Lehrerhandbuch EL-9650 Berechnungen (Englisch)

Lehrerhandbuch EL-9650 Berechnungen (Englisch)
SHARP CALC COVER
02.2.19 10:40 AM
Page 3
APPLYING
PRE-CALCULUS/CALCULUS
SHARP
U S I N G
T H E
EGraphing
L - 9Calculator
600
D A V I D P. L A W R E N C E
Applying
PRE-CALCULUS/CALCULUS
using the
SHARP EL-9600
GRAPHING CALCULATOR
David P. Lawrence
Southwestern Oklahoma State University
This Teaching Resource has been developed specifically for use with the
Sharp EL-9600 graphing calculator. The goal for preparing this book was
to provide mathematics educators with quality teaching materials that
utilize the unique features of the Sharp graphing calculator.
This book, along with the Sharp graphing calculator, offers you and your
students 10 classroom-tested, topic-specific lessons that build skills.
Each lesson includes Introducing the Topic, Calculator Operations, Method
of Teaching, explanations for Using Blackline Masters, For Discussion,
and Additional Problems to solve. Conveniently located in the back of
the book are 33 reproducible Blackline Masters. You’ll find them ideal
for creating handouts, overhead transparencies, or to use as student
activity worksheets for extra practice. Solutions to the Activities are
also included.
We hope you enjoy using this resource book and the Sharp EL-9600
graphing calculator in your classroom.
Other books are also available:
Applying STATISTICS using the SHARP EL-9600 Graphing Calculator
Applying PRE-ALGEBRA and ALGEBRA using the SHARP EL-9600 Graphing Calculator
Applying TRIGONOMETRY using the SHARP EL-9600 Graphing Calculator
Graphing Calculators: Quick & Easy! The SHARP EL-9600
CALCULUS USING THE SHARP EL-9600
i
CONTENTS
CHAPTER TOPIC
PAGE
1
Evaluating Limits
1
2
Derivatives
7
3
Tangent Lines
13
4
Graphs of Derivatives
19
5
Optimization
25
6
Shading and Calculating Areas Represented
by an Integral
31
Programs for Rectangular and Trapezoidal
Approximation of Area
37
8
Hyperbolic Functions
43
9
Sequences and Series
47
Graphing Parametric and Polar Equations
53
Blackline Masters
59
Solutions to the Activities
94
7
10
Dedicated to my grandma, Carrie Lawrence
Special thanks to Ms. Marina Ramirez and Ms. Melanie Drozdowski
for their comments and suggestions.
Developed and prepared by Pencil Point Studio.
Copyright © 1998 by Sharp Electronics Corporation.
All rights reserved. This publication may not be reproduced, stored in a retrieval
system, or transmitted in any form or by any means, electronic, mechanical,
photocopying, recording, or otherwise without written permission.
The blackline masters in this publication are designed to be used with appropriate
duplicating equipment to reproduce for classroom use.
First printed in the United States of America in 1998.
ii
CALCULUS USING THE SHARP EL-9600
Chapter one
EVALUATING LIMITS
Introducing the Topic
The concept of a limit is one of the basic building blocks of calculus.
An understanding of limits is also necessary when investigating the behavior
of a function near a vertical or horizontal asymptote and the end behavior of
functions in precalculus.
The limits you and your students consider in this chapter fall into one of three
categories:
•
lim f(x), the limit of a function f(x) as x increases without bound. This
x ∞
limit is an indicator of the positive end behavior of the function.
•
lim f(x), the limit of a function f(x) as x decreases without bound. This
x -∞
limit is an indicator of the negative end behavior of the function.
←
←
When either of these two limits exist; that is, the values of f(x) get closer and
closer to a specific number L as x gets larger and larger or as x gets smaller and
smaller, the line y = L, is a horizontal asymptote of the function.
lim f(x), the limit of a function f(x) as x gets very close to, but does not
x a
equal, the value x=a. This limit describes the behavior of the function
←
•
Evaluating Limits/CALCULUS USING THE SHARP EL-9600
1
near x=a rather than at x=a. The limit of f(x) as x approaches a exists
and equals L, written lim f(x)=l, provided that for all values of x in
x a
the domain of f(x), the values of f(x) get closer and closer to L as x
←
approaches a from each side of a.
When lim f(x) does not exist in the sense that the values of f(x) increase
x a
and/or decrease without bound as the values of x approach a, the line x=a is a
←
vertical asymptote of the function.
This chapter investigates graphical and numerical methods of evaluating limits,
provided those limits exist. These methods can, in many cases, give very
accurate approximations of limits. However, they do not prove the existence
of limits. You should consult a calculus text for methods of formal evaluation
of limits.
Calculator Operations
After turning your calculator on, prepare for the investigations in this chapter by
setting the calculator to floating point decimal display by pressing 2ndF
SET UP , touching C FSE, and double touching 1 Float Pt. Set the calculator to
rectangular coordinates by touching E COORD and double touching 1 Rect.
Press 2ndF
QUIT to exit the SET UP menu.
INVESTIGATING LIMITS GRAPHICALLY
Observing the graph of a function is useful for gathering information as to
whether or not a limit exists. If the limit does exist, a graph is helpful in providing
information that allows you to estimate the value of the limit or check an
algebraically determined value.
Consider, for instance, the function f(x) = (2x + 2)/(x 2 – 1). Press Y=
CL to access and clear the Y1 prompt. Press ENTER
CL to clear the
remaining prompts. Construct a graph of f(x) in the Decimal viewing window
+
2
2
➧
touch
▼
by first entering Y1= (2x + 2)/(x 2 – 1) with the keystrokes a/b
X/θ/T/n
x
2
–
2
X/θ/T/n
1 , and then press ZOOM , touch A ZOOM,
on the screen, and touch 7 Dec to see the graph.
Evaluating Limits/CALCULUS USING THE SHARP EL-9600
Notice that even though (2x + 2)/(x 2 – 1) is not defined at x = -1 (as evidenced by
the hole in the graph at that point), the functional values appear to be getting
closer and closer to -1. A careful observation of the graph leads to the following
estimates:
lim f(x) ≈ 0, lim f(x) ≈ -1, lim f(x) does not exist, and lim f(x) ≈ 0.
-∞
x 1
x ∞
x -1
It also appears that the line y = 0 is a horizontal asymptote and the line x = 1 is a
vertical asymptote for this function.
←
←
←
←
x
INVESTIGATING LIMITS NUMERICALLY
Tables of functional values sometimes provide more detailed information than a
graph when investigating limits. The Sharp EL-9600 has a TABLE feature to assist
you in constructing a numerical table of values. Press TABLE to access the
TABLE feature.
Notice the table provides the x values and their corresponding y values
according to Y1. You can change the table settings by pressing 2ndF
TBLSET .
Evaluating Limits/CALCULUS USING THE SHARP EL-9600
3
You can change the table start value and the table step value.
Verify the following values using the the TABLE feature.
x
x gets smaller and smaller →
-10
-50
-100
-250
-500
-1000
y
-.18182
.00797
-.00399
-.00200
-.03922
-.01980
-10,000
-.00020
y = f(x) appears to get closer and closer to 0
This provides evidence that lim- f(x) = 0.
x ∞
←
x
y
x approaches -1 from the left
1.05
1.01
1.001
.9756 .99502
.9995
-
1.0001
.9999
y=f(x) gets closer and closer to -1 from above
x approaches -1 from the right
.9999
.999
.99
.90
1.0001 1.0005
1.0051
1.0526
y=f(x) gets closer and closer to -1 from below
This provides evidence that lim- f(x) = -1.
x 1
←
Method of Teaching
Use Blackline Master 1.1 to create an overhead or handout for investigating the
limit of a function using a graph. Be certain students understand that this
method provides estimates of limits and does not constitute proof that a limit
exists or does not exist. Use Blackline Master 1.2 to create an overhead or handout for numerically investigating the limit of a function using the TABLE feature.
If students cannot establish a pattern using the values indicated on the Blackline
Masters, they should evaluate the function at other values of x until a pattern is
established or until they can determine that the function either increases or
decreases without bound. Use Blackline Master 1.3 to create a worksheet for the
students. Use the topics For Discussion to supplement the worksheets.
Using Blackline Master 1.3
Problem 1
At points where there is a jump discontinuity in the function (the limit from the
left differs from the limit from the right), you may find it necessary to set the
calculator to dot mode by pressing 2ndF
4
FORMAT , touching E STYLE1, and
Evaluating Limits/CALCULUS USING THE SHARP EL-9600
double touching 2 Dot. Do this whenever the pieces of the graph appear
connected at the point of discontinuity.
Problem 2
Discuss with students why the line x = 3 is a vertical asymptote and why the line
y = 1 is a horizontal asymptote for f(x). Also discuss the difference in the nature
of the two discontinuities at x = 3 and x = -3.
Problem 3
The equation in this example is called a logistics equation, and it represents limited population growth. Discuss with students why the independent variable in
▼
or
to trace the graph and estimate lim p(x) to
x ∞
be 10,000. The answer to the first question is obtained by using the TABLE
←
press TRACE , and press
▼
the function must be x for graphing. After drawing the graph, students can
feature or the CALC feature when viewing the graph. Students should let x take
on larger and larger values to obtain lim p(x)=10,000.
x ∞
←
For Discussion
A function f(x) is said to be continuous at a value x = a if:
•
f(a) exists,
•
lim f(x) exists, and
x a
•
lim f(x) = f(a)
x a
←
←
Discuss with students why the function in Blackline Master 1.3, f(x)=|x +
1|/(x+1), is continuous at all values of x except x=-1. Also discuss why the
function f(x)=(x2+2x – 3)/(x2 – 9), is continuous at all values except x = -3 and
x = 3, and why the function p(t)=10,000/(1 + 15e-t/4), is continuous at all values of t.
Evaluating Limits/CALCULUS USING THE SHARP EL-9600
5
Additional Problems
In problems 1-4, evaluate the limits using numerical methods. Use a graph of the
function to estimate the values of x where f(x) is not continuous.
1. lim- f(x) if f(x) = (8 + x3)/(x + 2)
x 2
←
2. lim f(x) if f(x) = (x – 2.5)/|2.5 – x|
x 2.5
←
3. lim f(x) and lim f(x) if f(x) = (1 + x)1/x
x 0
x ∞
←
←
4. lim f(x) if f(x)=(2300x)/(500 – x)
x ∞
←
5. Mary is taking a typing class. Suppose the number of words per
minute W that Mary can type after t weeks of practice is given by
the equation W(t)=85(1 – e-0.3t ).
a.
If the class lasts 6 weeks, how many words per minute can
Mary type at the end of the class?
b.
If Mary gets a job that requires her to type (and therefore
continue practicing), will she ever be able to type 100 words
per minute?
6
Evaluating Limits/CALCULUS USING THE SHARP EL-9600
Chapter two
DERIVATIVES
Introducing the Topic
The derivative is one of the fundamental tools of calculus used to study
functions and solve problems. The derivative of a function tells us the rate at
which the values of f(x) are changing as x changes. The ratio [f(x) – f(x)]/(x – a)
gives the average rate of change of a function f(x) with respect to the variable x.
Provided it exists, lim [f(x) – f(a)]/(x – a) is called the instantaneous rate of
x a
change of f(x) with respect to the variable x at a, or more simply, the derivative
f ’(a). If the limit does not exist, we say that f(x) is not differentiable at a.
←
Graphically, the average rate of change is the slope of the secant line joining the
points (x, f(x)) and (a, f(a)) for x ≠ /a, while the derivative f ’(a) gives the slope
of the tangent line to the function f(x) at x = a. Chapter 3 offers a program to
further investigate the geometrical interpretation of a derivative as the slope
of a the tangent line.
Calculator Operations
Prepare for the investigations in this chapter by setting the calculator to radian
measure by pressing 2ndF
SET UP , touching B DRG, and double touching
Derivatives/CALCULUS USING THE SHARP EL-9600
7
2 Rad. Set the calculator to floating point decimal display by touching C FSE
and double touching 1 Float Pt. Set the calculator to rectangular coordinates
by touching E COORD and double touching 1 Rect. Press 2ndF
QUIT to exit
the SET UP menu.
DERIVATIVES USING THE LIMIT DEFINITION
Consider the function f(x) = x2. To find the derivative of this function at x = 2
using the definition f ’(2) = lim [f(x) – f(2)]/(x – 2) = lim (x2 – 4)/(x – 2), provided
x 2
x 2
the limit exists, use the TABLE feature that was discussed in Chapter 1 to verify
←
←
the entries for the values of the quotient (x2 – 4)/(x – 2) in the table below:
(Table entries are rounded to five decimal places.)
