G Plus
11
Name _____________________________________________ Class ________ Date ________________
Activity 2 • Do Limits Take Sides?
13. Graph the function h(x)=
{
1 , x < –1
x+ 3
x2 + 3, –1< x< 2
–x + 9, x > 2
in the window
Sketch what you see on the axes.
14. Find each of the following limits and explain how you arrived at your conclusion
lim h(x)
______________________________________________________________
lim h(x)
______________________________________________________________
lim h(x)
______________________________________________________________
lim h(x)
______________________________________________________________
lim h(x)
______________________________________________________________
lim h(x)
______________________________________________________________
lim h(x)
______________________________________________________________
h. lim- h(x)
x 3
______________________________________________________________
a.
x→–3
b.
x→–3-
c.
x→–3+
d.
x→–1+
e.
f.
g.
x→–1
x→0
x→3
→
12
Calculus with the Casio fx-9750G Plus • Activity 2
Copyright © Casio, Inc.
Calculator Notes and Answers for Activity 2
How to graph a split defined function:
•
•
•
•
Enter each branch in its own Y= slot then create the restrictions by using putting
them in [lower, upper]
Example to graph f(x)= x – 4, x < 2 you would enter it as follows:
x – 1, x > 2
Y1 = x – 4, [lower, 2]
{
Y2 = x –1 , [2, upper] Note: The lower and upper can usually be just the min
and max of the viewing window if you only have two branches.
3.
The graph in standard window
Note: When students graph it they should
be very clear to indicate that there are
open circles at the endpoints of the "jump."
4.
As the input value gets closer to 2, the lower branch gets closer to –2, while the
upper branch gets closer to 1
5.
Should have the same results are in #4, but the numbers should be getting closer
to –2 and 1 respectively
6.
The function does not have a limit as x approaches 2 since the values are
different depending upon the direction you approach the input.
7.
One view of what happens as the cursor gets closer to 2, answers will vary.
Copyright © Casio, Inc.
Activity 2 • Calculus with the Casio fx-9750G Plus
13
Calculator Notes and Answers for Activity 2
8. Answers will vary, see above for some possible ordered pairs.
9. limit is 1
10. What is the limit of the function, as the input approaches 2 from values below 2
(or to the left of 2)?
11. The limit is –2. The explanations will vary, but a good explanation should cover
the fact that as the value "walks" along the function from values to the left of 2,
the input gets increasingly closer to –2.
12. Answer should be the same as 6.
13.
This is a representation of what the
student should sketch.
14a. xlim
h(x) = None, two different one sided limits
–3
→
b. xlim
h(x) = – (Note: while "none" is also acceptable, – is a more complete
→–3description of what is actually taking place.)
h(x) = (Note: while "none" is also acceptable, is a more complete
c. x lim
→–3+
description of what is actually taking place.)
h(x) = 4
d. x lim
→–1+
e. xlim
h(x) = None, two different one-sided limits
–1
→
f. xlim0 h(x) = 3
→
g. xlim3 h(x) = None, two different one-sided limits
→
h. xlim
h(x) = 12
3-
→
14
Calculus with the Casio fx-9750G Plus • Activity 2
Copyright © Casio, Inc.
Activity 3
A Graphical Look at Continuity
Teaching Notes
Topic Area: Derivatives and Continuity
Class Time: an exploratory introduction during the first 30 minutes of a class period
on the topic of continuity
Overview
This activity will have students explore the concept of continuity at a point. It will
also allow them to discover that simply having a limit at a point will not guarantee
that the function is also continuous.
It also explores the idea that a having a limit is a necessary, but not a sufficient condition to determine the continuity of a function at a point, and through all points.
Objectives
•
To develop a visual understanding of how limits and continuity relate
•
To be able to understand and communicate what it means for a function to be
continuous at a point
Getting Started
Using the Casio fx-9750G Plus, have students work in pairs or small groups
arranged prior to beginning the activity to maximize student involvement and
ownership of the results.
Prior to using this activity:
•
Students should be able to produce and manipulate graphs of functions
manually and with the graphing utility.
•
Students should be able to produce split defined (or piecewise) functions.
•
Students should have a basic understanding of the language of limits.
Ways students can provide evidence of learning:
•
Students should be able to produce graphs of functions and communicate
symbolically, graphically and verbally the relationship between having a limit
and being continuous.
Common mistakes to be on the lookout for:
•
Students may produce a graph on the calculator in such a way that the window
chosen may produce the appearance of a continuous function when, in fact, it is
not.
•
Students may confuse the pixel values with the actual function values.
Copyright © Casio, Inc.
Activity 3 • Calculus with the Casio fx-9750G Plus
15
Name _____________________________________________ Class ________ Date ________________
Activity 3 • A Graphical Look at Continuity
Introduction
This activity will have have you explore the concept of continuity at a point. It will
also allow you to discover that simply having a limit at a point will not guarantee
that the function is also continuous.
Using the Casio fx-9750G Plus you will be working in pairs or small groups.
Problems and Questions
Explore the behavior of the function f(x)= x2 – x – 6 around the vertex:
1. Go to the GRAPH menu and, in the viewing window, produce the graph of the
function f(x) and copy it to the axes.
2. Find and record the vertex of the function.
3. Making sure your zoom factors are set to 4 for both X and Y, trace to the vertex
and zoom in, record what you see.
4. What does it appear the value of xlim5 f(x) is? Explain why you arrived at that
→
answer.
_____________________________________________________________________________
_____________________________________________________________________________
16
Calculus with the Casio fx-9750G Plus • Activity 3
Copyright © Casio, Inc.
Name _____________________________________________ Class ________ Date ________________
Activity 3 • A Graphical Look at Continuity
5. Now explore the behavior of the split-defined function: g(x)=
Use the same viewing window as before.
{
x2 – x – 6, x< .5
–6, x = .5
x2 – x – 6, x>.5
Record what you see below.
6. What does it appear the value of xlim
g(x) is?
→.5
How does it compare to xlim
f(x) ?
.5
→
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
7. Now, trace to a value where x = .4, and zoom in, describe and record what you
see.
_____________________________________________________________________________
_____________________________________________________________________________
g(x) , lim - g(x) , lim g(x)
8. Find: xlim
x→.5
x→5
→.5+
9. Now find g(.5), how does this compare to your answers above?
_____________________________________________________________________________
_____________________________________________________________________________
10. Draw a conclusion about the relationship between limits and continuity.
_____________________________________________________________________________
_____________________________________________________________________________
Copyright © Casio, Inc.
Activity 3 • Calculus with the Casio fx-9750G Plus
17
Calculator Notes and Answers for Activity 3
1.
2. Vertex is (.5, -6.25) and can be found symbolically or using the MIN function in
the G-Solve folder.
3.
This is the screen sequence for the zoom.
Nothing unusual should be seen.
The vertex remains, the function is
continuous.
4. The limit is –6.25, the vertical value of the vertex. Answer will vary as to how it
was arrived at. Care should be taken to point out that simply tracing to a value is
not confirmation enough and can be tricky. Direct substitution is a valid
explanation. A good answer might also include a mention of "passing through" or
even a mention of continuity.
18
Calculus with the Casio fx-9750G Plus • Activity 3
Copyright © Casio, Inc.
Calculator Notes and Answers for Activity 3
5.
This provides a good look at the split
defined function.
The graph produced in the given window
will be as shown on the left.
The discontinuity will not be immediately
apparent from this graph.
6. The limit is –6.25. Answers may vary as students begin to get the idea that the
change in the definition of the function may be creating some problems,
although not with the limit. This is a good checkpoint for the understanding of
what it means to be a "limit."
7.
Copyright © Casio, Inc.
This is the screen sequence that produces
the desired screen.
Activity 3 • Calculus with the Casio fx-9750G Plus
19
Calculator Notes and Answers for Activity 3
8. All three limits are –6.25, although some students may try to refine the answers
to longer decimals. This provides another good opportunity to stress the idea of
"limit" as a value the function approaches.
9.
g(.5) = -6, a value different from the limit.
10. A good answer will include the fact that the function has a gap or a hole or a
jump (ie, a point of discontinuity at x = .5). The idea is to have them begin to
think about the fact that simply having a limit does not guarantee the continuity
of a function.
20
Calculus with the Casio fx-9750G Plus • Activity 3
Copyright © Casio, Inc.
Activity 4
Introduction to Derivatives
Teaching Notes
Topic Area: Derivatives
Overview
This activity will have students begin to connect the concept of slope and rate of
change to the derivative.
It also provides an introduction to the concept that the slope of a function extends
beyond linear slope, but that using the slope of a line can foster a discussion of average vs. instantaneous rates of change.
Objectives
•
To develop an understanding of the slope of a function that is not just linear
•
To be able to understand and communicate the visuals connected with the
average rate of change and the secant line to a function
Getting Started
Using the Casio fx-9750G Plus, have students work in pairs or small groups
arranged prior to beginning the activity to maximize student involvement and
ownership of the results.
Prior to using this activity:
•
Students should be able to produce and manipulate graphs of functions
manually and with the graphing utility.
•
Students should be able to use the statistics Menu to produce linear and
quadratic regression models.
•
Students should have a basic understanding of the language of limits.
•
Students should have an understanding of what a secant line is.
•
Students should have an understanding of slope as a rate of change.
Ways students can provide evidence of learning:
•
Students should be able to produce graphs of functions and communicate
symbolically, graphically, numerically and verbally the relationship between the
slope of a line, a function and an average rate of change
Common mistakes to be on the lookout for:
•
Not being able to relate the slope to a real world rate of change concept
•
Not being able to communicate the slope as the rate of change of output over
input
Copyright © Casio, Inc.
Activity 4 • Calculus with the Casio fx-9750G Plus
21
Name _____________________________________________ Class ________ Date ________________
Activity 4 • Introduction to Derivatives
Introduction
This activity provides an introduction to the concept that the slope of a function
extends beyond linear slope, but that using the slope of a line can foster a discussion of average vs. instantaneous rates of change.
Using the Casio fx-9750G Plus you will be working in pairs or small groups.