←x approaches 2 from the right
x approaches 2 from the left→
x
1.9
1.99
1.999
1.9999
2.0001
2.001
2.01
2.1
y
3.9
3.99
3.999
3.9999
4.0001
4.001
4.01
4.1
The quotient gets closer and closer
to 4 from below
The quotient gets closer and closer
to 4 from above
It certainly appears that lim (x2 – 4)/(x – 2) = f ’(2) = 4
x 2
←
DERIVATIVES USING THE d/dx FUNCTION
The calculator has a built-in function denoted by d/dx that uses numerical
methods to estimate the derivative of a function at a given value. The entry
form of the derivative function is d/dx (f (x), a). For instance, to estimate f ’(2)
+ –
for f (x) = x2 using the derivative function, press × ÷ MATH , touch A CALC,
double touch 05 d/dx( press X/θ/T/n x2 , 2 ) to input: d/dx(x2, 2).
Press ENTER to compute.
8
Derivatives/CALCULUS USING THE SHARP EL-9600
DERIVATIVES USING THE DERIVATIVE TRACE
Your calculator can also display values calculated by the derivative function on
the graphics screen as you trace the function. The values that are displayed are
the values calculated by the derivative function d/dx. To activate this y’ trace,
press 2ndF
FORMAT , touch D Y’, and double touch 1 ON. Press 2ndF
QUIT to exit the FORMAT menu. Now, press Y=
for Y1 by pressing X/θ/T/n
➧
touching 5, touching
CL and enter the function
2
x . Draw the graph by pressing WINDOW
EZ ,
on the screen, double touching 7 -5<x<5, and double
▼
touching 4 -10<Y<10. Press TRACE to activate the derivative trace and press
to observe the values calculated by the derivative trace as the trace cursor
moves along the graph.
Turn off the derivative trace by pressing 2ndF
double touch 2 OFF. Press 2ndF
FORMAT , touch D Y’, and
QUIT to exit the FORMAT menu.
Method of Teaching
Use Blackline Master 2.1 to create an overhead or handout for investigating the
derivative of a function using the limit definition. Use Blackline Master 2.2 to
create an overhead or handout for investigating the built-in derivative function
d/dx, and use Blackline Master 2.3 to create an overhead or handout for using the
derivative trace.
Be certain that students understand the d/dx function and the y ’ trace give
approximations to the derivative of the function at y, not the exact value of lim
x a
[f(x) – f(a)]/(x – a). Use the topics For Discussion to supplement the worksheets.
←
Derivatives/CALCULUS USING THE SHARP EL-9600
9
Using Blackline Master 2.1
The first problem demonstrated above under Calculator Operations is addressed
on Blackline Master 2.1 as Activity 1. In Activity 2, students should be able to
determine from the graph of (2x3 – 2)/(x – 1) that the limit as x approaches 1
exists. Be certain they notice that this quotient is not defined at x = 1, as
evidenced by the hole in the graph at x = 1.
Activity 3 asks the students to find f ’(-3). Discuss with them that the
limit definition of the derivative gives f ’(3)= lim [f (x) – f (-3)]/(x – 3) =
x -3
lim (2x3 + 54)/(x + 3) provided the limit exists. After determining from the graph
x -3
that the limit certainly seems to exist, students should create a table of values,
letting x approach -3 from both the left and right, to see that f ’(-3) appears to be 54.
←
←
Using Blackline Master 2.2
The second problem demonstrated above under Calculator Operations is
addressed on Blackline Master 2.2 as Activity 1. In Activity 2, students should
approximate the value of f ’(-1) with d/dx(2x3, 1). Activity 3 asks students to
determine that f ’(0) does not exist for f (x)=|x|, have them construct a graph of
y=|x|, and point out that the derivative of a function does not exist where the
graph comes to a “v”. This is a good example to show that continuity does not
imply differentiability since |x| is continuous, but not differentiable, at x = 0.
Using Blackline Master 2.3
The third problem demonstrated above under Calculator Operations is
addressed on Blackline Master 2.3 as Activity 1. In Activity 2, students should
look for a pattern in the y’ = f’(x) values as they relate to the x values or the
y = f(x) values to discover a rule for f’(x) . Discuss with students that this
method only provides an estimate for the rule for f’(x) , not a proof of that rule
or that f’(x) exists for all values of x. In Activity 3, students should realize that
f’(0) does not exist because the values of d/dx(x1/3/(x2+1), 0), as displayed by
the f’ trace, increase without bound as x approaches 0 from either side of 0.
10
Derivatives/CALCULUS USING THE SHARP EL-9600
For Discussion
The derivative of a function f(x) is often defined as:
f’(x) = lim [f(x + ∆x) – f(x)]/ ∆x.
∆x 0
If you substitute x=a+ ∆x in the definition of the derivative given at the beginning
←
of this chapter, f ’(a) = lim [f(x) – f(a)]/(x – a), you obtain f ’(a) = lim [f(a + ∆x) –
x a
∆x 0
f(a)]/ ∆x, the derivative of f(x) evaluated at x = a. Discuss with the students that
←
←
∆x used in this definition has the same meaning as the Dx used in the d/dx
function and the y’ trace.
Additional Problems
1. Use the limit definition of the derivative to determine if f ’(1) exists
for f(x) =2x – 3. If f ’(1) exists, find its value. If it does not exist, explain
why not.
2. Use the limit definition of the derivative to determine if f ’(1) exists
for f(x) =|x – 1|. If f ’(1) exists, find its value. If it does not exist,
explain why not.
3. Use values obtained with the derivative function d/dx to determine
if f ’(0) exists for f(x) = 2x/(x + 1). If f ’(0) exists, estimate its value.
If it does not exist, explain why not.
4. Use values obtained with the derivative function d/dx to determine
if f ’(0) exists for f(x) = sin (πx). If f ’(0) exists, estimate its value.
If it does not exist, explain why not.
5. For each of the following, enter the X parameters in the WINDOW
screen and then press ZOOM , and double touch 1 Auto to
construct a graph of the function y = f(x) over the indicated interval,
and then use the y’ trace to complete the table of values for
y’ = f ’(x). (Round table entries to 3 decimal places.) Use the
pattern you view in the table of values to determine a formula for f ’(x).
Derivatives/CALCULUS USING THE SHARP EL-9600
11
a. f(x)=e–x
(-6.3 < x < 6.3)
x
-4
-3
y’
x
-4
-3
y
y’
c. f(x)=ln x, x>0
(0 < x <12.6)
x
.5
1
-2
2
-1
-1
3
0
0
4
1
1
5
2
2
6
3
3
7
4
4
8
-2
12
y
b. f(x)= 1/2 x2
(-6.3 < x < 6.3)
Derivatives/CALCULUS USING THE SHARP EL-9600
y
y’
Chapter three
TANGENT LINES
Introducing the Topic
In this chapter, you and your students will learn how to program the Sharp
EL-9600 graphing calculator, execute the program and use the program to find
tangent lines to a curve. Further, you and your students will learn how to draw
a tangent line to a curve at a point. A tangent line to a curve is the line that
intersects the graph in only one point and its slope represents the slope of the
curve at that point. The TANGENT program computes the tangent line to a
curve at a particular point. The function is entered for Y1.
Calculator Operations
Turn the calculator on and press 2ndF
PRGM to enter the programming
menu. The menu consists of commands to execute, edit, and create new
programs. Touch C NEW and press ENTER to open a new program.
Tangent Lines/CALCULUS USING THE SHARP EL-9600
13
The calculator is now locked in ALPHA mode and is prepared to accept a
name for the new program. Name the new program TANGENT by pressing
T
A
N
G
E
N
T
ENTER .
You can now enter in the TANGENT program. Please note that you must press
ENTER at the end of each line. If you make a mistake, use the calculator’s
editing features to correct the error. Enter the following program:
PROGRAM
KEYSTROKES
(Input the point at which the tangent line is to be drawn)
Input X
2ndF
PRGM
A
3
X/θ/T/n
ENTER
(Find the slope of the tangent line)
d/dx(Y1, X)⇒M MATH
)
A
STO
0
5
ALPHA
VARS
M
A
ENTER
A
,
1
X/θ/T/n
ENTER
(Compute the point of intersection for the curve and tangent line)
VARS A ENTER 1 ( X/θ/T/n ) STO ALPHA
Y1(X)⇒Y
Y
ENTER
(Compute the y intercept)
-M•X+Y⇒B
(–) ALPHA
B
M
×
X/θ/T/n
1
ENTER
ENTER
(Display the equation for the line)
ClrT
14
2ndF
PRGM
C
Tangent Lines/CALCULUS USING THE SHARP EL-9600
+
ALPHA
Y
STO
ALPHA
Print “TANGENT 2ndF
PRGM
A
1
2ndF
LINE=
N
G
E
N
T
PRGM
1
2ndF
T
A
Print “Y=MX+B 2ndF
=
Print “M=
ALPHA
2ndF
=
X/θ/T/n
M
PRGM
1
2ndF
SPACE
PRGM
+
2
L
2
I
2
ALPHA
N
=
E
ALPHA
ALPHA
PRGM
2ndF
B
Y
ENTER
ALPHA
ENTER
ALPHA
M
ALPHA
B
ALPHA
ENTER
Print M
2ndF
PRGM
1
ALPHA
Print “B=
2ndF
PRGM
1
2ndF
=
M
ENTER
PRGM
2
ALPHA
ENTER
Print B
2ndF
PRGM
1
ALPHA
End
2ndF
PRGM
6
ENTER
Press 2ndF
PRGM
B
ENTER
QUIT to save the program and exit the editing mode.
for Y1. Enter f(x) = x3-x2+1 for Y1 by pressing Y=
–
X/θ/T/n
x
2
+
1
CL
X/θ/T/n
ab
3
▼
Before executing the TANGENT program, you need to enter the function of interest
ENTER .
Tangent Lines/CALCULUS USING THE SHARP EL-9600
15
Execute the TANGENT program by pressing 2ndF
PRGM , touching A EXEC,
and double touching TANGENT. Enter an X value for the point at which you
desire the tangent line to be found. Enter an X of 1 by pressing 1
ENTER .
You should then see the following equation for the tangent line to the curve
at x = 1.
You can repeat this process for other x values. Press ENTER to execute the
▼
If you receive an error statement, press
or
▼
program over and over again. Press CL to clear the screen.
to go to the line within the
program in which the error occurs. Compare your line with the correct one
above to find the error. Correct the error using the editing features of the
calculator and save the program by pressing 2ndF
QUIT . Try executing the
program again.
Method of Teaching
Use Blackline Masters 3.1 and 3.2 to create overheads for entering and
executing the TANGENT program. Go over in detail how to enter the program
and what the different program lines are doing. Have the students enter the
program and execute it (correcting any errors). Use Blackline Masters 3.3 to
create a worksheet for the students on how to draw tangent lines on a graph.
Use the topics For Discussion to supplement the worksheets.
Using Blackline Master 3.3
To draw the tangent line on a displayed graph, you must first enter the graph for
3
16
▼
Y1. Enter the function f(x) = x3 – x2 + 1 for Y1 by pressing Y=
–
X/θ/T/n
x2
+
1
ENTER .
Tangent Lines/CALCULUS USING THE SHARP EL-9600
CL
X/θ/T/n
ab
➧
Graph the function by pressing ZOOM , touching A ZOOM, touching
on the
screen, and touching 7 Dec.
DRAW , touching A DRAW,
double touching 5 T_line(, move the tracer right to x =1 by pressing
▼
Draw the tangent line at x = 1 by pressing 2ndF
repeatedly, and then press ENTER .
Tangent Lines/CALCULUS USING THE SHARP EL-9600
17
For Discussion
You and your students can discuss what occurs when the tangent line is horizontal.
Additional Problems
Find and graph the tangent lines for the following functions at the given point.
Remember to graph in the decimal window so your tracer will find the x integer.
1. f(x) = x 2 at x = 2
2. f(x) = sin x at x = 1
3. f(x)= tan x at x = -.5
4. f(x) = √ x at x = 1
5. f(x) = -(x – 2)2 + 1 at x = 2
18
Tangent Lines/CALCULUS USING THE SHARP EL-9600
Chapter four
GRAPHS OF DERIVATIVES
Introducing the Topic
In this chapter, you and your students will explore connections between the graph
of a function and the graph of the first and second derivatives of the function.
Calculator Operations
Prepare for the investigations in this chapter by setting the graphing calculator
to radian measure by pressing 2ndF
SET UP , touching B DRG, and double
touching 2 Rad. Set the calculator to floating point decimal display by touching
C FSE and double touch 1 FloatPt. Set the calculator to rectangular graphing by
touching E COORD, and double touching 1 Rect. Press 2ndF
QUIT to exit the
SET UP menu.
Also, choose sequential graphing by pressing 2ndF
FORMAT , touching
F STYLE2, and double touching 1 Sequen. Set the calculator to connected mode
by touching E STYLE1 and double touching 1 Connect. Press 2ndF
QUIT to
exit the FORMAT menu.
The built-in derivative function d/dx can be used to draw the graph of the
derivative of a function at all points where the derivative exists.
Graphs of Derivatives/CALCULUS USING THE SHARP EL-9600
19
Using the calculator, students can easily draw the graphs of many functions and
their derivatives to discover connections between the graphs of f(x), f ’(x), and
f ’’(x). For instance, suppose that f(x)=2x3 – 7x2 – 70x + 75. Press Y=
f(x) for Y1 by pressing 2
X/θ/T/n
+
7
X/θ/T/n
CL to clear additional Y prompts. Enter
ab
▼
access and clear Y1. Press ENTER
CL to
3
–
7
X/θ/T/n
x2
–
7
0
5 .
Graph the function by pressing WINDOW
EZ , touching 5, double touching
5 -10<X<10, and double touching 1 -500<X<500.
To determine the point at which the relative maximum occurs, press TRACE
2ndF CALC , and double touch 4 Maximum. The maximum occurs at X=
-2.4427, Y = 175.07. Find the point at which the relative minimum occurs with
2ndF
CALC , and double touch 3 Minimum. The minimum occurs at
X= 4.776, Y = -201.1084. Combining this information with a view of the graph,
we see that f(x) is increasing from -∞ to -2.4427 and from 4.776 to ∞ while f(x)
is decreasing for x between -2.4427 and 4.776.
Next, construct the graph of f ’(x). Press Y=
ENTER and input d/dx(Y1) in the
Y2 location with the keystrokes MATH , touch A CALC, double touch 05 d/dx (
press VARS
ENTER , touch A XY, double touch 1 Y1, and press )
Press GRAPH to obtain the graphs of f(x) and f ’(x).
20
Graphs of Derivatives/CALCULUS USING THE SHARP EL-9600
ENTER .
We now want to find the two x-intercepts of f(x). Press TRACE
▼
to place the tracer on the graph of the derivative. Then, press 2ndF
CALC
and double touch 5 X_Incpt to obtain X = 4.77606. Press 2ndF
CALC and
double touch 5 X_Incpt again to obtain the other x-intercept at X = -2.44273.
Comparing these values to the x-coordinates of the points at which the maxima
and minima of f(x) occur, we see they are almost identical. Calculus theory tells
us that these are exactly the same values. However, you may view a slight
difference in trailing decimal places due to the numerical approximation
routines used by the calculator. For convenience, we will round answers to
three decimal places.
Where is f ’(x) positive? The graph of the derivative is above the x-axis for
x<-2.443 and x>4.776. Notice this is where the graph of the function f(x) is
increasing. Where is f ’(x) negative? The graph of the derivative is below the
x-axis for -2.443<x<4.776. Notice this is where the graph of f(x) is decreasing.
Next, find the minimum point of f ’(x) by first making sure the trace cursor is
on the graph of the derivative, pressing 2ndF
CALC , and double touching
3 Minimum. The minimum of the derivative occurs at the point X = 1.167, Y =
-78.167. Look at the graph of the function of the derivative and observe that this
appears to be the point at which the function “bends a different way”; that is,
the point at which f(x) changes concavity and is called the point of inflection.
Find the point of inflection directly by moving the cursor to the original function
CALC , touching
➧
and pressing 2ndF
on the screen, and touching 7 Inflec.
Let’s now add the graph of the second derivative, f ’’(x), to the picture.
Press Y=
ENTER
ENTER , and input d/dx(Y2) in the Y3 location with the
keystrokes MATH , touch A CALC, double touch 05 d/dx( press VARS
ENTER , touch A XY, double touch 2 Y2, and press )
ENTER .
Press GRAPH to obtain the graphs of f(x), f ’(x), and f ’’(x).
Graphs of Derivatives/CALCULUS USING THE SHARP EL-9600
21
(Notice that the graph of f ’(x) takes longer to draw than the graph of f(x) and
that the graph of f ”(x) takes even longer to appear on the screen. This is
because the calculator is determining functional values of the derivatives using
numerical approximations before plotting the points and connecting them to
draw the graph. In fact, the graph of f ”(x) may at times appear “jagged” for this
reason. Students should realize that since this function is a cubic, its derivative
is a quadratic, and the second derivative is therefore a line. Your students may
sometimes prefer entering algebraically-calculated derivatives rather than using
the calculator-generated derivatives.)
Where is f ”(x) zero? After pressing TRACE
▼ ▼ to place the tracer on the
graph of the second derivative, press 2ndF
CALC and double touch 5 X_Incpt
to find that f ”(x)= 0 at X= 1.167. Calculus theory tells us that this is exactly the
x-value of the point where the function f (x) changes concavity; that is, the
inflection point.
▼
original function. Press 2ndF
▼
Find the y-value of this point by pressing
to move the cursor to the
CALC , double touch 1 Value, and enter 1.167
by pressing 1 • 1 6 7 ENTER . The calculator provides a y value of
-13.045. To the left of the point (1.167, -13.045) the function is concave down
(curved downward) and to the right of the inflection point, f (x) is concave up
(curved upward).
The connections we have discovered between the graphs of f (x), f ’(x), and f ”(x)
are summarized in the tables on the next page.
22
Graphs of Derivatives/CALCULUS USING THE SHARP EL-9600
Interval
f (x) is
f’(x) is
f ’’(x) is
x < -2.443
increasing, concave
down
positive
negative
-2.443 < x < 1.167
decreasing, concave
down
negative
negative
1.167 < x < 4.776
decreasing, concave
up
negative
positive
x > 4.776
increasing, concave
up
positive
positive
x-value of point
f (x) has
f ’(x) has
x = -2.443
relative maximum
x-intercept
x = 1.167
inflection point
minimum
x = 4.776
relative minimum
x-intercept
f ’’(x) has
—x-intercept
—-
Method of Teaching
Use Blackline Masters 4.1, 4.2, and 4.3 to create overheads or handouts for
investigating how the first and second derivatives of a function can be used to
find where the graph of the function is increasing/decreasing and changes
concavity. Also, use the Blackline Masters for investigating connections
between the graph of a function and its first and second derivatives. Use
Blackline Masters 4.4 to create a worksheet for the students. Inform the
students of inaccuracies in the calculator-generated values of higher derivatives
using the d/dx( operation, and they need to find the derivatives by hand to enter
them in Y2, etc.
Using Blackline Master 4.4
In Activity 1, students should trace the graph of Y3 to see that it lies on the xaxis except for at x=0 where f ’’(x) does not exist. When students use the d/dx
function to construct the graph of the second derivative of √ (9-X2) in Activity 2,
inaccuracies occur. In the two figures below, the one on the left shows the
graph of Y1= √ (9-X2), Y2=d/dx(Y1), and Y3=d/dx(Y2). The figure on the right
shows the graphs of Y1= √ (9-X2), Y2=-X/ √ (9-X2)= f ’(x) obtained by the power
and chain rules, and Y3=d/dx(Y2). Both figures were drawn using the default
viewing window.
Graphs of Derivatives/CALCULUS USING THE SHARP EL-9600
23
For Discussion
The derivative function d/dx is very reliable when graphing the first derivative
of a function. However, if you want it to graph second and/or higher order
derivatives that are computed from a calculator-generated derivative,
inaccuracies may sometimes result due to limitations of the numerical
approximation techniques and the technology.
Additional Problems
Construct the graphs of f(x) , f ’(x), and f ’’(x) for each of the following functions.
Use these graphs of the first and/or second derivatives to identify where the
function f(x) is increasing, decreasing, concave up, concave down, and where
any relative maxima or minima occur.
24
1.
f(x) = 5x2 – x3 + 4x – 2
2.
f(x) = e-x2
3.
f(x) = √ x
4.
f(x) = |x3|
Graphs of Derivatives/CALCULUS USING THE SHARP EL-9600
Chapter five
OPTIMIZATION
Introducing the Topic
In this chapter, you and your students will learn to apply procedures for finding
maxima and minima to solve “real-world” problems. The calculator’s CALC
function can be used to approximate such values with a high degree of accuracy
in a precalculus course, and finding exact values of maxima and minima is one
of the most important applications of first derivatives in a calculus course.
Calculator Operations
Prepare for the investigations in this chapter by setting the calculator to radian
measure by pressing 2ndF
SET UP , touching B DRG, and double touching
2 Rad. Set the calculator to floating point decimal display by touching C FSE
and double touch 1 FloatPt. Set the calculator to rectangular graphing by
touching E COORD, and double touching 1 Rect. Press 2ndF
QUIT to exit
the SET UP menu.
Instructions given in this chapter are appropriate for either a precalculus or
calculus course. However, if you are using this manual in a calculus course, you
can have students enter the function in each problem in Y1, d/dx(Y1) in Y2, and
graph the function and the calculator-generated derivative in an appropriate
Optimization/CALCULUS USING THE SHARP EL-9600
25
viewing window. Students should then enter their algebraically-determined
derivative in Y3 and press GRAPH . If only two graphs are observed, it is
very probable that the algebraically-determined derivative has been correctly
computed.
Consider the following application. A new product was introduced through a
television advertisement appearing during the Super Bowl. Suppose that the
proportion of people that purchased the product x days after the advertisement
appeared is given by f(x)= (5.3x)/(x2 + 15). When did maximum sales occur and
what proportion of people purchased the product at that time?
To answer this question using a graph of f(x), first find a suitable domain
and range for the problem situation. Since x is the number of days after the
advertisement appeared, x ≥ 0, and because y is a proportion, 0 ≤ y ≤ 1. Next,
press Y=
CL to clear the Y1 prompt. Press ENTER
CL to clear additional
X/θ/T/n
▼
prompts. Enter f(x) in the Y1 location with the keystrokes a/b
X/θ/T/n
x
2
+
1
5
•
3
5 . Let’s examine the graph for the first 25
days after the advertisement appeared. Press WINDOW , enter Xmin= 0,
Xmax= 25, Xscl= 5, Ymin= 0, Ymax= 1, Yscl= 1. Press GRAPH to obtain:
When did maximum sales occur and what proportion of people purchased
the product at that time? Press 2ndF
CALC and double touch 4 Maximum to
find X = 3.873, Y= 0.684. We see from the graph that a relative (local) maximum
occurs at this point. Is it the absolute maximum? Press WINDOW and change
Xmax to 100. Press GRAPH to view the graph of the function. Repeat the
procedure for Xmax = 500. The graph certainly does not appear to have any
functional values greater than Y= 0.684.
26
Optimization/CALCULUS USING THE SHARP EL-9600
(Calculus students should realize that since f(x) is a continuous function, the
absolute maximum occurs at a point where either f ’(x)= 0 or f ’(x) does not exist
or at an endpoint of the interval. They can therefore obtain the definite answer
that the absolute maximum occurs at X= 3.873, Y= 0.684.) You may wish to have
students express their answers to problems in this chapter in sentence form. If
so, an appropriate answer is “The maximum sales occurred 3.873 days after the
advertisement first appeared, and the proportion of people that purchased the
product at that time is .684.”
Let’s look at another example. A metal container with no top in the form of a
right circular cylinder is being designed to hold 185 in3 of liquid. If the material
for the container costs 14¢ per square inch and the cost of welding the seams
around the circular bottom and up the side cost 5¢ per inch, find the radius of
the container with the smallest cost. What is the minimum cost?
You will need these formulas:
Cylinder:
Circular Base:
Volume= πr2h
Area= πr2
Lateral Surface Area= 2πrh
Circumference= 2πr
Now, the area of the container = area of lateral surface + area
of bottom = 2 πrh + πr2
length of welds = h + 2πR
$ cost of container = .14(area of container) + .05(length of welds)
= .14(2πrh + πr2) + .05(h + 2πr)
Next, we know that 183 = πr2h, so h = 183/(πr2).
Substituting this in the cost equation, we find $ cost of container
= .14(2πr(183/(πr2)) + πr2) + .05(183/(πr2) + 2πr).
Recall that the independent variable in the graphing mode must be called x.