Problems and Questions
1. Calculate the slope of the line connecting the points (2,5) and (5,2)?
____________________________________
2. Describe the meaning of the slope you just found in terms of input and
output.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
3. Now calculate the slope of the line connecting the points (-1,8) and
(11,-4)
____________________________________
4. What conclusions, if any, can you draw about these 4 points? Explain.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
5. Name two other points that would share the same characteristics as these
points? Explain your choices.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
6. If, at the end of his first year of employment, Mike’s annual salary was $42,000
and at the end of his 3rd year of employment with the same company, Mike’s
annual salary was $49,000.
What conclusion could you draw about the growth of Mike’s salary over that
period of time? Explain.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
22
Calculus with the Casio fx-9750G Plus • Activity 4
Copyright © Casio, Inc.
Name _____________________________________________ Class ________ Date ________________
Activity 4 • Introduction to Derivatives
7. Given the same data as above, if Mike were to stay with the same company for
10 years, predict what his salary should be at the end of those 10 years.
Explain.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
8. What if Mike’s actual salary after 10 years was $100,000? How does that agree
with your prediction from above? How does that compare to the rate of growth
you used in your prediction in item #7?
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
9. Create a good model using the data at the end of the first, third and tenth year
salaries. Record the result here and explain why you chose your model.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
10. Using your model from item 9, what would you say that average change in
Mike’s salary was between years 4 and 10? Between years 4 and 9? Between
years 4 and 6?
Explain how you arrived at your answers.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
11. How might you estimate the rate that Mike’s salary would be growing at the end
of the 5th year with the company?
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
12. Now find the equation of the secant line connecting the points (4.9, 58767) and
(5.1, 59972)
____________________________________
Copyright © Casio, Inc.
Activity 4 • Calculus with the Casio fx-9750G Plus
23
Name _____________________________________________ Class ________ Date ________________
Activity 4 • Introduction to Derivatives
13. Graph the model you created in item 9, and the equation of the line from item
12 in the following viewing window:
Copy the graph and explain what you see:
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
14. The derivative of a function at a point (also known as the instantaneous rate of
change) is the same as the slope of the line tangent to the function at that point.
Based upon your exploration what could you estimate the derivative of your
salary model to be at the end of the 5th year? And how does that translate to
Mike’s salary growth rate during that same time period?
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
Extension
Given the function f(x)= 3x2 – 2x + 1 find a good estimate for the equation of the
line tangent to f(x) at x = 2. Explain your process and how accurate you think you
are.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
24
Calculus with the Casio fx-9750G Plus • Activity 4
Copyright © Casio, Inc.
Calculator Notes and Answers for Activity 4
1. -1
2. Answers will vary: A complete answer should include a mention of the relative
change of a decrease in output by 1 for every increase in the input of 1.
3.
-1
4. Answers will vary. However all should include mention that they have the same
slope. Plotting the points using the STAT mode will also show that they are on
the same line. Care should be taken to point out that JUST because they share a
slope does not put them on the same line.
5. Answers will vary. Any other points that have slopes of –1 will work, however, if
the answer given to #4 includes the co linearity of the points, then the additional
points chosen should also be on that same line.
6. Answers will vary, but should include a mention that his salary has raised an
average of $3500 per year over the time period in question.
7. $73,500 This answer can be found by either using the slope or creating the
equation of the line connecting the points (1,42000) and (3, 49000) and
extrapolating.
8. That actual salary would be greater, thus the growth rate will have had to have
been greater at some point for that to take place. If a numerical comparison of
the growth rates are attempted, it must be made clear by the student what they
are using to create that new comparison and they should be prompted to explain
why they have made that choice.
9. A good answer should be the creation of the quadratic equation that results from
using the three points (1,42000) , (3, 49000) and (10, 100000).
10. Answers will vary. Most students will likely find the values of the model
associated with 4, 6, 9, and using the given value at 10 and find the slopes of the
respective secant lines.
Some students may begin to suggest that because of the function behavior, these
secant values are not good predictors.
Between 4 and 10: Average increase is $7706 per year
Between 4 and 9: Average increase is $7286 per year
Between 4 and 6: Average increase is $6024 per year
Copyright © Casio, Inc.
Activity 4 • Calculus with the Casio fx-9750G Plus
25
Calculator Notes and Answers for Activity 4
11. Answers will vary. Some students might take the growth between 4 and 5
[$5603] and then 5 and 6 [6445] and take the average [$6024] some may begin to
estimate closer, perhaps anticipating the question asked in item 12, some may
estimate over an even closer slope interval. Care should be taken to make sure
that the students continue to use slope and discuss rate of change and not simply
plug 5 into some model and use the output for the answer to the question.
12. y = 6025x + 29244.50
13.
While answers will vary, a good answer should point out that the parabola is the
model of the actual data and the line is the secant line connecting the two given
points. Some answers may being to bring up the concept of the tangent line and
it’s very close relationship to the curve at the point of tangency.
14. The actual value of the derivative at 5, to the nearest cent is $6023.81 This is
close to the secant line slopes as the student gets closer and closer to 5 from
either side. Here a discussion of limits as it pertains to the finding of the slope is
also a good extension.
Extension
Answers will vary, the actual answer is y = 10x – 11 .
Care should be taken to be sure that students don’t simply use the calculator function
to create the line without being able to communicate the connection between the
slope of the secant line/tangent line and the value of the function at x = 2. The student
estimation of accuracy will depend upon their process.
26
Calculus with the Casio fx-9750G Plus • Activity 4
Copyright © Casio, Inc.
Calculator Notes and Answers for Activity 4
How to do a regression on the Casio fx-9750 Plus
1.
From the MENU press 2 (STAT)
2.
Input the x-values into List 1, and the y-values into List 2.
3.
Press F2 (CALC), then F3(REG).
4.
Your basic menu choices then become: F1(linear), F2(med-med line),
F3(quadratic), F4(cubic), F5(quartic), F6(next page).
5.
After choosing the model you want, the next screen will produce the values and
the general model.
Copyright © Casio, Inc.
Activity 4 • Calculus with the Casio fx-9750G Plus
27
Calculator Notes and Answers for Activity 4
You also have the option of graphing the points, creating and copying the model from
there.
a)
Start at the STAT menu, put the values in the lists as you need, this time press
F1(GRPH), then choose F1(GPH1). The calculator will set a proper window and
plot the points.
b)
You now have the same model choices along the F1-F6 keys.
c)
After you make your choice it will create the model and give you the options to
draw it, and or copy it to the function grapher.
d)
Choose F5 (COPY) and it will take you to the Y= screen where you can choose the
place you want to put it and press EXE to store the entire function which you can
then access at any time by going to the GRAPH section from the main MENU.
e)
If you choose DRAW it will draw the model through the points you’ve graphed.
(b)
(c)
(d)
(e)
(Accessing the GRAPH section and the newly stored function)
28
Calculus with the Casio fx-9750G Plus • Activity 4
Copyright © Casio, Inc.
Activity 5
Being Locally Linear
Teaching Notes
Topic Area: Derivatives and Slope
Class Time: an exploration during the first part of a class period while connecting the
slope of a function to the derivative
Overview
This activity will begin to bring home the point that as the behavior around a single
point on a differentiable function is examined, the function will "flatten out" and very
much resemble the behavior of a line drawn through the point of interest. The example given should motivate a discussion of what it means to be locally linear with
regard to a differentiable function.
Objectives
•
To connect the much earlier concept of linear slope to the examination of the
rate of change of a function and the idea of what a derivative is
•
To be able to understand and communicate the visual and numerical ideas of
linear slope and its relationship to the instantaneous rate of change of any
function
Getting Started
Using the Casio fx-9750G Plus, have students work in pairs or small groups
arranged prior to beginning the activity to maximize student involvement and
ownership of the results.
Prior to using this activity:
•
Students should be able to produce and manipulate graphs of functions
manually and with the graphing utility.
•
Students should have an understanding of "decimal" and "standard" window
and how to easily produce them.
•
Students should be able to use Zoom features of the graphing utility to examine
specific parts of the graph.
•
Students should have an understanding of slope of a line as a rate of change.
Ways students can provide evidence of learning:
•
Students should be able to produce graphs of functions and communicate
changes taking place to the appearance of a function as they zoom in on a
particular value.
Common mistakes to be on the lookout for:
•
Not understanding the zoom process and what is taking place
Copyright © Casio, Inc.
Activity 5 • Calculus with the Casio fx-9750G Plus
29
Name _____________________________________________ Class ________ Date ________________
Activity 5 • Being Locally Linear
Introduction
This activity will begin to bring home the point that as the behavior around a single point on a differentiable function is examined, the function will "flatten out"
and very much resemble the behavior of a line drawn through the point of interest.
Using the Casio fx-9750G Plus you will be working in pairs or small groups.
Problems and Questions
1. Graph the function y = x2 – 2x – 3 in the viewing window.
Record the results below.
2. What can you say about the slope of the function over the viewing window?
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
3. Set your zoom factors to:
30
Calculus with the Casio fx-9750G Plus • Activity 5
Copyright © Casio, Inc.
Name _____________________________________________ Class ________ Date ________________
Activity 5 • Being Locally Linear
Trace to x = 2 and zoom in at that point. Record your results below.
4. Using the trace function, record both the x and y values immediately above and
below x = 2:
x
y
2
-3
5. Find the equation of the line connecting the first and third points in your table
above.
____________________________________
6. Graph the line along with the original function in the last window you have and
record the results below.
7. Zoom in on both at x = 2 and describe what you see.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
Copyright © Casio, Inc.
Activity 5 • Calculus with the Casio fx-9750G Plus
31
Name _____________________________________________ Class ________ Date ________________
Activity 5 • Being Locally Linear
8. As the behavior of a function is examined closer and closer to a particular point
of interest, in many cases the function begins to "flatten out", ie, become
approximately linear over a very small neighborhood around the particular point
of interest.
This behavior is called being "locally linear" and for this small interval can be
very closely approximated by examining the behavior of the line tangent to the
graph at the particular point of interest. With this in mind, examine the graph
of y = Sin(x), with the settings in radian mode, in the same window used at the
beginning of this activity.
Record what you see below.
Then, change the setting to degree mode and explain why the results change in
light of this activity.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
32
Calculus with the Casio fx-9750G Plus • Activity 5
Copyright © Casio, Inc.