Press Y= and clear any previously-entered functions with CL , and either type
in the $ cost of container in its current form with r = x or type in the simplified
Optimization/CALCULUS USING THE SHARP EL-9600
27
π
+
1
a/b
•
9
2
•
4
1
5
X/θ/T/n
▼
5
2ndF
+
π
•
1
X/θ/T/n
4
x
X/θ/T/n
π
2ndF
2
▼
x
2
▼
a/b
▼
version, 51.24/x + .14πx2 + 9.15/(πx2) + .1πx, in Y1 with the keystrokes
+
•
1
2ndF
X/θ/T/n . What window settings do we use? Since x is the radius of the
container, we know that x > 0. The cost Y1 will also be greater than 0. A little
experimenting leads to a viewing window of 0< x < 10 and 0 < y< 50. Enter the
viewing window and press GRAPH to view the graph of the cost function:
To find the minimum cost, press 2ndF
CALC and double touch 3 Minimum
to obtain X= 3.799, Y= 21.231. Changing Xmax to increasingly larger values
shows that the costs continue to increase past this point. Therefore, the
radius of the container with the smallest cost is 3.799 inches. The minimum
cost of the container is $21.23.
Method of Teaching
Use Blackline Master 5.1 to create an overhead or handout for investigating a
maximization problem. Use Blackline Master 5.2 to create an overhead or handout for investigating a minimization problem. Use Blackline Master 5.3 to create
an overhead or handout for further investigation of optimization problems.
Caution students that they should always check their answers to “real-world”
problems to see if they make sense. Use the topics For Discussion to supplement
the worksheets.
28
Optimization/CALCULUS USING THE SHARP EL-9600
Using Blackline Master 5.3
The problems discussed above under Calculator Operations are shown on
Blackline Master 5.1 and 5.2. Blackline Master 5.3 provides two additional
activities. In Activity 2, students need to realize that part of the region that
is along the wall of the house requires no fence. They then need to minimize
the perimeter function f(w) = 2w + 675/w. Students should express this function
in terms of x, draw the graph in a viewing window such as 0 < x < 75, 0 < y < 200,
and press 2ndF
CALC and double touch 3 Minimum to find the minimum
width.
Some students may choose to work with the function f(l ) = 2(675/l ) + l.
The method is the same.
For Discussion
Discuss the terms relative (local) maxima or minima and absolute maximum
or minimum with students. Calculator-generated graphs only show a portion of
the graph of a function, so we can just verify that relative maxima or minima
exist. While calculus students have methods to justify that the y-coordinate of
a point is an absolute maximum or minimum, precalculus students can only
form an “educated” opinion by observing the behavior of the graph for
increasing values of x.
Additional Problems
1. The sum of two whole numbers is 52. What is the smallest possible
value of the sum of their squares?
2. Rancher Johnson has 250 meters of fencing. What is the largest
possible area of a rectangular corral that he can enclose with the
fencing? Allow 3 meters for a gate (on one of the longer sides of the
corral) that is not made from the fencing.
Optimization/CALCULUS USING THE SHARP EL-9600
29
3. The formula h = -16t2 + vot + ho gives the height h feet above the
ground that an object propelled vertically upward from an initial
height ho feet with an initial velocity of vo feet per second is t
seconds after it is propelled upward. (Assume air resistance is
negligible.)
a.
Find the maximum height attained by a toy rocket that is shot
vertically upward from ground level at an initial velocity of 30
feet per second.
b.
Find the time it takes a ball that is thrown vertically upward
with an initial velocity of 12 feet per second from a cliff that is
200 feet above ground level to reach its maximum height.
4. A voltage, measured in volts, applied to a certain electronic circuit
for t seconds is given by the equation v(t) = 1.5 ecos(2t). What is
the maximum voltage during a 3 second time interval?
5. A rectangular bin, designed to hold 16 cubic feet of grain, is to be
constructed with a square base and no top. The cost for the base of
the bin is 20¢ per square foot and the cost for each of the sides is
12¢ per square foot. Find the dimensions that minimize
construction costs.
30
Optimization/CALCULUS USING THE SHARP EL-9600
Chapter sIX
SHADING AND CALCULATING AREAS
REPRESENTED BY AN INTEGRAL
Introducing the Topic
In this chapter, you and your students will learn to use the calculator’s
numerical integrate function to approximate the definite integral of a function
over a specified interval. Calculus students learn that the connection between
definite integrals and the geometric concept of area is this:
If f is a continuous function for a ≤ x ≤ b and f(x) ≥ 0 for a ≤ x ≤ b, the area of the
b
region between y = f(x) and the x-axis from x = a to x = b is given by f(x)dx.
a
∫
An example of a function and the region satisfying these conditions is shown in
the figure below:
Shading and Calculating Areas Represented by an Integral/CALCULUS USING THE SHARP EL-9600
31
Calculator Operations
Prepare for the investigations in this chapter by setting the calculator to radian
measure by pressing 2ndF
SET UP , touching B DRG, and double touching 2
Rad. Set the calculator to floating point decimal display by touching C FSE and
double touch 1 FloatPt. Set the calculator to rectangular graphing by touching
E COORD, and double touching 1 Rect. Press 2ndF
QUIT to exit the SET UP
menu. Let’s use the calculator to find an estimate of 1 2x dx.
0
∫
Access the numerical integrate function by pressing
+ –
× ÷
CL
MATH , touch
A CALC, and double touch 06 ∫ to see:
The blinking cursor is on the lower box asking for input of the lower limit of
by pressing 1
▼
integration. Enter 0. Press ▲ and input 1 for the upper limit of integration
. Next, press 2
X/θ/T/n to input the integrand. The
expression is incomplete and will result in an error message without the “dx”,
so press MATH and double touch 07 dx. Press ENTER to view:
32
Shading and Calculating Areas Represented by an Integral/CALCULUS USING THE SHARP EL-9600
You can interpret this result geometrically by graphing the function over the
indicated interval and shading the region whose area is represented by the
integral. Press Y=
CL to access and clear the Y1 prompt. Clear additional
prompts by pressing ENTER
2
CL . Enter f(x) in Y1 with the keystrokes
X/θ/T/n . Press WINDOW and enter Xmin= 0 and Xmax= 1. Draw the
graph by pressing ZOOM , touching A ZOOM, and double touching 1 Auto.
Next, to shade the region whose area is the value of the definite integral, press
2ndF
DRAW , touch G SHADE, and double touch 1 Set to access the shading
Since Y1= 2X is the function “on the top,” press
▼
screen:
to move to the upper
bound, and touch Y1 on the screen to place Y1 in the position. Since we are
only dealing with one function, leave the lower bound location empty. Press
GRAPH to view the shaded region:
Notice that the area of the region between f(x)= 2x and the x-axis from x = 0 to x = 1
is the area of the shaded triangle. This area equals 1/2 base • height = 1/2 (1)(2) = 1.
Shading and Calculating Areas Represented by an Integral/CALCULUS USING THE SHARP EL-9600
33
Therefore, the calculator’s approximation to
∫0 2x dx gives, in this instance,
1
the exact value of the area. You will generally find the numerical approximation
given by the calculator is fairly accurate for most functions you use in a
beginning calculus course.
You can also use these ideas to find the area between two curves.
If f and g are continuous functions for a ≤ x ≤ b andf(x) ≥ g(x) for a ≤ x ≤ b, the
area of the region between y = f(x), y = g(x), and the vertical lines x = b and x = b
b
is given by f(x) – g(x) dx.
a
∫
An example of two functions and a region satisfying these conditions is shown in
the figure below:
The area of the shaded region equals
The area also equals
∫a f(x) – g(x) dx.
b
∫a f(x)dx – ∫a g(x)dx.
b
b
Students can choose either form of entry, but instructions in this chapter are
given for the single integral.
Let’s calculate and draw a graph of the area of the region between f(x) =
5x - x2 + 12 and g(x) = ex + 5. First, construct a graph to verify that the conditions
for using an integral to calculate the area in this situation are satisfied. Press
2ndF
DRAW , touch G SHADE, and double touch 2 INITIAL. You should do
this before beginning each new problem. Return to and clear the Y prompts by
pressing Y=
CL . Clear additional prompts if necessary. Input f(x) in Y1 with
the keystrokes 5
34
X/θ/T/n
–
X/θ/T/n
x2
+
1
2
ENTER .
Shading and Calculating Areas Represented by an Integral/CALCULUS USING THE SHARP EL-9600
ex
X/θ/T/n
▼
Input g(x) in Y2 with the keystrokes 2ndF
+
5 .
A little experimenting leads to a viewing window such as the one obtained
with -6.3 ≤ x ≤ 6.3, -10 ≤ y ≤ 30. Your viewing window should clearly show the
region between f(x) and g(x) and, if applicable, display the intersections of the
functions. Enter the viewing window and press GRAPH to view the graphs.
Shade the region between the two curves by pressing 2ndF
DRAW , touch
bottom,” pressing
▼
G SHADE, and double touch 1 SET. Touch Y2 since Y2 is the function “on the
, and touch Y1 since Y1 is the function “on the top.”
Press GRAPH to view the shaded region:
Next, find the limits of integration. Press 2ndF
CALC and double touch
2 Intsct. Do this twice to obtain the x-coordinates of the two points of
intersection. You should obtain X = -1.09375 and X = 2.583471.
+ –
× ÷
CL , press MATH , touch A CALC, double touch 06 ∫, enter -1.09375
in the lower box, press ▲ , enter 2.583471 in the upper box, press
2
function “on the top,” 5x – x + 12, press –
▼
Press
, enter the
( , and then enter the function “on
the bottom,” e x + 5. Close the parentheses by pressing ) and complete the
integral expression by pressing MATH and double touch 07 dx. Press ENTER
to obtain the area 20.343776.
Method of Teaching
Use Blackline Master 6.1 to create an overhead or handout for investigation
of shading a region and calculating an area represented by an integral.
Use Blackline Master 6.2 to create an overhead or handout for investigating
shading the region between two curves and calculating its area.
Shading and Calculating Areas Represented by an Integral/CALCULUS USING THE SHARP EL-9600
35
Use Blackline Master 6.3 to create an overhead or handout for investigating the
average value of a function. Use the topics For Discussion to supplement the
worksheets.
Using Blackline Master 6.3
In order to better explain the concept of average value, you can tell students
that the average value of a non-negative continuous function f(x) over the
interval a ≤ x ≤ b equals the height of the rectangle whose base is b – a and
whose area is the same as the area under the graph of f from a to b.
For Discussion
Explain to the students the concept of “negative” and “positive” area and
discuss how this might affect calculations and shading.
Additional Problems
In problem 1-3, use the calculator’s numerical integration function to obtain an
approximation to the specified area. Draw a graph and shade the region whose
area you have approximated:
1. the area bounded by f(x)= x3 – 2x and the x-axis for 2.4 < x < 5.8.
2. the area bounded by f(x)= e-x and the x-axis for -2 < x < 1.
3. the area between f(x)= sin x, g(x)= |cos x|, and the vertical line
x = π/4 and x =3π/4.
4. Find the average value of the function f(x) = 1/x over the interval
1 ≤ x ≤ 5.
3
5. Find x dx.
-1
∫
36
Shading and Calculating Areas Represented by an Integral/CALCULUS USING THE SHARP EL-9600
Chapter seven
PROGRAMS FOR RECTANGULAR
AND TRAPEZOIDAL APPROXIMATION
OF AREA
Introducing the Topic
In this chapter, you and your students will review how to program the Sharp
graphing calculator, execute a program, and use a program to calculate the area
under a curve using rectangular and trapezoidal approximations. Simpson’s
approximation was used in the last chapter.
The RECTAPP program approximates the definite integral of a function using
rectangles. When doing rectangle approximation, three estimates can be found.
These are found by creating rectangles from the left endpoints of the given
partition, right endpoints, and midpoint. The continuous function is entered
within the program. The lower limit of the definite integral is entered as ‘A,’ the
upper limit as ‘B,’ and the number of intervals in the partition as ‘N.’
The TRAPAPP program approximates the definite integral of a function using
trapezoids (trapezoid rule). Once again, the continuous function is entered
within the program, the lower limit of the definite integral is entered as ‘A,’ the
upper limit as ‘B,’ and the number of intervals in the partition as ‘N.’
Rectangular and Trapezoidal Approximation of Area/CALCULUS USING THE SHARP EL-9600
37
Calculator Operations
Turn the calculator on and set to radian mode by pressing 2ndF
touch B DRG, double touch 2 Rad. Press 2ndF
menu. Press 2ndF
SET UP ,
QUIT to exit the SET UP
PRGM to enter programming mode. Touch C NEW
to enter a new program, followed by ENTER . Name the new program
RECTAPP by pressing R
E
C
T
A
P
P
followed by ENTER .
You can now enter in the RECTAPP program. Remember to press ENTER
at the end of each line. If you make a mistake, use the calculator’s editing
features to make corrections. Enter the following program:
PROGRAM KEYSTROKES
(Input the number of intervals in the partition)
Input N
2ndF
PRGM
A
3
ALPHA
N
ENTER
(Input the lower limit of the definite integral)
Input A
2ndF
PRGM
Y1(A)⇒L
VARS
A
L
3
ALPHA
ENTER
A
1
A ENTER
( ALPHA A
)
STO
ALPHA
ENTER
(Input the upper limit of the definite integral)
Input B
Y1(B)⇒R
2ndF
PRGM
3
ALPHA B ENTER
VARS ENTER 1 ( ALPHA B ) STO
(Compute the width of intervals in the partition)
( ALPHA B – ALPHA A )
(B–A)/N⇒W
ALPHA
W
ENTER
A
+
÷
ALPHA
ALPHA
N
R ENTER
STO
(Compute first midpoint)
A+W/2⇒X
ALPHA
Y1(X)⇒M
VARS
ALPHA W ÷ 2 STO X/θ/T/n ENTER
1 ( X/θ/T/n ) STO ALPHA M
ENTER
ENTER
A+W⇒X
ALPHA
A
+
ALPHA
W
STO
X/θ/T/n
ENTER
(Calculate the approximations)
Label LOOP
2ndF
ALPHA
38
PRGM
B
1
2ndF
A-LOCK
L
O
O
P
ENTER
Rectangular and Trapezoidal Approximation of Area/CALCULUS USING THE SHARP EL-9600
Y1(X)+L⇒L
VARS
ENTER
ALPHA
Y1(X)+R⇒R
L
VARS
1
R
X+W/2⇒X
X/θ/T/n
Y1(X)+M⇒M
VARS
X/θ/T/n
)
+
ALPHA
L
STO
(
X/θ/T/n
)
+
ALPHA
R
STO
ENTER
ENTER
ALPHA
(
1
ENTER
ALPHA W ÷ 2 STO X/θ/T/n ENTER
ENTER 1 ( X/θ/T/n ) + ALPHA M STO
+
ALPHA
M
ENTER
X+W/2⇒X
X/θ/T/n
+
ALPHA
If X<BGoto
2ndF
PRGM
B
3
X/θ/T/n
LOOP
2ndF
PRGM
B
2
2ndF
ClrT
2ndF
PRGM
C
1
ENTER
Print “LEFT=”
2ndF
PRGM
A
1
2ndF
L
F
E
T
=
W
2ndF
÷
2ndF
PRGM
1
ALPHA
Print “MID=”
2ndF
PRGM
1
2ndF
D
2ndF
Print W*M
PRGM
2
2ndF
PRGM
1
ALPHA
Print “RIGHT=” 2ndF
PRGM
1
2ndF
G
H
T
=
2ndF
2ndF
PRGM
1
ALPHA
End
2ndF
PRGM
6
ENTER
Press 2ndF
MATH
A-LOCK
PRGM
2
×
W
PRGM
F
L
2
O
ALPHA
O
2ndF
P
B
ENTER
A-LOCK
ENTER
ALPHA
2
5
ENTER
2ndF
L
ENTER
A-LOCK
M
I
ENTER
×
W
PRGM
PRGM
Print W*R
X/θ/T/n
STO
PRGM
Print W*L
=
2
2
2
W
ALPHA
2ndF
M
ENTER
A-LOCK
R
I
ENTER
×
ALPHA
R
ENTER
QUIT to save the program and exit the editing mode.
Method of Teaching
Use Blackline Masters 7.1 through 7.4 to create overheads and worksheets for
entering and executing the RECTAPP and TRAPAPP programs. Go over in detail
how to enter the program and what the different program lines are doing. Have
the students enter the programs and execute them (correcting any errors).
Use the topics For Discussion to supplement the worksheets.
Rectangular and Trapezoidal Approximation of Area/CALCULUS USING THE SHARP EL-9600
39
Using Blackline Masters 7.1-7.4
Enter the function you wish to approximate, say f(x) = cos x, by pressing
Y=
CL
cos
X/θ/T/n . Press ENTER
CL to clear additional prompts.
To execute the RECTAPP program, press 2ndF
PRGM
A , highlight the
‘RECTAPP’ program, and press ENTER . You will be prompted to enter the
number of intervals desired ‘N,’ followed by lower limit ‘A’ and upper limit ‘B.’
Enter ‘N’ of 5 by pressing 5
ENTER , followed by ‘A’ of 0 and ‘B’ of 1.
You should then see the following display of left, midpoint and right rectangular
approximations for the area under the cosine function between 0 and 1 with 5
intervals in the partition.
You can repeat this process for a different number of partitions, different limits,
or another function. Press ENTER to execute the program over and over again.
▼
If you receive an error statement, press
or
▼
Press CL to exit the program.
to go to the line within the
program in which the error occurs. Compare your line with the correct one
above to find the error. Correct the error using the editing features of the
calculator and save the program by pressing 2ndF
QUIT . Try executing
the program again.
Repeat the process to enter the following TRAPAPP program.
40
Rectangular and Trapezoidal Approximation of Area/CALCULUS USING THE SHARP EL-9600
PROGRAM
KEYSTROKES
(Input the number of intervals in the partition)
Input N
2ndF
PRGM
A
3
ALPHA
N
ENTER
(Input the lower limit of the definite integral)
Input A
2ndF
PRGM
Y1(A)⇒L
VARS
A
ALPHA
3
ALPHA
ENTER
L
A
A ENTER
( ALPHA A
1
)
STO
ENTER
(Input the upper limit of the definite integral)
Input B
2ndF
PRGM
Y1(B)⇒R
VARS
ENTER
0⇒Z
R
ENTER
0
STO
3
ALPHA B ENTER
1 ( ALPHA B ) STO
ALPHA
Z
ENTER
(Compute the width of intervals in the partition)
( ALPHA B – ALPHA A )
(B–A)/N⇒W
A+W⇒X
ALPHA
W
ENTER
ALPHA
A
+
ALPHA
ALPHA
W
STO
÷
ALPHA
X/θ/T/n
N
STO
ENTER
(Calculate the approximation)
Label LOOP
2ndF
PRGM
ALPHA
2Y1(X)+Z⇒Z
2
1
2ndF
A-LOCK
ENTER
ALPHA
Z
1
If X<BGoto
2ndF
PRGM
3
X/θ/T/n
2ndF
PRGM
2
2ndF
ClrT
2ndF
PRGM
C
1
ENTER
Print “TRAP=”
2ndF
PRGM
A
1
2ndF
T
A
=
Print (W/2)
O
+
ALPHA
X/θ/T/n
ENTER
P
+
ALPHA
P
(L+Z+R)
2ndF PRGM
( ALPHA L
End
2ndF
PRGM
)
Z
ENTER
X/θ/T/n
R
O
X/θ/T/n
(
X+W⇒X
LOOP
L
ENTER
VARS
STO
B
W
STO
MATH
A-LOCK
PRGM
F
5
L
ALPHA
O
2
O
2ndF
P
B
ENTER
A-LOCK
2ndF PRGM 2 ENTER
1 ( ALPHA W ÷ 2 )
+
ALPHA
6
ENTER
Z
+
ALPHA
R
)
ENTER
Rectangular and Trapezoidal Approximation of Area/CALCULUS USING THE SHARP EL-9600
41
Execution of the TRAPAPP program for an ‘n’ of 5, an ‘a’ of 0, and a ‘b’ of 1
should result in the trapezoidal approximation of .838664209 for the area
under the cosine function between 0 and 1 with 5 intervals in the partition.
For Discussion
You and your students can discuss how you can change the functions,
the relationships between the approximations, and the changes in the
approximations when the number of intervals is increased for a fixed
function and interval.
42
Rectangular and Trapezoidal Approximation of Area/CALCULUS USING THE SHARP EL-9600
Chapter eight
HYPERBOLIC FUNCTIONS
Introducing the Topic
In this chapter, you and your students will learn how to graph and evaluate
hyperbolic functions on the Sharp EL-9600. The hyperbolic functions are
defined the same as trigonometric functions, except the hyperbolic functions
use a hyperbola instead of a circle. In trigonometry, the coordinates of any
point on the unit circle can be defined (x, y) = (cos t, sin t) where t is the
measure of the arc from (x, y) to (1, 0). In radians, the central angle = t and
twice the area of the shaded region equals t.
Whereas, with hyperbolic functions the coordinates of any point on the unit
hyperbola can be defined as (x, y) = (cosh t, sinh t). Once again, t is the measure
of the arc from (x, y) to (1, 0) and t equals twice the area of the shaded region.
Hyperbolic Functions/CALCULUS USING THE SHARP EL-9600
43
The four additional hyperbolic functions are defined in terms of hyperbolic sine
and hyperbolic cosine. The six hyperbolic functions are expressed as y = sinh x,
y = cosh x, y = tanh x, y = csch x, y = sech x, and y = coth x.
Calculator Operations
Turn the calculator on and press Y=
Press ENTER
CL to access and clear the Y1 prompt.
CL to remove additional expressions. The calculator should
be setup in radian measurement with rectangular coordinates. To complete
this setup, press 2ndF
SET UP touch B DRG, double touch 2 Rad, touch
E COORD, and double touch 1 Rect. Press 2nd
QUIT to exit the SET UP
menu and return to the Y prompts.
To enter the hyperbolic sine function ( y = sinh x ) for Y1, press MATH ,
➧
touch A CALC, touch
on the screen, double touch 15 sinh, and press
X/θ/T/n . Enter the viewing window by pressing ZOOM , touching F HYP,
and double touching 1 sinh x.
44
Hyperbolic Functions/CALCULUS USING THE SHARP EL-9600
The remaining five hyperbolic functions can be graphed in a similar manner.
Method of Teaching
Use the Blackline Master 8.1 to create an overhead for graphing the hyperbolic
sine function. Go over in detail how hyperbolic functions are derived and how
to graph them. Follow the hyperbolic sine function demonstration with the
hyperbolic cosine demonstration discussed in Using Blackline Master 8.1 and
appearing on Blackline Master 8.1.
Next, use the Blackline Masters 8.2 and 8.3 to create worksheets for the
students. Have the students graph the remaining four hyperbolic functions.
Use the topics For Discussion to supplement the worksheets.
Using Blackline Master 8.1
The problem discussed above under Calculator Operations is presented on
Blackline Master 8.1. The following problem also appears on the Blackline
Master. Graph the hyperbolic cosine function by pressing Y=
CL to remove
the hyperbolic sine function.
To enter the hyperbolic cosine function ( y = cosh x ) for Y1, press MATH ,
➧
on the screen, double touch 16 cosh x, press X/θ/T/n . Use the same
viewing window as before (-6.5 < x < 6.5 by -10 < y < 10) and press GRAPH to
touch
view the graph.
Hyperbolic Functions/CALCULUS USING THE SHARP EL-9600
45
For Discussion
You and your students can discuss how the hyperbolic functions differ from the
trigonometric functions and why. Where do the hyperbolic functions increase
and decrease? What are their domain and ranges? Does periodicity exist?
Why or why not?
EVALUATION OF HYPERBOLIC FUNCTIONS
Evaluating hyperbolic functions is easy on the Sharp EL-9600 graphing
calculator. First, access the home or computational screen by pressing
+ –
× ÷
CL .
Second, enter the expression to be evaluated, for example sinh 1. Enter sinh 1
pressing 1
46
➧
by pressing MATH , touching
on the screen, double touching 15 sinh, and
ENTER .
Hyperbolic Functions/CALCULUS USING THE SHARP EL-9600
Chapter nine
SEQUENCES AND SERIES
Introducing the Topic
In this chapter, you and your students will learn how to create a table for and
graph of a sequence. You will also examine recursive sequences, including the
Fibonnoci sequence. In addition, you will learn how to calculate the partial sum
of a series.
Calculator Operations
Turn the calculator on and set the calculator to sequence mode by pressing
2ndF
SET UP , touching E COORD, and double touching 4 Seq. Press
2ndF
QUIT
Y= to access the sequence prompts. Clear any sequences by
pressing CL .
Sequences and Series/CALCULUS USING THE SHARP EL-9600
47
Enter the sequence generator an = n2 – n for u(n) by pressing X/θ/T/n
x2 – X/θ/T/n ENTER . Enter n1 = 1 for u(nMin) by pressing 1
ENTER .
View a table of sequence values by pressing TABLE .
Graph the sequence by first setting the format to time and dot modes. Do
this by pressing 2ndF
FORMAT , touching G TYPE, double touching 2 Time,
touching E STYLE1, and double touching 2 Dot. Press 2ndF
QUIT to exit
the FORMAT menu. Enter the viewing window by pressing WINDOW
double touching 3, double touching 5 -1<X<10, and double touching
2 -10<Y<100.
48
Sequences and Series/CALCULUS USING THE SHARP EL-9600
EZ ,
Enter the recursive sequence generator an = an – 1 + 2n for u(n) by pressing
Y= CL 2ndF u ( X/θ/T/n – 1 ) + 2 X/θ/T/n ENTER .