Calculator Notes and Answers for Activity 5
1.
2. Answers will vary: A good answer should minimally contain comments about
the slope changing throughout the behavior of the function. [A more "advanced"
answer would contain comments about the slope changing from negative to
positive, and perhaps even mentioning where the slope is zero.
3.
4.
x
y
1.975
-3.049375
2
-3
2.025
-2.949375
Copyright © Casio, Inc.
Activity 5 • Calculus with the Casio fx-9750G Plus
33
Calculator Notes and Answers for Activity 5
5. y = 2x – 6.999375
6.
7.
This is what the calculator will show.
A good answer will include comments that the line and the function begin to be
very "close together" around the value of x = 2. Some students with greater
insight might begin to discuss the line being very close to tangent (Care should
be taken to point out that while it "looks" pretty tangent, the line being discussed
is not tangent, but a secant line in a very small neighborhood of x = 2)
For some students an extra zoom or two might clarify the idea being presented.
8.
in radian mode
in degree mode
The goal here is for students to realize that if the mode is changed to degree, they are now
looking at a graph that is being produced over only a neighborhood +6.3 degrees away from
Sin(0) thus creating a graph very close to y = 0 for that interval.
Note, students should also be encouraged to zoom around the Sine graph at any point and be
asked to communicate the fact that relatively few zooms will produce a very "linear"
looking graph. All explanations should be accompanied by a description of the window that
is producing the viewed result.
34
Calculus with the Casio fx-9750G Plus • Activity 5
Copyright © Casio, Inc.
Activity 6
Continuity Meets Differentiability
Teaching Notes
Topic Area: Derivatives and Continuity
Class Time: an exploration during the first half of a class period to point out visually
that continuity is a necessary but not sufficient condition for differentiability.
Overview
This activity will begin to extend the idea of local linearity and derivative. It will also
connect those concepts to continuity and point out that continuity is a necessary but
not sufficient condition for differentiability. The connections will be made visually
using the idea of local linearity (or what happens when it’s missing). Symbolic derivatives will, where appropriate, be used to support these findings.
Objectives
•
To connect the ideas of slope, local linearity and differentiability
•
To be able to understand and communicate the idea that continuity alone
does not guarantee that a function has a derivative
Getting Started
Using the Casio fx-9750G Plus, have students work in pairs or small groups
arranged prior to beginning the activity to maximize student involvement and
ownership of the results.
Prior to using this activity:
•
Students should be able to produce and manipulate graphs of functions
manually and with the graphing utility.
•
Students should have had an introduction to basic symbolic derivatives to
make an easier connection to the visuals.
•
Students should be able to use Zoom features of the graphing utility to
examine specific parts of the graph, including setting the zoom factors.
•
Students should have an understanding of slope of a function at a point as the
visual presentation of the derivative.
Ways students can provide evidence of learning:
•
Students should be able to produce graphs of functions and communicate
why a certain function may not have a derivative at a certain point.
•
Students should be able to, where appropriate, back up their graphical
presentation with symbolic analysis.
Common mistakes to be on the lookout for:
•
Not understanding the zoom process and what is taking place
•
Not being able to communicate the concept of derivative verbally
•
Entering the rational exponents incorrectly resulting in the calculator
producing a graph different that the one desired
Copyright © Casio, Inc.
Activity 6 • Calculus with the Casio fx-9750G Plus
35
Name _____________________________________________ Class ________ Date ________________
Activity 6 • Continuity Meets Differentiability
Introduction
This activity will begin to extend the idea of local linearity and derivative. It will
also connect those concepts to continuity. The connections will be made visually
using the idea of local linearity (or what happens when it’s missing).
Problems and Questions
Explore the behavior of the function: y = x2/3 around the value x = 2.
1. Sketch the graph of the function y = x2/3 in the INIT default viewing window.
Record the graph below and describe what you see.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
2. Trace to the value of x = 2 and with your zoom factors set to 4 for X and Y, zoom
in twice. Record what you see and explain what is going on.
3
2
1
1
2
3
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
36
Calculus with the Casio fx-9750g Plus • Activity 6
Copyright © Casio, Inc.
Name _____________________________________________ Class ________ Date ________________
Activity 6 • Continuity Meets Differentiability
3. Using trace, fill in the following values for the function accurate to 5
decimal places.
x
y
2.0125
Pt 1
2.00625
Pt 2
2
Pt 3
1.99375
Pt 4
1.9875
Pt 5
4. Calculate slopes of Pt 1 & Pt 2, then Pt 2 & Pt 3, then Pt 3 & Pt 4, Then Pt 4
& Pt 5 and record them as Slope 1, Slope 2, Slope 3, and Slope 4:
Slope 1
Slope 2
Slope 3
Slope 4
5. What do your results indicate? Explain how the graph you saw either agrees
or disagrees with those results.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
6. Now let’s examine the same function around the point x = 0 by graphing the
function in the INIT window, tracing to x = 0, and with the zoom factors still
set at 4, zoom in twice. Record the graph below.
Copyright © Casio, Inc.
Activity 6 • Calculus with the Casio fx-9750G Plus
37
Name _____________________________________________ Class ________ Date ________________
Activity 6 • Continuity Meets Differentiability
Describe what you see.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
7. Using trace, fill in the following values for the function accurate to 5 decimal
places.
x
y
-0.0125
Pt 1
-0.00625
Pt 2
0
Pt 3
0.00625
Pt 4
0.0125
Pt 5
8. Repeat the same slope procedure as before: Calculate slopes of Pt 1 & Pt 2, then
Pt 2 & Pt 3, then Pt 3 & Pt 4, Then Pt 4& Pt 5 and record them as Slope 1,
Slope 2, Slope 3, and Slope 4:
Slope 1
Slope 2
Slope 3
Slope 4
9. What do these results indicate? Compare them to the results from the
exploration of the graph around x = 2.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
38
Calculus with the Casio fx-9750g Plus • Activity 6
Copyright © Casio, Inc.
Name _____________________________________________ Class ________ Date ________________
Activity 6 • Continuity Meets Differentiability
10. While continuity is a necessary condition for a function to have a derivative at
that same point, it is not a sufficient condition as these two examples indicate.
The function explored is both continuous and differentiable at x = 2, however, it
is continuous but NOT differentiable at x = 0. Use symbolic derivatives to
support the visual evidence found in these explorations.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
11. Can you come up with some other simple functions that might provide
places where the function is continuous and differentiable at one point in
its domain, and continuous but NOT differentiable at another point?
Copyright © Casio, Inc.
Activity 6 • Calculus with the Casio fx-9750G Plus
39
Calculator Notes and Answers for Activity 6
1.
Descriptions will vary. A good answer will include a statement about there
being a hard corner at x = 0.
2.
The graph should be virtually linear, while descriptions will vary, there should
be a comment about the "straightening out" of the function. Answers may
include comments about seeing a good "linear approximation" of the function at
x = 2. There should also be comments regarding the continuity around x = 2.
3.
Using trace, fill in the following values for the function accurate to 5 decimal
places.
x
y
2.0125
1.59401
Pt 1
2.00625
1.59071
Pt 2
2
1.58740
Pt 3
1.99375
1.58409
Pt 4
1.9875
1.58078
Pt 5
4.
40
Slope 1
0.528
Slope 2
0.5296
Slope 3
0.5296
Slope 4
0.5296
Calculus with the Casio fx-9750g Plus • Activity 6
Copyright © Casio, Inc.
Calculator Notes and Answers for Activity 6
5.
A good answer will include statements about the slopes being the same and the
graph becoming linear around the point x = 2. The graph should show a picture
that is highly linear in the small neighborhood of x = 2.
6.
In stark contrast to the prior exploration, a good answer should include
comments about the graph NOT straightening out, (becoming locally linear).
There should also be some comments about the continuity being maintained.
7.
x
y
-0.0125
.05386
Pt 1
-0.00625
.03393
Pt 2
0
0
Pt 3
0.00625
.03393
Pt 4
0.0125
.05386
Pt 5
8.
Slope 1
-3.188
Slope 2
-5.4288
Slope 3
5.4288
Slope 4
3.188
9. Answers will vary. A good answer should include a direct comparison
indicating that the graph is not becoming locally linear around x = 0, while it
did "straighten out" around x = 2. The idea that the graph is continuous at
both x = 0 and x = 2 should be discussed.
2
10. dy
3
x
dx
( ) = 23 x
-1
3
A good answer will point out that the derivative at x = 2 exists (and = .52913,
very close to the value found in the exploration). However, the derivative at x = 0
does not exist (division by 0). In fact, repeated zooming around x = 0 will
continue to provide the same slope with different signs on either side of x = 0.
Copyright © Casio, Inc.
Activity 6 • Calculus with the Casio fx-9750G Plus
41
Activity 7
Derivative Behavior of Polynomials
Teaching Notes
Topic Area: Derivatives
Overview
This activity will lead students to make connections between the behavior of some
well known polynomial functions and their derivatives. They will be asked to plot the
functions, confirm the expected behavior using the grapher and then overlay the
derivative, confirming again using the grapher.
Objectives
•
To be able to express verbally and graphically the behavior of some well known
functions
•
To be able to understand and communicate the behavior of the derivative of
these well known functions to the function itself
•
To make sure that students can express the behavior of the derivative as
producing output values relative to the SLOPE of the original function, and not
simply compare output values to output values
Getting Started
Using the Casio fx-9750G Plus, have students work in pairs or small groups
arranged prior to beginning the activity will provide students with opportunity to
exchange ideas.
Prior to using this activity:
• Should be able to produce and manipulate graphs of functions manually
and with the graphing utility
•
Should have had an introduction to basic symbolic derivatives to make an
easier connection to the visuals
•
Should have a basic understanding of the transformations of polynomial
functions
•
Should be able to use the Casio fx-9750G Plus to graph a derivative
How to graph a derivative on the Casio fx-9750G Plus.
• In the GRAPH Menu, in the Y= (entry) screen
42
•
Press OPTN.
•
Press F2 (CALC).
•
Press F1 (d/dx)
•
Entry syntax: d/dx (function, x)
Example:
Calculus with the Casio fx-9750g Plus • Activity 7
Copyright © Casio, Inc.