Enter a1 = 1 by pressing 1
ENTER
.
View a table of sequence values by pressing TABLE .
Graph the sequence in the same window as above by pressing GRAPH .
Sequences and Series/CALCULUS USING THE SHARP EL-9600
49
Method of Teaching
Use Blackline Master 9.1 to create an overhead or worksheet for entering a
sequence, generating a table for the sequence, and graphing the sequence.
Use Blackline Master 9.2 to create an overhead or worksheet for entering a
recursive sequence, generating a table, and graphing. Use Blackline Master 9.3
to create a worksheet on entering the Fibonocci sequence, making a table
for it, and graphing it. Use the topics For Discussion to supplement the
worksheets.
Using Blackline Master 9.3
The Fibonnoci sequence is the sequence where the previous two terms are
added together to form the next term. The first two terms of the sequence are
a1 = 1 and a2 = 1. Enter the Fibonnoci sequence generator an = an – 1 + an – 2
for u(n) by pressing Y= CL 2ndF u ( X/θ/T/n – 1 ) + 2ndF
u
(
2ndF
X/θ/T/n
{ 1 ,
–
1
2
)
ENTER . Enter a1 = 1 and a2 = 1 by pressing
2ndF } ENTER .
Notice that upon entry, the comma disappears from the sequence screen.
View a table of sequence values by pressing TABLE .
50
Sequences and Series/CALCULUS USING THE SHARP EL-9600
Graph the sequence in the same window as before by pressing GRAPH .
Find the partial sum of a series, ∑ 1/X, for the first 10 terms. Return the SET UP
to rectangular mode by pressing 2ndF
double touching 1 Rect. Press 2ndF
SET UP , touching E COORD, and
QUIT to exit the SET UP menu. First
find the sequence of the first 10 terms of series ∑ 1/X by first pressing
CL
+ –
× ÷
2ndF
LIST , touching A OPE, and double touching 5 seq( . Enter the
generator 1/X by pressing 1 ÷ X/θ/T/n , . Enter the lower and upper
pressing ENTER . Press
▼
bounds for the sequence by pressing 1
,
1
0
) . Find the sequence by
to see more of the sequence.
Sequences and Series/CALCULUS USING THE SHARP EL-9600
51
touching 6 cumul, and pressing 2ndF
ANS
LIST , double
ENTER . Press
▼
Find the partial sum of the first 10 terms by pressing 2ndF
to move
right in the sequence of partial sums until the last term is seen. This is the
partial sum of the first 10 terms.
For Discussion
You and your students can discuss how to store a sequence into a list, and
perform other calculations of the list.
52
Sequences and Series/CALCULUS USING THE SHARP EL-9600
Chapter ten
GRAPHING PARAMETRIC
AND POLAR EQUATIONS
Introducing the Topic
In this chapter, you and your students will learn how to graph polar (R = f(θ))
and parametric equations on the Sharp EL-9600. The ordered pairs of polar
functions are defined as a ‘R’ (radius, distance from origin) in terms of an angle
measurement in radians (measured from the initial ray beginning at the origin
and passing through (1,0)). Whereas, the ordered pairs of a parametric function
are defined as ‘X’ and ‘Y,’ but both ‘X’ and ‘Y’ are in terms of another variable ‘T’
(usually time).
Calculator Operations
Turn the calculator on and press 2ndF
SET UP , touch E COORD, and
double touch 3 Polar to change to polar mode. While in the SET UP menu,
the calculator should be setup in radian mode. To complete this setup, touch
B DRG, and double touch 2 Rad.
Graphing Parametric and Polar Equations/CALCULUS USING THE SHARP EL-9600
53
Press 2ndF
QUIT to exit the SET UP menu. Make sure calculator is in
connected mode by pressing 2ndF
FORMAT , touching E STYLE1, and
double touching 1 Connect. While in the FORMAT menuset the calculator to
display polar coordinates when tracing by touching B CURSOR and double
touching 2 PolarCoord. Also, set the calculator to display the expression
during tracing by touching C EXPRESS and double touching 1 ON.
Press 2ndF
QUIT to exit the FORMAT menu. Press Y= to access the R1
prompt. Press CL to remove an old R1 expression. Press ENTER
CL to
clear additional R prompts.
Enter the polar function r = 2(1 – cos θ) for R1, by pressing 2 ( 1 –
cos X/θ/T/n ) . Notice, when in polar mode the X/θ/T/n key provides a
θ for equation entry.
54
Graphing Parametric and Polar Equations/CALCULUS USING THE SHARP EL-9600
Now, graph the polar function in the Decimal viewing window by pressing
➧
ZOOM , touching A ZOOM, touching
on the screen, and touching 7 Dec.
This particular shape of curve is called a cardoid. Trace the curve by pressing
TRACE . Notice the expression is displayed at the top of the screen.
Graphing Parametric and Polar Equations/CALCULUS USING THE SHARP EL-9600
55
Method of Teaching
Use Blackline Masters 10.1 and 10.2 to create overheads for graphing polar
and parametric equations. Go over in detail how to change modes, adjust the
format, and graph equations. Use Blackline Masters 10.3 and 10.4 to create
a worksheets for the students. Have the students graph the four equations
(2 polar, 2 parametric). Use the topics For Discussion to supplement the
▼
and
▼
worksheets. Engage the trace feature by pressing TRACE , followed by
. Tracing will assist you in discussing the topics.
Using Blackline Master 10.2
The problem discussed above under Calculator Operations is presented on
Blackline Master 10.1. The following discussion appears on Blackline Master
10.2. Turn the calculator on and press 2ndF
SET UP , touch E COORD,
and double touch 2 Param to change to parametric mode. Press 2ndF
QUIT to exit the SET UP menu. Make sure calculator is set to display
rectangular coordinates when tracing by pressing 2ndF
FORMAT , touching
B CURSOR and double touching 1 RectCoord. Press 2ndF
QUIT to exit the
FORMAT menu.
To enter the parametric function X1T = 2(cos T)3, Y1T = 2(sin T)3, press
Y= CL 2 ( cos X/θ/T/n ) ab 3 ENTER CL 2 ( sin X/θ/T/n
) ab 3 ENTER . Notice, when in parametric mode the X/θ/T/n key
provides a T for equation entry.
56
Graphing Parametric and Polar Equations/CALCULUS USING THE SHARP EL-9600
Now, graph the parametric function in the Decimal viewing window by pressing
➧
ZOOM , touching A ZOOM, touching
on the screen, and touching 7 Dec. Press
TRACE and notice the expression and T values now appear on the range screen.
For Discussion
You and your students can use the calculator to discuss how functions are
generated from t = 0 or θ = 0 through their maximum (use the trace feature).
Additional Problems
Graph the following equations:
1. R1 = 2sin (3θ) (three leaved rose or propeller)
Decimal viewing window.
2. R1 = 3 sin θ (circle)
Decimal viewing window.
3. X1T = sin T
Y1T = cos (2T)
Default viewing window.
4. X1T = 3T2
Y1T = T3 – 3T
- √ 3 < t < √ 3, 0 < x < 12.6, -3.1 < y < 3.1
Graphing Parametric and Polar Equations/CALCULUS USING THE SHARP EL-9600
57
CONTENTS OF
REPRODUCIBLE BLACKLINE MASTERS
Use these reproducible Blackline Masters to create handouts, overhead
transparencies, and activity worksheets.
EVALUATING LIMITS
BLACKLINE MASTERS 1.1 - 1.3
60 - 62
DERIVATIVES
BLACKLINE MASTERS 2.1 - 2.3
63 - 65
TANGENT LINES
BLACKLINE MASTERS 3.1 - 3.3
66 - 68
GRAPHS OF DERIVATIVES
BLACKLINE MASTERS 4.1 - 4.4
69 - 72
OPTIMIZATION
BLACKLINE MASTERS 5.1 - 5.3
73 - 75
SHADING AND CALCULATING AREAS
REPRESENTED BY AN INTEGRAL
BLACKLINE MASTERS 6.1 - 6.3
76 - 78
PROGRAMS FOR RECTANGULAR AND TRAPEZOIDAL
APPROXIMATION OF AREA
BLACKLINE MASTERS 7.1 - 7.4
79 - 82
HYPERBOLIC FUNCTIONS
BLACKLINE MASTERS 8.1 - 8.3
83 - 85
SEQUENCES
BLACKLINE MASTERS 9.1 - 9.3
86 - 88
GRAPHING PARAMETRIC AND POLAR EQUATIONS
BLACKLINE MASTERS 10.1 - 10.4
KEYPAD FOR THE SHARP EL-9600
89 - 92
93
Blackline Masters/CALCULUS USING THE SHARP EL-9600
59
NAME _____________________________________________________ CLASS __________ DATE __________
1.1
EVALUATING LIMITS GRAPHICALLY
1. Set the calculator to floating point decimal display by pressing 2ndF
SET UP , touching C FSE, and double touching 1 Float Pt. Set the calculator
to rectangular coordinates by touching E COORD and double touching
1 Rect. Press 2ndF
QUIT to exit the set up menu.
(2x + 2)
2. Consider the function f(x) = (x 2 – 1) . Press Y= CL to access and clear the
Y1 prompt. Press ENTER CL to clear the remaining prompts.
(2x + 2)
➧
Y1= (x 2 – 1) with the keystrokes a/b 2 X/θ/T/n + 2
x2 – 1 , and then press ZOOM , touch A ZOOM, touch
▼
3. Construct a graph of f(x) in the Decimal viewing window by first entering
X/θ/T/n
on the
screen, and touch 7 Dec to see the graph.
(2x + 2)
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
4. Notice that even though (x 2 – 1) is not defined at x = -1 (as evidenced by the
hole in the graph at that point), the functional values appear to be getting
closer and closer to -1. A careful observation of the graph leads to the
following estimates:
lim f(x) = 0,
x → -∞
lim f(x) = -1,
x → -1
lim f(x) does not exist,
x→1
and lim f(x) = 0.
x→∞
5. It also appears that the line y = 0 is a horizontal asymptote and the line x = 1
is a vertical asymptote for this function.
60
Blackline Masters/CALCULUS USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
1.2
EVALUATING LIMITS NUMERICALLY
1. Tables of functional values sometimes provide more detailed information
than a graph when investigating limits. The Sharp EL-9600 has a TABLE
feature to assist you in constructing a numerical table of values.
Press TABLE to access the TABLE feature.
2. Notice the table provides the x values and their corresponding y-values
TBLSET . You can change the table start value and the table step value.
Verify the following values using the the TABLE feature.
x gets smaller and smaller→
x
-10
y
-.18182
-50
-.03922
-100
-.01980
-250
-500
-1000
-10,000
.00797
-.00399
-.00200
-.00020
y = f(x) appears to get closer and closer to 0
This provides evidence that lim f(x) = 0.
x→∞
x approaches -1 from the left
x
-1.05
1.01
y
-.9756
-.99502
x approaches -1 from the right
-1.001
-.9995
-1.0001
-.9999
y = f(x) gets closer and closer to -1 from above
-.9999
-.999
-1.0001
-1.0005
-.99
-1.0051
-.90
-1.0526
y = f(x) gets closer and closer to -1 from below
This provides evidence that lim f(x) = -1.
x → -1
Blackline Masters/CALCULUS USING THE SHARP EL-9600
61
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
according to Y1. You can change the table settings by pressing 2ndF
NAME _____________________________________________________ CLASS __________ DATE __________
1.3
EVALUATING LIMITS
(x + 1)
in a decimal viewing window.
(x + 1)
Trace the graph for values of x less than x = -1. Describe the values of
f(x) for x < -1.
1. Consider the graph of f(x) =
________________________________________________
Describe the values of y = f(x) for x > -1.
________________________________________________
Do you feel this behavior of the function continues for all values of x;
that is, as x gets smaller and smaller and as x gets larger and larger?
________________________________________________
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
(x2 + 2x – 3)
2. Use the TABLE feature with f(x) =
(x2 – 9) to estimate the following limits.
lim f(x) = __________
x → -∞
lim f(x) = __________
x → -3
lim f(x) = __________
x→3
lim f(x) = __________
x→∞
3. The population of fish in a certain small lake at time t months after it is
10,000
stocked is given by p(t) =
. Due to a limited supply of food
-t/4
(1 + 15e )
and oxygen, the growth of the population is limited. Find the size of the
population 20 days after the lake is stocked and the limiting value of the
size of the population of fish.
62
Blackline Masters/CALCULUS USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
2.1
DERIVATIVES USING THE LIMIT DEFINITION
Activity 1
1. Set the calculator to radian measure by pressing 2ndF
SET UP , touching
B DRG, and double touching 2 Rad. Set the calculator to floating point
decimal display by touching C FSE and double touching 1 Float Pt. Set the
calculator to rectangular coordinates by touching E COORD and double
touching 1 Rect. Press 2ndF
QUIT to exit the SET UP menu.