Name _____________________________________________ Class ________ Date ________________
Activity 7 • Derivative Behavior of Polynomials
Introduction
This activity will begin to extend the idea of local linearity and derivative. It will
also connect those concepts to continuity. The connections will be made visually
using the idea of local linearity (or what happens when it’s missing).
The derivative of a function represents the behavior of the slope of the function at
each point along its domain. The goal of this activity is to have you able to make
the connections to the picture of the function and the picture of the behavior of the
slope of the function.
Problems and Questions
1.
Draw the graphs of the following functions in the window.
y = 2x
2.
y = 2x + 5
y = 2x – 3
Describe the behavior of the slope of each function.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
3.
Using your calculator, draw the function y = 2x – 3 and the graph of its slope
on the same axes. Copy it below.
4. Does this agree with what you expected to see? Explain.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
Copyright © Casio, Inc.
Activity 7 • Calculus with the Casio fx-9750G Plus
43
Name _____________________________________________ Class ________ Date ________________
Activity 7 • Derivative Behavior of Polynomials
5.
Given the general equation of a linear function: ax + by = c , generalize the
relationship between the linear function and its derivative. Provide some
examples to support your hypotheses.
6.
Using the same window as before, draw the graph of: y = x2 on the axes
shown below. Confirm the behavior on your calculator.
7.
Describe the behavior of the slope of the function over the following intervals:
• (– 0)
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
• (0, )
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
44
Calculus with the Casio fx-9750g Plus • Activity 7
Copyright © Casio, Inc.
Name _____________________________________________ Class ________ Date ________________
Activity 7 • Derivative Behavior of Polynomials
8.
Based upon your knowledge of what a derivative is, what would you say the
derivative of the function is when x = 0? Explain.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
9.
Sketch the function over again on the axes provided below and then overlay
what you think the behavior of the derivative would look like.
10. Use your grapher to produce the picture of the actual derivative, does it agree
with the graph you produced manually?
11. Now try the same procedure with the following function:
y = –3(x–2)2 +2
The function and derivative manually:
Copyright © Casio, Inc.
The function and derivative by calculator:
Activity 7 • Calculus with the Casio fx-9750G Plus
45
Name _____________________________________________ Class ________ Date ________________
Activity 7 • Derivative Behavior of Polynomials
12. Given the general equation of a quadratic function: y = –ax2 + bx +c,
generalize the relationship between the quadratic function and its derivative.
Provide some examples to support your hypotheses.
13. Given that the general form of a polynomial is y = –anxn + an–1xn–1 + ... + a0
make a general statement about any polynomial function and its derivative.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
14. Provide one fourth degree example to support your conclusion.
46
Calculus with the Casio fx-9750g Plus • Activity 7
Copyright © Casio, Inc.
Calculator Notes and Answers for Activity 7
1.
2.
Answers will vary, but the goal is to have students discuss that the slopes are all
the same (lines are parallel). A likely answer will also include a comment that
the slope = 2 for each line.
3.
4.
Answers will vary. A complete answer should contain a statement regarding the
fact that the slope is constant therefore the graph of the derivative should be a
horizontal line.
5.
Answers will vary. A complete answer should contain a statement that the slope
of the line will always be a horizontal line. y = –a/b
All provided examples should contain linear functions and horizontal lines as
the derivative sketches. Students thinking farther ahead may start with a
horizontal line as an example and then show the line y = 0 as the derivative.
6.
7.
(– 0) A complete answer should cover the fact that in this entire interval the slope
is negative but changing. Some answers may include statements about the slope
"slowing down" or being smaller or less as the interval approaches 0 [alternately
may include statements about the slope "speeding up" as the interval moves away
from zero]. Care should be taken that the students are talking about the behavior
of the slope relative to the values of x.
(0, ) A complete answer should cover the fact that in this entire interval the slope
is positive but changing. Some answers may include statements about the slope
"speeding up" or being larger or more as the interval moves away from 0.
Copyright © Casio, Inc.
Activity 7 • Calculus with the Casio fx-9750G Plus
47
Calculator Notes and Answers for Activity 7
8. A complete answer should include a statement that the derivative = 0 @ x = 0.
The explanation could use the difference quotient/limit approach using small
values from the graph, or could make the connection that the tangent is
horizontal at x = 0.
9.
10. Same graph as above. They should agree.
11.
Both graphs should agree, if not, further discussion needs to take place about
the derivative representing the picture of the slope.
12. A complete answer should contain statements that the derivative of a quadratic
function will always be linear. Students should be very clear that the line exits
above the x-axis when the slope of the function is positive, has a root at the
vertex of the parabola, and exits below the x-axis when the slope is negative.
Examples should be consistent. Require them to verbalize their support choices.
13. The goal is to have students recognize that the derivative of any polynomial will
be another polynomial of one degree less. Good answers will also contain state
ments consistent with the fact that the derivative graph is above the x-axis when
the slope of the function is positive, has a root at any vertex, and exits below the
x-axis when the slope is negative. This might require further investigation. This
is also a good lead into the power rule for derivatives.
14. Answers will vary, one example provided here:
48
Calculus with the Casio fx-9750g Plus • Activity 7
Copyright © Casio, Inc.
Activity 8
Derivative Behavior of Common
Trigonometric Functions
Teaching Notes
Topic Area: Derivatives
Overview
This activity will lead students to making connections between the behaviors of some
well known trigonometric functions and their derivatives. They will be asked to plot
the functions, confirm the expected behavior using the grapher and then overlay the
derivative, confirming again using the grapher.
The derivative of a function represents the behavior of the slope of the function at
each point along its domain. The goal of this activity is to have students make the
connections to the picture of the function and the picture of the behavior of the slope
of the function.
Objectives
•
To be able to express verbally and graphically the behavior of some well known
trigonometric functions
•
To be able to understand and communicate the behavior of the derivative of
these well known functions to the function itself
•
To make sure that students can express the behavior of the derivative as
producing output values relative to the SLOPE of the original function, and not
simply compare output values to output values
Getting Started
Using the Casio fx-9750G Plus, have students work in pairs or small groups
arranged prior to beginning the activity to share ideas.
Prior to using this activity:
•
Should be able to produce and manipulate graphs of functions manually and
with the graphing utility
•
•
Should have a basic understanding of the behavior and appearance of basic
trigonometric functions
Should be able to use the Casio fx-9750G Plus to graph a derivative
How to graph a derivative on the Casio fx-9750G Plus.
• In the GRAPH Menu, in the Y= (entry) screen
•
Press OPTN.
•
Press F2 (CALC).
•
Press F1 (d/dx)
•
Entry syntax: d/dx (function, x)
Copyright © Casio, Inc.
Example:
Activity 8 • Calculus with the Casio fx-9750G Plus
49
Name _____________________________________________ Class ________ Date ________________
Activity 8 • Derivative Behavior of Common
Trigonometric Functions
Introduction
This activity will have you making connections between the behavior of some well
known trigonometric functions and their derivatives. You will be asked to plot the
functions, confirm the expected behavior using the grapher and then overlay the
derivative, confirming again using the grapher.
The derivative of a function represents the behavior of the slope of the function at
each point along its domain. The goal of this activity is to have you make the connections to the picture of the function and the picture of the behavior of the slope
of the function.
Problems and Questions
Make sure the calculator is in radian mode.
1. Draw the graph of y = Sin(x) in the default initial window and record it here.
2. Describe the slope of the function over the interval [0, 2π] .
_____________________________________________________________________________
_____________________________________________________________________________
3. Using your understanding of derivative as slope, sketch the function, y = Sin(x)
and its derivative over the interval [0, 2π] .
4. Using your calculator, produce the same graphs as above. Do the graphs
produced agree with what you expected to see? Explain.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
50
Calculus with the Casio fx-9750g Plus • Activity 8
Copyright © Casio, Inc.
Name _____________________________________________ Class ________ Date ________________
Activity 8 • Derivative Behavior of Common
Trigonometric Functions
5. Draw the graph of y = 2Sin(x), in the interval [0, 2π] and record it here. Then
using your knowledge of slope, overlay the graph of the slope function (the
derivative).
6. Have your calculator produce the graph of the derivative. Does it agree with your
sketch? Explain.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
7. Draw the graph of y = Sin(2x), in the interval [0, 2π] and record it here. Then
using your knowledge of slope, overlay the graph of the slope function (the
derivative).
8. Have your calculator produce the graph of the derivative. Does it agree with your
sketch? Explain.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
Copyright © Casio, Inc.
Activity 8 • Calculus with the Casio fx-9750G Plus
51
Name _____________________________________________ Class ________ Date ________________
Activity 8 • Derivative Behavior of Common
Trigonometric Functions
9. Draw the graph of y = Cos(x) in the default initial window and record it here.
10. Describe the slope of the function over the interval [0, 2π] .
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
11. Using your understanding of derivative as slope, sketch the function, y = Cos(x)
and its derivative over the interval [0, 2π] .
12. Using your calculator, produce the same graphs as above. Do the graphs
produced agree with what you expected to see? Explain.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
13. Draw the graph of y = 2Cos(x), in the interval [0, 2π], record it here. Then
using your knowledge of slope, overlay the graph of the slope function (the
derivative).
52
Calculus with the Casio fx-9750g Plus • Activity 8
Copyright © Casio, Inc.
Name _____________________________________________ Class ________ Date ________________
Activity 8 • Derivative Behavior of Common
Trigonometric Functions
14. Have your calculator produce the graph of the derivative. Does it agree with your
sketch? Explain.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
15. Draw the graph of y =Cos(2x), in the interval [0, 2π] , record it here. Then
using your knowledge of slope, overlay the graph of the slope function (the
derivative).
16. Have your calculator produce the graph of the derivative. Does it agree with your
sketch? Explain.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
17. Compare and contrast the behaviors of the derivatives of the Sine and Cosine
functions.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
18. Given the general functions y = A • Sin (x) and y = Sin (Bx), and using the
calculator, explore their derivative behaviors for additional values of A and B.
Do the same for the Cosine functions and draw a general set of conclusions of
the effects of A and B on the derivative behavior. Can you come up with a
general symbolic rule using these results? If so, what is it?