2. Consider the function f(x)2= x 2 at x = 2 where f ’(2) =
(x – 4)
[f(x) – f(2)]
lim
= lim
.
x→2 (x – 2)
x→2 (x – 2)
(x 2 – 4)
and use the TABLE feature to verify the following table.
(x – 2)
x
1.9 1.99
1.999 1.9999
2.0001 2.001
2.01
2.1
y
3.9 3.99
3.999 3.9999
4.0001 4.001
4.01
4.1
Activity 2
1. Consider the function f(x) =2 x 3 at x = 1 where f ’(1) equals
[f(x) – f(1)]
(2x3 – 2)
lim
= lim
.
(x – 1)
x→1
x→1 (x – 1)
2. Enter Y1 =
(2x 3 – 2)
and use the TABLE feature to build the following table.
(x – 1)
x
.999
.9
.99
.9999
1.0001 1.001 1.01 1.1
y
(2x3 – 2)
It appears that lim (x – 1) = f ’(1)=
x→1
.
Activity 3
1. Consider the function f(x) =2 x 3 at x = -3. f ’(-3)=_____
Blackline Masters/CALCULUS USING THE SHARP EL-9600
63
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
3. Enter Y1=
NAME _____________________________________________________ CLASS __________ DATE __________
2.2
DERIVATIVES USING THE d/dx FUNCTION
Activity 1
1. The calculator has a built-in function denoted by d/dx that uses numerical
methods to estimate the derivative of a function at a given value. The entry
form of the derivative function is d/dx (f(x), a).
+ –
÷
2. Estimate f ’(2) for f(x) = x2 using the derivative function. Press ×
touch A CALC, double touch 05 d/dx(, press X/θ/T/n x2 , 2
2
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
input: d/dx (X , 2). Press ENTER to compute.
Activity 2
1. Estimate f ’(1) for f(x) = 2x 3 using the derivative function.
2. f ’(1) = __________
Activity 3
1. Investigate f ’(0) for f(x) = |x| using the derivative function.
Complete the following table.
x
-.1
.01
-.001
-.0001
.0001 .001
.01
.1
f ’(x)
What is your conclusion regarding f ’(0)?
_____________________________________________________
64
Blackline Masters/CALCULUS USING THE SHARP EL-9600
MATH ,
) to
NAME _____________________________________________________ CLASS __________ DATE __________
2.3
DERIVATIVES USING THE DERIVATIVES TRACE
Activity 1
1. Activate the derivative trace by pressing 2ndF
double touch 1 ON. Press 2ndF
2. Press Y=
FORMAT , touch D Y', and
QUIT to exit the FORMAT menu.
CL and enter the function for Y1 by pressing X/θ/T/n
EZ , touching 5, touching
➧
3. Draw the graph by pressing WINDOW
x2 .
on the
4. Press TRACE to activate the trace and press
▼
screen, double touching 7 -5<x<5, and touching 4 -10<Y<10.
to the values for the derivative.
Activity 2
1. Enter f(x) for Y1 and construct the graph by pressing ZOOM , touching
on the screen, and touching 7 Dec. Press ZOOM ,
touching A ZOOM, and double touching 1 Auto.
2. Complete the following tables of values. Use the pattern you view in the
table of values to determine a formula for f ’(x).
a. f(x) = e2x
x
-3
y
y’
b. f(x) = 2.5 x 2
-2
x
-3
-2
-1
-1
0
0
1
1
2
2
3
3
y
y’
Remember to turn off the derivative trace.
Blackline Masters/CALCULUS USING THE SHARP EL-9600
65
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
➧
A ZOOM, touching
NAME _____________________________________________________ CLASS __________ DATE __________
3.1
TANGENT LINES
Enter and execute a program for finding a tangent line to a curve
at a given point.
1. Turn the calculator on and press 2ndF
PRGM to enter the
programming menu.
2. Touch C NEW and press ENTER to open a new program.
3. Name the new program TANGENT by pressing T
A
N
G
E
N
T
ENTER .
4. Press ENTER at the end of each line. If you make a mistake, use the
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
calculator's editing features to correct the error. Enter the following program:
PROGRAM
KEYSTROKES
Input X
d (Y1,X)⇒M
dx
2ndF
MATH
STO
Y1(X)⇒Y
0
ALPHA
A
A
3
5
VARS
M
X/θ/T/n
A
ENTER
ENTER
A
,
1
X/θ/T/n
)
ENTER
ENTER
1
(
X/θ/T/n
)
STO
ALPHA
ENTER
(–)
B
66
A
VARS
Y
-M•X+Y⇒B
PRGM
ALPHA
M
×
X/θ/T/n
+
ALPHA
Y
STO
ALPHA
ENTER
ClrT
2ndF
PRGM
C
1
ENTER
Print
2ndF
PRGM
A
1
2ndF
"TANGENT
ALPHA
LINE=
L
Print
2ndF
"Y=MX+B
=
I
T
N
A
E
=
PRGM
ALPHA
N
M
G
E
PRGM
2
T
SPACE
N
2ndF
ENTER
1
2ndF
X/θ/T/n
PRGM
+
2
ALPHA
Blackline Masters/CALCULUS USING THE SHARP EL-9600
ALPHA
B
Y
ENTER
ALPHA
NAME _____________________________________________________ CLASS __________ DATE __________
3.2
TANGENT LINES
Continue to enter the TANGENT program.
2ndF
=
PRGM
1
2ndF
2ndF
PRGM
1
ALPHA
Print "B=
2ndF
PRGM
1
2ndF
2ndF
PRGM
1
ALPHA
End
2ndF
PRGM
6
ENTER
M
ALPHA
2
ALPHA
B
ALPHA
B
ENTER
QUIT to save the program and exit the editing mode.
2
+
1
CL
X/θ/T/n
ab
3
–
ENTER .
7. Execute the TANGENT program by pressing 2ndF
PRGM , touching
A EXEC, and double touching TANGENT.
8. Enter an X of 1 by pressing 1
ENTER . You should then see the equation
for the tangent line to the curve at x = 1.
9. You can repeat this process for other x values. Press ENTER to execute the
▼
10. If you receive an error statement, press
or
▼
program over and over again.
to go to the line within the
program in which the error occurs. Compare your line with the correct one
above to find the error. Correct the error using the editing features
of the calculator and save the program by pressing 2ndF
QUIT .
Try executing the program again.
Blackline Masters/CALCULUS USING THE SHARP EL-9600
67
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
x
M
ENTER
PRGM
6. Enter f(x) =x 3 – x 2+1 for Y1 by pressing Y=
X/θ/T/n
ALPHA
ENTER
Print B
5. Press 2ndF
2
ENTER
Print M
=
PRGM
▼
Print "M=
NAME _____________________________________________________ CLASS __________ DATE __________
3.3
TANGENT LINES
Draw the tangent line to a function at a given point.
1. To draw the tangent line on a displayed graph, you must first enter the
CL
X/θ/T/n
ab
3
▼
graph for Y1. Enter the function f(x) = x3 – x2 + 1 for Y1 by pressing Y=
–
X/θ/T/n
x2
+
1
ENTER .
3. Draw the tangent line at x = 1 by pressing 2ndF
DRAW , touching
A DRAW, double touching 5 T_line(, move the tracer right to x =1 by
pressing
68
▼
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
the screen, and touching 7 Dec.
repeatedly, and then press ENTER .
Blackline Masters/CALCULUS USING THE SHARP EL-9600
➧
2. Graph the function by pressing ZOOM , touching A ZOOM, touching
on
NAME _____________________________________________________ CLASS __________ DATE __________
4.1
DERIVATIVES
1. Set the graphing calculator to radian measure, floating point decimal display,
and rectangular graphing by pressing 2ndF
SET UP , touching B DRG,
double touching 2 Rad, touching C FSE, double touch 1 FloatPt, touching
E COORD, and double touching 1 Rect. Press 2ndF
QUIT to exit the
SET UP menu.
2. Choose sequential graphing and connected mode by pressing 2ndF
FORMAT , touching F STYLE2, double touching 1 Sequen, touching
E STYLE1 and double touching 1 Connect. Press 2ndF
QUIT to exit
the FORMAT menu.
3. Consider f(x) = 2x 3 – 7x 2 – 70x + 75. Press Y=
pressing 2
7
X/θ/T/n
ab
3
–
7
X/θ/T/n
x2
–
7
0
X/θ/T/n
5 .
4. Graph the function by pressing WINDOW EZ , touching 5, double touching
5 -10 < X < 10, and double touching 1 -500 < Y < 500.
5. To determine the point at which the relative maximum occurs, press
TRACE
2ndF
CALC , and double touch 4 Maximum.
6. Find the point at which the relative minimum occurs by pressing 2ndF
CALC , and double touch 3 Minimum.
7. Combining this information with a view of the graph, we see that f(x) is
increasing from -∞ to -2.4427 and from 4.776 to ∞ while f(x) is decreasing
for x between -2.4427 and 4.776.
Blackline Masters/CALCULUS USING THE SHARP EL-9600
69
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
+
CL to clear additional Y prompts. Enter f(x) for Y1 by
▼
Press ENTER
CL to access and clear Y1.
NAME _____________________________________________________ CLASS __________ DATE __________
4.2
GRAPHS OF DERIVATIVES
1. Graph f ’(x) by pressing Y=
d
ENTER and input dx (Y1) in the Y2 location
with the keystrokes MATH , touch A CALC, double touch 05 d/dx( , press
VARS
ENTER , touch A XY, double touch 1 Y1, and press )
ENTER .
Press GRAPH to obtain the graphs of f(x) and f ’(x).
2. We now want to find the two x-intercepts of f ’(x). Press TRACE
▼
place the tracer on the graph of the derivative. Then, press 2ndF
and double touch 5 X_Incpt. Press 2ndF
to
CALC
CALC and double touch
5 X_Incpt again to obtain the other x-intercept.
3. Comparing these values to the x-coordinates of the points at which the
maxima and minima of f(x) occur, we see they are almost identical.
4. Where is f ’(x) positive? Notice this is where the graph of f(x) is increasing.
5. Next, find the minimum point of f ’(x) by first making sure the trace cursor is
on the graph of the derivative, pressing 2ndF
CALC , and double touching
3 Minimum.
6. Look at f(x) and observe that this appears to be the point at which the
function “bends a different way.”
7. Find the point of inflection directly by moving the cursor to the original
function and pressing 2ndF
CALC , touching
➧
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
Where is f ’(x) negative? Notice this is where f(x) is decreasing.
double touching 7 Inflec.
70
Blackline Masters/CALCULUS USING THE SHARP EL-9600
on the screen, and
NAME _____________________________________________________ CLASS __________ DATE __________
4.3
GRAPHS OF DERIVATIVES
1. Graph the second derivative by pressing Y=
ENTER
ENTER , and input
d/dx (Y2) in the Y3 location with the keystrokes MATH , touch A CALC,
double touch 05 d/dx( , press VARS
2 Y2, and press )
ENTER , touch A XY, double touch
ENTER .
2. Press GRAPH to obtain the graphs of f(x), f ’(x), and f ’’(x). Where is f ’’(x).
zero? After pressing TRACE
derivative, press 2ndF
▼
▼ to place the tracer on the second
CALC and double touch 5 X_Incpt to find that
f ’’(x). = 0 at X= 1.167.
3. Find the y-value of this point of inflection by pressing ▲
cursor to the original function. Press 2ndF
▲ to move the
CALC , double touch 1 Value,
and enter 1.167.
f ’’(x) are summarized in the tables below.
Blackline Masters/CALCULUS USING THE SHARP EL-9600
71
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
4. The connections we have discovered between the graphs of f(x), f ’(x), and
NAME _____________________________________________________ CLASS __________ DATE __________
4.4
GRAPHS OF DERIVATIVES
In each activity below, press Y= and enter f(x) in Y1, d/dx (Y1)
in Y2, and d/dx (Y2) in Y3.
Activity 1
1. Choose the dot graphing mode by pressing 2ndF
E STYLE1, and double touch 2 Dot. Press 2ndF
➧
FORMAT menu. Press ZOOM , touch
FORMAT , touch
QUIT to exit the the
on the screen, and touch 7 Dec
to set the Decimal viewing window.
2. Use the graphs of f(x), f ’(x), and f ’’(x) to fill in the information in the tables
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
below for the function f(x)= |x+1|.
Activity 2
1. Continue with the dot graphing mode and Decimal viewing window. Use the
graphs of f(x), f ’(x), and f ’’(x) to fill in the information in the tables below for
the function f(x) = √(9 – x2).