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
Copyright © Casio, Inc.
Activity 8 • Calculus with the Casio fx-9750G Plus
53
Calculator Notes and Answers for Activity 8
1.
2.
Answers will vary. A Good answer will include statements that the slope is
π
3π
positive (increasing) over the intervals 0, 2 and 2 , 2π and negative
(decreasing) over the interval π , 3π .
(
2
2
)
[ ) (
]
A well thought out answer should also include statements that the slope is = 0 at
the vertices.
3. Answers may vary but should look like the graph the calculator produces for
question #4.
4.
If the graphs do not agree in 3 & 4, discussion should take place regarding the
differences.
5.
The drawn in derivative should look like the result from #6.
6.
54
Calculus with the Casio fx-9750g Plus • Activity 8
Copyright © Casio, Inc.
Calculator Notes and Answers for Activity 8
7.
8.
Here there is the first real difference that might cause some confusion. The
amplitude of the slope is different than the amplitude of the original function. It
is difficult to arrive at this just from a graph, a student whose hand sketched
graph includes this, has likely already used the symbolic rules or has used some
function values to get the actual slopes.
9. .
10. Good answers will be similar to the response to the Sin function indicating that
the slope is positive (increasing) over the interval (π, 2π)and negative
(decreasing) over the interval (0, π) . A well thought out answer should also
include statements that the slope is = 0 at the vertices.
11.
12. Should see same as the answer shown to #11 above.
Copyright © Casio, Inc.
Activity 8 • Calculus with the Casio fx-9750G Plus
55
Calculator Notes and Answers for Activity 8
13. The graph of y=2Cos(x)
14. The function and its derivative
15.
The Graph of y = Cos(2x)
16. The function and its derivative
56
Calculus with the Casio fx-9750g Plus • Activity 8
Copyright © Casio, Inc.
Calculator Notes and Answers for Activity 8
17. Answers will vary. Complete answers should include statements that the Sine
function derivative produces graphs that look behave like the Cosine function,
while the Cosine function derivative produces graphs that seem to be the
opposite (negation) of the Sine function.
There should also be mention that changing the amplitude of the function is
consistent with the amplitude of the derivative, but changing the period of the
function is consistent with the period of the derivative, but also changes the
amplitude of the derivative.
18. y = A • Sin(x)→ dy = A • Cos(x)
dx
dy
y = Sin(Bx)→
= B • Cos(Bx)
dx
y = A • Cos(x)→ dy = -A • Sin(x)
dx
dy
y = Cos(Bx)→
= -B • Sin(Bx)
dx
Copyright © Casio, Inc.
Activity 8 • Calculus with the Casio fx-9750G Plus
57
Activity 9
Looking at Relationships
Teaching Notes
Topic Area: Derivatives and Graphs
Overview
A great deal of information about a function can be found by analyzing the behavior
of the first and second derivatives. This activity will provide a graphical examination
of the relationships between the function and its derivatives.
Objectives
•
Be able to explain information about the graph of a function based on the first
and second derivatives
•
Know that the derivative of a function is positive when the function increases,
and negative when the function decreases
•
Know that a positive second derivative means the function is concave upward
and a negative second derivative means the function is concave downward
Getting Started
Using the Casio fx-9750G Plus, students can work this activity independently or in
pairs..
Prior to using this activity:
•
Students should be able to take basic symbolic derivatives.
•
Students should know the terms relative minimum and relative maximum.
•
Students should be able to produce the graph of a derivative and second
derivative from the calculator.
Ways students can provide evidence of learning:
•
Students should be able to explain how the first derivative yields information
about the increasing/decreasing nature of the function.
•
Students should be able to explain how the second derivative yields information
about the concavity of the graph.
Common mistakes to be on the lookout for:
•
Students may understand where a function is increasing or decreasing but they
may misinterpret that on the graph as thinking the function is always above or
below the x-axis instead of the graph of the derivative being positive/negative.
•
58
The speed at which the calculator shows a second derivative graph is relatively
slow. Some students may conclude there is no graph being produced.
Calculus with the Casio fx-9750g Plus • Activity 9
Copyright © Casio, Inc.
Name _____________________________________________ Class ________ Date ________________
Activity 9 • Looking at Relationships
Introduction
This activity will provide a graphical examination of the relationships between the
function and its first and second derivatives.
The increasing/decreasing nature of a function can be examined by the
positive/negative behavior of its derivative. Similarly, the upward/downward
concavity can be examined by the positive/negative behavior of its second
derivative.
Problems and Questions
1. Graph the function y = 2x3 – 3x2 – 12x + 4 in the window.
2. Record the results here:
3. At what x-values does it appear the function reaches its relative minimum and
maximum values?
____________________________________
4. Using the G-Solve functions, confirm those values and find the minimum and
maximum function values.
5. Record the domain interval/intervals where the function increases.
____________________________________
6. Record the domain interval/intervals where the function decreases.
____________________________________
7. Explain what kind of values would you expect the derivative to have over the
interval where the function increases.
____________________________________
Copyright © Casio, Inc.
Activity 9 • Calculus with the Casio fx-9750G Plus
59
Name _____________________________________________ Class ________ Date ________________
Activity 9 • Looking at Relationships
8. Using the d/dx from the OPTN menu, set Y2 to produce the graph of the
derivative, graph both the function and the derivative together and record here.
(Be sure to LABEL on your sketch which is the function and which is the
derivative.)
9. From the graph, what are the y-values of the derivative where the original
function has a relative maximum or minimum?
____________________________________
10. Explain the nature of the y-values of the derivative over the interval(s) where the
original function increases.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
11. Explain the nature of the y-values of the derivative over the interval(s) where the
original function decreases.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
12. When the derivative crosses the x-axis explain what happens to the graph of the
original function?
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
13. Over what x interval(s) does the derivative increase?
____________________________________
60
Calculus with the Casio fx-9750g Plus • Activity 9
Copyright © Casio, Inc.
Name _____________________________________________ Class ________ Date ________________
Activity 9 • Looking at Relationships
14. Over what x interval(s) does the derivative decrease?
____________________________________
15. If the first derivative is increasing, do you expect the second derivative to be
positive or negative? Explain.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
16. If the first derivative is decreasing, do you expect the second derivative to be
positive or negative? Explain.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
17. Using the d2/dx2 from the OPTN menu, set Y3 to produce the graph of the
second derivative, graph both the first and second derivative together and record
here. (Be sure to LABEL on your sketch which is the first and which is the
second derivative).
18. Do the graphs produced match your expectations? If not, explain any differences
you see.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
Copyright © Casio, Inc.
Activity 9 • Calculus with the Casio fx-9750G Plus
61
Name _____________________________________________ Class ________ Date ________________
Activity 9 • Looking at Relationships
19. The graph of a function is concave down when the graph of the first derivative is
decreasing. Sketch the portion of the original function that is concave down and
record it here.
Explain what is true about both the first and second derivatives over the interval
you just sketched.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
20. A point of inflection is a point where the concavity changes. Based upon your
exploration what is the point of inflection for the original graph? Explain how
you know.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
62
Calculus with the Casio fx-9750g Plus • Activity 9
Copyright © Casio, Inc.
Calculator Notes and Answers for Activity 9
2.
3. Relative Max at x =-1, Relative Min at x = 2
4. Using the G-Solve functions, confirm those values and find the minimum and
maximum function values.
5. Increases: (–, –1) (2, )
6. Decreases: (–1, 2)
7. Positive values
8.
Der
Function
9. Y = 0 in both cases
10. Where the function is increasing, the y-values of the derivative are positive
11. Where the function is decreasing, the y-values of the derivative are negative
12. The function reaches a relative extreme point
13. (.5, )
14. (–, .5)
15. Positive. It should follow the same behavior as the relationship between the
original function and its 1st derivative
16. Negative, same reason as above.
Copyright © Casio, Inc.
Activity 9 • Calculus with the Casio fx-9750G Plus
63
Calculator Notes and Answers for Activity 9
17.
1st
Der.
2nd Der.
18. Answers may vary, but they should match.
19.
The first derivative is decreasing, the second derivative is negative.
20. The point of inflection is (.5, -2.5). The concavity will change at the root of the
second derivative which is x = .5, that produces a y-value of –2.5 in the original
function.
64
Calculus with the Casio fx-9750g Plus • Activity 9
Copyright © Casio, Inc.
Activity 10
Looking at Slope Fields
Teaching Notes
Topic Area: Slope Fields
Overview
A Slope Field is a visual representation of the solution to a differential equation created by creating a series of small linear approximations to the slope at certain points.
In this activity students will sketch some slope fields and then confirm their appearance using the calculator. This can be used as a first introduction to the idea of Slope
Fields and also as a general introduction to antidifferentiation.
Objectives
•
Understand what a slope field represents in terms of dy/dx
•
Create and explain a slope field from a given differential equation
Getting Started
Using the Casio fx-9750G Plus and the Slope Field Program, have students begin
the activity independently and then share and discuss their results with another
student.
Prior to using this activity:
•
Students should have a working knowledge of differentiation and be conversant
with the language of differential equations.
•
It is not necessary for students to know any symbolic antidifferentiation
methods.
Ways students can provide evidence of learning:
•
Students should be able to sketch their own slope fields for a given differential
equation over specific grid points using pencil and paper.
Common mistakes to be on the lookout for:
•
Students may misunderstand that the graph being produced is the graph of the
solution to dy/dx (the graph of the antiderivative) and be confused when the
given equation does not seem to fit the slope field.
•
For example, the slope field of the expression dy/dx = x, correctly drawn will
produce a parabolic fit, students may incorrectly expect a linear fit.
Copyright © Casio, Inc.
Activity 10 • Calculus with the Casio fx-9750G Plus
65
Activity 10
Slope Field Program
This program can be found in the download section of
the Casio Education website:
http://www.casioeducation.com
Notes:
•
The program stores the slope field in Pic 1 automatically.
•
The differential equation must be stored in Y1 in the GRAPH menu prior to
executing the program.
•
The optimal window is the INIT window, set while in the GRAPH menu, prior to
executing the program.
•
When the slope field has been created on the calculator press AC/on key to break
out of the program.
To overlay the slope field on the graph of a proposed solution graph:
•
Execute the program, press AC/ON when done.
•
Return to GRAPH menu.
•
Put the proposed solution graph in Y1.
•
Draw the graph.
•
Press OPTN.
•
Press F1 (Pict).
•
Press F2 (Rcl).
•
Press F1 (Pic 1).
The picture of the slope field will be placed on the graph of the proposed solution for
comparison.
66
Calculus with the Casio fx-9750G Plus • Activity 10
Copyright © Casio, Inc.
Name _____________________________________________ Class ________ Date ________________
Activity 10 • Looking at Slope Fields
Introduction
A Slope Field is a visual representation of the solution to a differential equation
created by a series of small linear approximations to the slope at certain points.
In this activity students will sketch some slope fields and then confirm their
appearance using the calculator.
An equation like dy/dx = 2x which contains a derivative is called a "differential
equation". The problem becomes finding a function, y in terms of x, when we are
given its derivative. Note the phrasing: we are looking for "a solution" and not "the
solution". This is due to the fact that the slope will allow us to arrive at a family of
curves and missing some initial condition value we will not be able to arrive at a
single solution but should be able to make some conclusions as to the appearance
of the family of solutions. If we examine the plot of the slope over a series of grid
points we end up with a Slope Field.
If given the differential equation dy/dx = f(x,y) a plot of short line segments with
slopes f(x,y) over specific grip points produces a slope field. This slope field will
give you a look at the behavior of the solution to the original differential equation.
Problems and Questions
1. Fill in the accompanying table representing the slope of dy/dx = 2x for the grid
points shown, [for example at the point (0,0) you should a slope of 0, at (1,2) a
slope of 2 etc].
(x,y)
dy/dx
(x,y)
dy/dx
(x,y)
dy/dx
(x,y)
dy/d
(x,y)
-6, -2
-6, -1
-6, -0
-6, 1
-6, 2
-5,-2
-5,-1
-5,-0
-5,1
-5,2
-4,-2
-4,-1
-4,-0
-4,1
-4,2
-3,-2
-3,-1
-3,-0
-3,1
-3,2
-2,-2
-2,-1
-2,-0
-2,1
-2,2
-1,-2
-1,-1
-1,-0
-1,1
-1,2
0,-2
0,-1
0,-0
0,1
0,2
1,-2
1,-1
1,-0
1,1
1,2
2,-2
2,-1
2,-0
2,1
2,2
3,-2
3,-1
3,-0
3,1
3,2
4,-2
4,-1
4,-0
4,1
4,2
5,-2
5,-1
5,-0
5,1
5,2
6,-2
6,-1
6,-0
6,1
6,2
Copyright © Casio, Inc.
dy/dx
Activity 10 • Calculus with the Casio fx-9750G Plus
67
Name _____________________________________________ Class ________ Date ________________
Activity 10 • Looking at Slope Fields
2. Sketch a small line segment with the slopes calculated, centered at each grid
point.
For example at the point (1,2) which will have a slope of 2, this is what
you should sketch:
And for the points (-1,2) (0,2) and (1,2) (with slopes of –2, 0, 2 respectively) you
should see this:
3. What familiar family of curves does this slope field seem to indicate is the
solution to the given differential equation? Explain.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
68
Calculus with the Casio fx-9750G Plus • Activity 10
Copyright © Casio, Inc.
Name _____________________________________________ Class ________ Date ________________
Activity 10 • Looking at Slope Fields
4. Using the Slope Field program, have the calculator produce the slope field for
dy/dx = 2x. How does it compare with what you hand sketched?
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
5. Have the calculator draw one member of the family of curves you think is best
represented by the slope field, then have it overlay the slope field on the graph
produced. Record what you see here:
6. Look at the graph the calculator produced. How does it compare to what you
expected to see?
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
Extension:
•
Use the following differential equations and generate a slope field for each.
•
Have the calculator generate the slope field
•
Draw a conclusion about the general solution to the differential equation.
a) dy/dx = x2
Copyright © Casio, Inc.
Activity 10 • Calculus with the Casio fx-9750G Plus
69
Name _____________________________________________ Class ________ Date ________________
Activity 10 • Looking at Slope Fields
b) dy/dx = Sin(x)
c) dy/dx = ex
70
Calculus with the Casio fx-9750G Plus • Activity 10
Copyright © Casio, Inc.
Calculator Notes and Answers for Activity 10
1.
(x,y)
dy/dx
(x,y)
dy/dx
(x,y)
dy/dx
(x,y)
dy/d
(x,y)
dy/dx-
-6, -2
-12
-6, -1
-12
-6, -0
-12
-6, 1
-12
-6, 2
12
-5,-2
-10
-5,-1
-10
-5,-0
-10
-5,1
-10
-5,2
-10
-4,-2
-8
-4,-1
-8
-4,-0
-8
-4,1
-8
-4,2
-8
-3,-2
-6
-3,-1
-6
-3,-0
-6
-3,1
-6
-3,2
-6
-2,-2
-4
-2,-1
-4
-2,-0
-4
-2,1
-4
-2,2
-4
-1,-2
-2
-1,-1
-2
-1,-0
-2
-1,1
-2
-1,2
-2
0,-2
0
0,-1
0
0,-0
0
0,1
0
0,2
0
1,-2
2
1,-1
2
1,-0
2
1,1
2
1,2
2
2,-2
4
2,-1
4
2,-0
4
2,1
4
2,2
4
3,-2
6
3,-1
6
3,-0
6
3,1
6
3,2
6
4,-2
8
4,-1
8
4,-0
8
4,1
8
4,2
8
5,-2
10
5,-1
10
5,-0
10
5,1
10
5,2
10
6,-2
12
6,-1
12
6,-0
12
6,1
12
6,2
12
2. The slope field should reflect a parabolic shape.
3. Students should recognize this as a family of parabolas. However, the apparent
vertical nature of the horizontal extremities could throw some students off. This
presents a good opportunity to discuss the nature of graphical approximations.
4. If the original field was drawn correctly, they should see a graph very similar to
what they hand sketched.
5. Answers will vary based upon the chosen graph. Shown here are the graphs for
y = x2 and y = x2 – 2.
Copyright © Casio, Inc.
Activity 10 • Calculus with the Casio fx-9750G Plus
71
Calculator Notes and Answers for Activity 10
6.
If done correctly they should be very similar. If not, this presents a good
opportunity to discuss why their graphs are not accurate.
Extension:
a)
a cubic family of curves
1
(actual family: y = 3 x2 + c)
b)
also a trigonometric family
(actual family: y = -Cos(x) + c)
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Calculus with the Casio fx-9750G Plus • Activity 10
Copyright © Casio, Inc.
Calculator Notes and Answers for Activity 10
c)
an exponential family
(actual family y = ex + c)
Copyright © Casio, Inc.
Activity 10 • Calculus with the Casio fx-9750G Plus
73
Activity 11
Riemann Sums
Teaching Notes
Topic Area: Anti Derivatives, Integration
Overview
This activity will present students with the tools to calculate and analyze Riemann
sums. They will hand sketch rectangles and use that to approximate the area under
a curve. They will then use the calculator to perform increasing numbers of calculations to observe the convergence of upper and lower Riemann Sums with regular
partitions as the size of the partitions decrease and the number of rectangles
increase.
Objectives
•
Calculate Riemann Sums
•
Develop an understanding of when Riemann sum approximation will be over
or under the actual value of a definite integral
•
Observe the convergence of the upper and lower Riemann sum values as the
number of rectangles increases
Getting Started
Using the Casio fx-9750G Plus, have students work in pairs or small groups
arranged prior to beginning the activity. Students should have a working knowledge of the derivative function as a model for representing average and instantaneous change and should be able to use the Sum and Sequence commands.
Prior to using this activity:
•
Students should have a working knowledge of the derivative function as a model
for representing average and instantaneous change.
•
Students should be able to use the Sum and Sequence commands.
•
It is not necessary for students to know any symbolic antidifferentiation
methods.
Ways students can provide evidence of learning:
•
Students should be able to sketch their own upper and lower rectangles for a
given function over specified intervals.
•
Students should be able to explain the difference in the upper and lower
Riemann Sums.
Common mistakes to be on the lookout for:
•
Students may, in calculating the sums by hand, forget to multiply the series sum
by the base width.
•
74
Students should be given the programs only after they have demonstrated ability
to construct a Riemann Sum on their own.
Calculus with the Casio fx-9750G Plus • Activity 11
Copyright © Casio, Inc.
Activity 11
Riemann Sum Programs
Riemann Sum Drawing Program:
Teaching Notes
These programs can be found in the download section
of the Casio Education website:
http://www.casioeducation.com
Riemann Sum Calculation program: (Calculates both the left and right RSum)
Copyright © Casio, Inc.
Activity 11 • Calculus with the Casio fx-9750G Plus
75
Name _____________________________________________ Class ________ Date ________________
Activity 11 • Riemann Sums
Introduction
A Riemann Sum is a method of approximation for calculating the area under a
curve. When a function represents change, the area under the curve represents the
accumulation of that change. For instance, if you have curve measuring velocity
over time, the sums of those velocities over specific time intervals represents the
distance traveled during that time period.
This activity will present you with the tools to calculate and analyze Riemann
sums. You will hand sketch rectangles and use that to approximate the area under
a curve. You will then use the calculator to perform increasing numbers of calculations to observe the convergence of upper and lower Riemann Sums with regular
partitions as the size of the partitions decrease and the number of rectangles
increase.
Problems and Questions
1. Assume you are on cruise control driving down a clear highway at a constant
rate of 60 miles per hour. Record the graph of your velocity over the first 5
hours on the graph below.
70
60
50
40
30
20
10
1
2
3
4
5
2. Given these conditions, how far would you have gone over the first hour, two
hours, and three hours?
1 hour:
____________________________________
2 hours: ____________________________________
3 hours: ____________________________________
How did you calculate these results?
_____________________________________________________________________________
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Calculus with the Casio fx-9750G Plus • Activity 11
Copyright © Casio, Inc.
Name _____________________________________________ Class ________ Date ________________
Activity 11 • Riemann Sums
3. What is the area under the curve over the first hour? Two hours? Three Hours?
1 hour:
____________________________________
2 hours: ____________________________________
3 hours: ____________________________________
4. Explain why the answers to item #2 and item #3 agree.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
5. Now assume you have a rate of change that is being modeled by the function
y = x2 over the interval [0,5]
Sketch the function on these axes:
6. Now, it is not as easy to find the area under the curve over the first three hours
as the rate of change itself is changing at each point on the interval. We can,
however, approximate the area using rectangles. What if we were to put three
rectangles, each one unit long, under the curve, with measuring the height of
the rectangles at the RIGHT or upper endpoint of the interval. Sketch those
rectangles and record here:
7. Calculate the area represented by those three rectangles. Does it seem that
approximation will be more or less than the actual area? Why?
_____________________________________________________________________________
_____________________________________________________________________________
Copyright © Casio, Inc.
Activity 11 • Calculus with the Casio fx-9750G Plus
77
Name _____________________________________________ Class ________ Date ________________
Activity 11 • Riemann Sums
8. Now sketch the same rectangles, this time using the LEFT or lower endpoint to
mark the height of the triangles.
9. Calculate the total area represented by these lower rectangles. Does it seem that
this approximation will be more or less than the actual area? Why?
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
10. Leaving the interval the same, now change from 3 to 6 equally spaced rectangles.
(What will happen to the width of the rectangle in this case?), sketch the
rectangles and record the areas first for the RIGHT endpoint, then the LEFT
endpoint rectangles.
11.
Right Endpoint
Left Endpoint
Area: _________________________
Area: _________________________
Using the RSUMCALC program, find the areas over the same intervals for 12,
18, and 36 equally spaced rectangles. (Remember to be aware of what it does to
the width of each rectangle.)
Number of
partitions
78
Upper
Riemann Sum
Calculus with the Casio fx-9750G Plus • Activity 11
Lower
Riemann Sum
Copyright © Casio, Inc.
Name _____________________________________________ Class ________ Date ________________
Activity 11 • Riemann Sums
12. What happens to the width of each rectangle as the number of rectangles
increase?
_____________________________________________________________________________
13. What happens to the areas as the number of rectangles increase? Why do you
suppose this happens?
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
14. In your own words explain what these Upper and Lower Riemann Sums
represent.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
Extension:
Calculate the Upper and Lower Riemann Sums for each of the following functions
over the interval indicated and for the number of partitions indicated:
Function
1
x
Interval
Number of
partitions
[1,4]
4
Upper
Riemann Sum
Lower
Riemann Sum
8
12
16
3x+5
[0, 6]
3
6
10
20
Sin(x)
(radian
mode)
[0, π]
2
4
8
16
Copyright © Casio, Inc.
Activity 11 • Calculus with the Casio fx-9750G Plus
79
Calculator Notes and Answers for Activity 11
1.
70
60
50
40
30
20
10
1
2
3
4
5
2. 60 Miles, 120 Miles, 180 Miles.
Rate x time = distance
3. 60 units2, 120 units2, 180 units2
4. Answers will vary, a complete answer should contain statements about the
area being an accumulation of the change.
5.
6.
7. (1)2 • 1 + (2)2 • 1 + (3)2 • 1 =14
Answers should clearly state the area is more than what is covered, by the
function, as the rectangles are above the function in each case.
80
Calculus with the Casio fx-9750G Plus • Activity 11
Copyright © Casio, Inc.
Calculator Notes and Answers for Activity 11
8.
(Note: Some students may be confused in not seeing the three rectangles, they
need to be reminded that the left endpoint here is at x = 0 so there will be no
rectangle shown.)
9. (0)2 • 1 + (1)2 • 1 + (2)2 • 1 = 5
Answers should clearly state the area is less than what is covered by the
function, as the rectangles are below the function in each case.
10. Right endpoint
Area:
11.
Left Endpoint
11.375
Area: 6.875
Number of
partitions
Upper
Riemann Sum
Lower
Riemann Sum
12
10.16
7.91
18
9.76
8.26
36
8.62
7.92
12. The widths get smaller
13. The areas begin to converge toward a common value. The "why" answers will
vary, but a good answer should contain a statement about the error decreasing
as the amount of rectangles increase.
14. Answers will vary, but should indicate statements about increasingly accurate
approximations of the area under a curve. Some students who may have seen
the process of integration might connect the Riemann Sum to the definite
integral over that same interval.
Copyright © Casio, Inc.
Activity 11 • Calculus with the Casio fx-9750G Plus
81
Calculator Notes and Answers for Activity 11
Extension:
Function
1
x
3x+5
Sin(x)
(radian
mode)
Interval
Number of
partitions
Upper
Riemann Sum
Lower
Riemann Sum
[1,4]
4
1.71
1.15
8
1.54
1.26
12
1.48
1.30
16
1.46
1.32
3
66
102
6
75
93
10
78.6
89.4
20
81.3
86.7
2
1.57
1.57
4
1.90
1.90
8
1.82
1.97
16
1.99
1.99
[0, 6]
[0, π]
Note: Care should be taken to remind students that there is no firm "rule" for whether
the left or right endpoint rule will give the best approximation, it depends upon the
function. The goal of this activity is simply to bring out the tools for the use of
Riemann Sums.
82
Calculus with the Casio fx-9750G Plus • Activity 11
Copyright © Casio, Inc.
Appendix
Overview of the Calculator Functions
This is not meant to be an exhaustive tutorial. This overview is to provide
a starting point when beginning to use the Casio fx-9750G Plus graphing
calculator.
Run Function
The Casio fx-9750G Plus is just like any other calculator when in the run
function. There are a few extra functions it offers that other calculators do not.
Fraction calculations:
Using the ab/c key, you can enter numbers as
fractions and do any normal mathematical
operations. The EXE will give you answers
in fraction or mixed number format. You
can change from fraction format to a
decimal format by pressing the F↔D key
and vice versa. You may also change the
mixed number to an improper fraction by
pressing SHIFT ab/c for d/c. This function
will only work if the expression is entered in
fraction format originally.
d/c
ab/c
Editing:
If you need to make a change in a previous
calculation, use the deep recall to retrieve
the equation. In the Run function, press the
AC/ON to clear the screen. Then use the Up
Arrow key to scroll through the previous
equations until you come to the one you
want to edit. Press the left arrow key to make the changes desired. *Note: The
previous calculations will be lost as soon as you exit the Run function.
Probability:
In the Run function, press the OPTN (option) menu. Press the F6 (arrow right)
for more options. Then press the F3(PROB) function. This will allow you to do
factorials, permutations, combinations, and random numbers.
Copyright © Casio, Inc.
Appendix • Calculus with the Casio fx-9750G Plus
83
Statistics Function
Entering Data:
If there are statistics in the lists, press F6 for more options, then press
F4(DEL-A) and F1(YES). Enter the data in List 1 by entering the number and
pressing EXE. When you press EXE the number will show up in the list. After
the list is complete, you can begin calculating the statistics.
Statistics:
Press F2 to show the CALC options. Check the F6(SET) for the setup of the
calculations. Make sure that the 1Var XList is set at List 1 by highlighting it and
pressing F1. Press EXIT to get back to the previous screen and press F1(1VAR).
Arrow down to see other statistics.
Exit out:
Press EXIT twice to get back to the statistics lists.
Box and Whiskers:
Using the same data as in a histogram, you can make a box and whiskers graph.
From the graph option menu, press F6(SET). Arrow down to graph type and set
to Box. Press EXIT to get back to the previous screen. Check to see if graph 1 is
selected in the SEL option, and press F6(DRAW). Statistics can be seen from here
by pressing F1(1VAR). SHIFT F1(Trace) will allow you to trace the graph for
quartile ranges using the right arrow key. Press EXIT twice to return to the STAT
window.
Scatter Plot:
Enter data in lists 1 and 2 as done above. Press F1(GRPH) options then F6(SET).
Arrow down to graph type and set at SCATter. Make sure that XList and YList
are set at List1 and List2 respectively. Press EXIT and then press F1 to see
Grph1. If the graph does not appear, EXIT and press F4(SEL) to make sure only
Graph 1 is turned on.
Line of Best Fit:
On the screen with the graph are options for different kinds of regressions. Press
the regression you think will fit the data the best and the calculator will give you
a regression analysis. Notice the choices at the bottom of the screen. If you press
the F5(COPY), it will copy the line to the GRAPH function so that you can see
the graph later, if needed. After you press F5(COPY), press EXE to store the
graph and it will take you back to the regression information. Press F6(DRAW)
function to see the graph.
84
Calculus with the Casio fx-9750G Plus • Appendix
Copyright © Casio, Inc.
Matrix Function
Entering Matrices:
In the matrix function, you will see a list of matrices you can enter. With the first
available matrix highlighted, enter the number of rows needed first and then the
number of columns. Press EXE after each entry. After the second EXE, the actual
matrix will show up on the screen. When you enter the matrix, the numbers will go
across first and then down to the next row. From this screen, you can do basic
matrix operations by pressing F1(R-OP). You can add and switch rows here.
Matrix Operations:
Most other basic matrix operations can be done in the Run function. Press
OPTN in the Run function and then press F2(MAT) for matrix operations. You
can add and subtract matrices, find the determinant, transpose, augment, send a
matrix to lists, and finding the identity of a matrix. After the function has been
entered, press EXE to see the resulting matrix.
List Function
Sorting data:
The data entered in the statistics lists will show up here as well. If you would like
to erase the lists, highlight the list and press F4(DEL-A). Then enter the new data
as before. You may sort the data by pressing F1. It will prompt you by asking how
many lists, then it will prompt you to select a list. When sorting more than one
list, the calculator will ask for a base list and a second or third list. You can sort
the data in ascending (SRT-A) order or descending order (SRT-D).
Other Operations:
Press the OPTN key for more options in the List function. These options include
moving matrices, finding the minimum, maximum, mean, median, mode, and
product of lists.
Copyright © Casio, Inc.
Appendix • Calculus with the Casio fx-9750G Plus
85
Graph Function
Entering Equations:
When you enter the graph function, the graph edit screen is up. Type in an
equation using the XT or the letter X button for the variable X. You can change
the Y= to an inequality by pressing F3(TYPE) before entering in the equation.
Press F6 for more options for the equation. Press EXIT to get back out to the
original options. Press F6(DRAW) to see the graph.
View of the Graph:
If the graph cannot be seen, press SHIFT F3(V-window). Press F3(STD) for a
standard 20-interval view, press F1(INIT) for initial settings, and F2(TRIG) for a
standard trigonometry function. You may also set the setting manually and store
that setting by pressing F4(STO). Press EXIT to get back to the graph-edit
screen. You may view more than one graph at a time by pressing F1 on each
graph you want to show. This is how you can solve systems of equations.
Dual Screen:
In the graph function, press SHIFT(SETUP). Arrow down to Dual Screen and
switch it to Graph by pressing F1. Press EXIT to get to the previous screen.
Notice when you press DRAW, whatever picture of the graph was last seen will
be on the left, and the frame on the right will be blank. Zoom in on the graph
and that part of the graph will show up on the right. You may also use the dual
screen for a graph and a table. Go into the setup again and change the dual
screen to G to T for Graph to Table. Press EXIT to get back to the edit screen.
The graph will appear with a blank table. Press SHIFT F1(Trace) to trace values
on the graph. To put those values in the table press EXE. Put multiple values in
the table by pressing EXE repeatedly.
Solving for a Specific Point:
After you graph two inequalities you can find the point of intersection of those
graphs by using the G-solve function. Press F5(G-Solv) (you do not have to press
the SHIFT key if the graphs are already on the screen). Press F5(ISCT). The
calculator will think for a few seconds and then trace the graph to the
intersection point for you. You can also find the roots, maximum, minimum,
y-intercept, y and x coordinates for given values, and definite integrals.
86
Calculus with the Casio fx-9750G Plus • Appendix
Copyright © Casio, Inc.
Dynamic Function
Built in List of Equations:
The dynamic function is used to demonstrate the effect of changing certain
variables in an equation. There are seven built in functions that are common in
basic algebra and trigonometry. To get to this list, press F5(B-IN) after you are
in the Dynamic function. To choose one of the built-in functions, highlight and
press F1(SEL).
Setting the Variable and Speed:
After choosing the equation in question, press F4(VAR) to choose the active
variable. Highlight the variable you want to change and enter in values for the
other variables. Notice the arrow to the right of the Dynamic Variable. That is
the speed indication. Press F3(SPEED) to change the speed of the dynamic
function. Highlight the desired speed and press F1(SEL). Each speed has a
different symbol. This symbol is shown next to the variable on the previous
screen. Press EXIT to return to that screen.
Setting the range:
In the dynamic variable screen, press F2(RANG). Set the start and end of the
range you want to show. The pitch is the interval between numbers in range.
Press EXIT to get to the previous screen.
Viewing the dynamic graph:
In the Dynamic Variable screen, press F6(DYNA). When you do this, the
calculator will say "One Moment Please" while loading the graph. Once the graph
is on the screen, press EXE to see the change in the variable. When you press the
EXE, notice the active variable will change.
Entering equations manually:
If the equation you need is not in the built-in list you can enter the equation in
manually. Use the alpha key and letters to enter variables. The variables entered
in manually will be available in the dynamic function.
Copyright © Casio, Inc.
Appendix • Calculus with the Casio fx-9750G Plus
87
Table Function
Entering Equations:
Equations that have been entered into the graph function or dynamic function
will show up here. To delete the equation, press F2, F1. Enter an equation. Set
the type and range. Press F6(TABL) to see the table. The x-values are located in
the first column and the y-values in the second. In the table you can type any
value for x and press EXE and the calculator will calculate the y-value.
Viewing a Graph of the function:
The two choices at the bottom of the table screen are G-CON and G-PLT. Press
F5(G-CON), a connected graph, and press F6(G-PLT), a scatter graph of the
function.
Recursion Function
Inputting Recursion Formulas:
Go into the Recursion function by pressing 8. Once in the function, you must
select what type of recursion function you would like. Press F3(TYPE). Select
the type of recursion by pressing the corresponding F1, F2, or F3. This will
choose between sequences of one, two, and three terms respectively. Two enter
in a equation press F4 for variable options. After the equation is entered, press
EXE to store.
Creating a Table:
You must first set the range of the table. Set the range by pressing F5(RANG).
The range specifies the start and ending value for the variable n, where a and b
should start, and where the pointer starting point is on the graph. Press F2(a1),
then enter in your values for start, end, and a1. The variable n will go in increments of 1. EXIT back out to the equation and press F6(TABL) to see the table.
The options for the table appear across the bottom of the screen.
You have four options: to delete the recursion formula table, to draw a connected
line graph of the formula, to draw a plot type graph of the formula, or to draw a
graph and analysis of the convergence/divergence of the graph (WEB). FORM
takes you back to the formula.
88
Calculus with the Casio fx-9750G Plus • Appendix
Copyright © Casio, Inc.
Conics Function
Graphing a Conic:
In the Main Menu, press 9 for Conics. The first screen you come to will be a
choice of conic equations already input into the calculator memory. Choose one
by arrowing and highlighting the chosen equation and press EXE. Then enter
the values in for the variables in the equation as listed. After each value entered,
press EXE to store and move to the next value. Press F6(DRAW) to see the graph
of the conic equation. Note, the graph of a circle may not necessarily show up as
a circle because the view window needs to be set manually (Choose a X-value that
is twice as large as the Y-value to get a perfect circle).
Equation Function
Solving Systems of Equations:
To solve for a system of equations, select the equation function by pressing the
XT button. Choose F1 for Simultaneous equations. Choose the number of
unknowns you have by pressing the corresponding function key. Enter in the
values for each of variables in the equations. After the values are entered in, press
F1(SOLV). The calculator will solve for the unknown values. REPT at the
bottom of the screen will take you back to the previous screen.
Solving for a Variable in Polynomials:
You can solve for a variable in polynomials up to the third degree. Press F2 for
Polynomial when you enter the equation function. The calculator will ask you to
specify the degree of the polynomial by pressing either F1 for 2 or F2 for 3. Enter
in the values for the polynomial in the matrix shown and then press F1(SOLV).
The two values for X will show up in a matrix.
Solving Equations:
Enter into the equation solver by pressing F3 for Solver. To enter the equation
in question, you may enter numbers, alpha-characters, and symbols. If you do
not put an equals sign in the equation the calculator will assume that the
equation is to the left of the equals sign and that a zero follows it. To specify a
letter or number other than zero, type SHIFT, = and then type the value. After
the equation is entered, press EXE to store. The variables in the equation will
Copyright © Casio, Inc.
Appendix • Calculus with the Casio fx-9750G Plus
89
show up on the screen. Enter the known values and press EXE to store each one.
Highlight the unknown value and then press F6(SOLV). The value for the
unknown will be shown as well as the value for the left-hand side and right-hand
side of the equation to show how accurate the answer is.
Program Function
Running Programs:
After you are in the program function, highlight the program you would like to
run and press F1(EXE).
Other Options:
In this function you can also EDIT a program, create a NEW program, DEL a
program, or DEL-All programs. If you press F6 for more options, you can also
find (SRC) a program or rename (REN) a program.
Time Value of Money Function
Doing Financial Calculations:
In the financial function you have the ability to calculate several variables
using simple interest, compound interest, cash flow, amortization, conversion,
cost, selling, price, margin, and day and date calculation. There are many
abbreviations in the different modes of the function:
APR: annual percentage rate
BAL: balance of principal after installment
C/Y: compounding periods per year
Csh: list for cash flow
Cst: cost
D: number of days
d1: date 1
d2: date 2
EFF: effective interest rate
FV: future value
I%: periodic/ annual interest rate
INT: interest portion of installment
IRR: internal rate of return
Mrg: margin
n: number of compound periods
NFV: net future value
90
NPV: net present value
P/Y: installment periods per year
PBP: pay back period
PM1: first installment
PM2: second installment
PMT: payment
PRN: principal portion of installment
PV: present value
Sel: selling price
SFV: simple future value
SI: Simple Interest
SINT: total interest from installment PM1 to
installment PM2
SPRN: total principal from installment PM1 to
installment PM2
Calculus with the Casio fx-9750G Plus • Appendix
Copyright © Casio, Inc.
Link Function
Transmitting:
The link function is used to transmit and receive data from other calculators.
Calculators can share information from the program list, tables, graphs, lists, and
statistics. By pressing F1 you can transmit data to another calculator. The
calculator will ask for the type of transmission you are making. At this point, F1
will allow you to select what you want to transmit. The calculator will give you a
list that consists of lists, matrices, files, graphs, pictures, variables, and receive 1
and receive 2. The receive options are for receiving 1 list or 2 lists simultaneously.
Select from this list what to transmit and press F6 for the transmission.
Receiving:
The only thing required for receiving data is the press F2(REC). The transmitting
calculator must do all the work!
Image Set Mode:
In the Link function menu, F6 is the image set mode. The images are sent by
pressing the F↔D key. *Note that the F÷D key will not change a fraction to a
decimal or vice versa if the image set mode is set to monochrome. The shift between
decimals to fraction can occur only if the image set mode is turned off.
Contrast Function
Setting the Contrast:
You can adjust the contrast of the screen by using the left and right arrow keys.
Press the right arrow key to darken the contrast and the left arrow key to lighten
the contrast.
Copyright © Casio, Inc.
Appendix • Calculus with the Casio fx-9750G Plus
91
Memory Function
Memory Usage:
To check memory usage, select it by highlighting it and pressing EXE. You can
delete entire sections of the memory as listed only. You cannot delete individual
lists or programs from here. This tool is useful to see where you have memory
used and how much memory you have left on the calculator. The calculator will
give you the option of backing out before you erase any section.
Resetting Memory:
Highlight the Reset option and press EXE. This option will reset the entire
memory of the calculator. This will clear all programs and any statistics, graphs,
matrices, lists, tables, and equations you have entered. The calculator will again
let you back out if you accidentally press Reset.
92
Calculus with the Casio fx-9750G Plus • Appendix
Copyright © Casio, Inc.
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