72
Blackline Masters/CALCULUS USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
5.1
OPTIMIZATION
1. Set the calculator to radian measure by pressing 2ndF
SET UP , touching
B DRG, and double touching 2 Rad. Set the calculator to floating point
decimal display by touching C FSE and double touch 1 FloatPt. Set the
calculator to rectangular graphing by touching E COORD, and double
touching 1 Rect. Press 2ndF
QUIT to exit the SET UP menu.
2. A new product was introduced through a television advertisement appearing
during the Super Bowl. Suppose that the proportion of people that
purchased the product x days after the advertisement appeared is given by
f(x) =
(5.3x)
. When did maximum sales occur and what proportion of
(x2 + 15)
people purchased the product at that time?
3. Press Y=
CL to clear the Y1 prompt. Press ENTER
CL to clear
×
3
X/θ/T/n
X/θ/T/n
x2
+
1
5 .
4. Let’s examine the graph for the first 25 days after the advertisement
appeared. Press WINDOW , enter Xmin = 0, Xmax = 25, Xscl = 5, Ymin = 0,
Ymax = 1, Yscl = 1.
5. Press GRAPH to view the graph.
6. When did maximum sales occur and what proportion of people
purchased the product at that time? Press 2ndF
CALC and
double touch 4 Maximum.
7. Is it the absolute maximum? Press WINDOW and change Xmax to 100.
Press GRAPH to view the graph of the function. Repeat the procedure for
Xmax= 500.
8. The maximum sales occurred __________ days after the advertisement first
appeared, and the proportion of people that purchased the product at that
time is __________ .
Blackline Masters/CALCULUS USING THE SHARP EL-9600
73
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
5
▼
additional prompts. Enter f(x) in the Y1 location with the keystrokes a/b
NAME _____________________________________________________ CLASS __________ DATE __________
5.2
OPTIMIZATION
1. A metal container with no top in the form of a right circular cylinder is being
designed to hold 183 in3 of liquid. If the material for the container costs 14¢
per square inch and the cost of welding the seams around the circular
bottom and up the side cost 5¢ per inch, find the radius of the container
with the smallest cost. What is the minimum cost?
2. The area of the container is equal to the area of lateral surface + area of
bottom (2πrh + πr 2). The length of welds is equal to h + 2πR. The cost of
the container is equal to .14(area of container) + .05(length of welds).
Therefore, cost is equal to .14(2πrh + πr 2) + .05(h + 2πr).
183
.
We know that 183 = πr 2h, so h =
(πr 2)
5
+
2ndF
1
•
a/b
π
9
2
4
•
1
5
X/θ/T/n
2ndF
π
+
•
1
X/θ/T/n
4
x
π
2ndF
2
▼
x
2
▼
a/b
▼
4. Press Y= and clear any previously-entered functions with CL . Enter the
simplified expression 51.24 + .14πx 2 + 9.152 + .1πx, in Y1 with the
(πx )
x
keystrokes:
▼
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
3. Substituting this in the cost equation, we find cost of the container is
.14(2πr( 1832 ) + πr 2) + .05( 1832 ) + 2πr).
(πr )
(πr )
+
X/θ/T/n
•
1
X/θ/T/n .
5. Set the viewing window to 0 < x < 10 and 0 < y < 50. Enter the viewing
window and press GRAPH to view the graph of the cost function.
6. To find the minimum cost, press 2ndF
CALC and double touch
3 Minimum. The radius of the container with the smallest cost is
__________ inches. The minimum cost of the container is $__________.
74
Blackline Masters/CALCULUS USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
5.3
OPTIMIZATION
In each activity, choose a viewing window appropriate for the
problem situation. Use the graph of the function to find the
optimal solution.
Activity 1
A rumor about a new dress code spreads through a school. The proportion of
students who have heard the rumor at time t days since the rumor started is
6.4t
given by p(t) = 2
.
(t + 12)
1. Because t is the number of days since the rumor started, let t ≥ ____.
Because p(t) is a proportion, let ____ ≤ p(t) ≤ ____.
2. Graph the proportion of students who heard the rumor.
Indicate your viewing window in the blanks provided:
Ymin = ____, Ymax = ____, Yscl = ____.
3. What is the largest proportion of students who heard the rumor? _____
4. How many days after the rumor started had the largest proportion
heard? _____
Activity 2
Lee is going to put a fence around part of the back yard. In order to have
room for a tool shed and the family dog, a rectangular region whose area is
approximately 675 square feet needs to be fenced. The back of the house is to
be used as one side of the fenced-in portion.
1. Sketch the region to be fenced, indicating what your variable(s) represent.
Write an equation representing the amount of fence Lee will buy.
_____________________________________
2. What dimensions will minimize the amount of fence Lee will buy?
_______________feet by_______________feet
Blackline Masters/CALCULUS USING THE SHARP EL-9600
75
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
Xmin = ____, Xmax = ____, Xscl = ____,
NAME _____________________________________________________ CLASS __________ DATE __________
6.1
SHADING AND CALCULATING AREAS
REPRESENTED BY AN INTEGRAL
1. Set the calculator to radian measure by pressing 2ndF
SET UP , touching
B DRG, and double touching 2 Rad. Set the calculator to floating point
decimal display by touching C FSE and double touch 1 FloatPt. Set the
calculator to rectangular graphing by touching E COORD, and double
touching 1 Rect. Press 2ndF
2. Find an estimate of
∫0 2x dx.
1
3. Integrate a function by pressing
double touch 06 ∫.
QUIT to exit the SET UP menu.
+ –
× ÷
MATH , touch A CALC, and
. Next, press 2
X/θ/T/n to input the integrand. Enter the “dx” by
pressing MATH and double touching 07 dx. Press ENTER to compute.
5. Shade the region by first pressing Y=
CL to access and clear the Y1
prompt. Clear additional prompts by pressing ENTER
6. Enter f(x) in Y1 with the keystrokes 2
CL .
X/θ/T/n . Press WINDOW and
enter Xmin = 0 and Xmax = 1. Draw the graph by pressing ZOOM , touching
A ZOOM, and double touching 1 Auto.
7. Shade the region by first pressing 2ndF
DRAW , touch G SHADE, and
double touch 1 Set to access the shading screen.
8. Since Y1= 2X is the function “on the top,” press
▼
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
▼
4. Enter 0 for the lower limit. Press ▲ , input 1 for the upper limit, and press
, and touch Y1 on the
screen. Since we are only dealing with one function, leave the lower bound
location empty. Press GRAPH to view the shaded region.
76
Blackline Masters/CALCULUS USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
6.2
SHADING AND CALCULATING AREAS
REPRESENTED BY AN INTEGRAL
1. Calculate and draw a graph of the area of the region between
f(x) = 5x – x 2 + 12 and g(x) = ex + 5.
2. First, press 2ndF
DRAW , touch G SHADE, and double touch 2 INITIAL.
Return to and clear the Y prompts by pressing Y=
CL . Clear additional
prompts if necessary.
X/θ/T/n
–
X/θ/T/n
2
ENTER . Input g(x) in Y2 with the keystrokes 2ndF
▼
3. Input f(x) in Y1 with the keystrokes 5
+
ex
x2
+
1
X/θ/T/n
5 .
4. Enter the viewing window -6.3 < x < 6.3 and -10 < y < 30. Your viewing
applicable, display the intersections of the functions. Press GRAPH to view
the graphs.
5. Shade the region between the two curves by pressing 2ndF
DRAW , touch
the bottom," pressing
▼
G SHADE, and double touch 1 SET. Touch Y2 since Y2 is the function "on
, and touch Y1 since Y1 is the function “on the top."
Press GRAPH to view the shaded region.
6. Next, find the limits of integration. Press TRACE
2ndF
CALC and double
touch 2 Intsct. Do this twice to obtain the x-coordinates of the two points of
intersection.
+ –
× ÷
CL , press MATH , touch A CALC, double touch 06 ∫, enter the
lower bound, press ▲ , enter the upper bound, press
▼
7. Press
, enter the function
2
“on the top,” 5x – x + 12, press – ( , enter the function “on the bottom,”
ex + 5, press ) , press MATH and double touch 07 dx. Press ENTER to
obtain the area.
Blackline Masters/CALCULUS USING THE SHARP EL-9600
77
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
window should clearly show the region between f(x) and g(x) and, if
NAME _____________________________________________________ CLASS __________ DATE __________
6.3
SHADING AND CALCULATING AREAS
REPRESENTED BY AN INTEGRAL
The average value of a continuous function y = f(x) over the
interval from x = a to x = b is given by ( 1 ) b f(x) dx.
(b–a) a
∫
Activity 1
1. Use the calculator’s numeric integration function to approximate
0
(x+2)2 dx. _______________
-4
∫
2. Graph the region whose area this integral represents. Shade the region.
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
Sketch the shaded region below.
3. Find the average value of f(x) = (x+2)2 over the interval -4 ≤ x ≤ 0. _________
Activity 2
The temperature at time t, measured in hours since midnight over
a 24-hour period on a certain day in June at Newark airport, is
approximated by the function f(t) = 80 + 10sin(.26t – 2.3) degrees
Fahrenheit.
1. Enter f(t) in Y1. Construct an autoscaled graph over the interval 0 ≤ t ≤ 24.
2. Find the minimum and maximum temperatures over a 24-hour period and
the approximate hour t which they occur.
minimum=______°F at______, maximum=______°F at______.
3. Find the average temperature over the 24-hour period.______°F
78
Blackline Masters/CALCULUS USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
7.1
PROGRAM FOR RECTANGULAR
APPROXIMATION OF AREA
Set up and enter a program:
1. Turn the calculator on and set to radian mode by pressing 2ndF
touch B DRG, double touch 2 Rad. Press 2ndF
SET UP ,
QUIT to exit the SET UP
menu.
2. Press 2ndF
PRGM to enter programming mode. Touch C NEW to enter a
new program, followed by ENTER . Name the new program RECTAPP by
pressing R
E
C
T
A
P
P followed by ENTER .
3. You can now enter in the RECTAPP program. Remember to press ENTER at
the end of each line. If you make a mistake, use the calculator's editing
PROGRAM
KEYSTROKES
Input N
2ndF
PRGM
A
3
Input A
2ndF
PRGM
3
ALPHA
Y1(A)⇒L
VARS
A
L
ENTER
2ndF
PRGM
Y1(B)⇒R
VARS
ENTER
A+W/2⇒X
A
N
A
ENTER
(
1
ENTER
ALPHA
A
)
STO
ALPHA
ENTER
Input B
(B–A)/N⇒W
ALPHA
R
ENTER
(
ALPHA
3
B
ALPHA
1
(
–
ALPHA
ALPHA
W
ENTER
ALPHA
A
+
ALPHA
B
ENTER
ALPHA
W
A
÷
B
)
STO
)
÷
ALPHA
2
STO
ALPHA
N
STO
X/θ/T/n
ENTER
Blackline Masters/CALCULUS USING THE SHARP EL-9600
79
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
features to make corrections. Enter the following program:
NAME _____________________________________________________ CLASS __________ DATE __________
7.2
PROGRAM FOR RECTANGULAR
APPROXIMATION OF AREA
Entering a program (cont.)
Y1(X)⇒M
VARS
M
ENTER
ALPHA
Label LOOP
2ndF
A
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
ALPHA
W
VARS
B
ENTER
L
VARS
1
1
ENTER
1
R
ENTER
X+W/2⇒X
X/θ/T/n
+
ALPHA
Y1(X)+M⇒M
VARS
ALPHA
2ndF
A-LOCK
O
O
P
+
ALPHA
L
STO
(
X/θ/T/n
)
+
ALPHA
R
STO
ENTER
W
(
1
X+W/2⇒X
X/θ/T/n
+
ALPHA
If X<BGoto
2ndF
PRGM
3
LOOP
MATH
÷
2
5
ALPHA
O
P
ENTER
W
÷
2
B
PRGM
C
1
ENTER
Print "LEFT="
2ndF
PRGM
A
1
2ndF
L
F
PRGM
1
+
ALPHA
STO
2ndF
2ndF
=
)
X/θ/T/n
ENTER
M
X/θ/T/n
STO
ENTER
X/θ/T/n
F
T
STO
X/θ/T/n
ClrT
2ndF
L
ENTER
)
ENTER
E
X/θ/T/n
X/θ/T/n
M
O
STO
(
ALPHA
Print W•L
STO
ENTER
ALPHA
L
)
ENTER
ALPHA
80
+
PRGM
ALPHA
Y1(X)+R⇒R
X/θ/T/n
ENTER
A+W⇒X
Y1(X)+L⇒L
(
1
2ndF
PRGM
PRGM
ALPHA
PRGM
W
2
×
Blackline Masters/CALCULUS USING THE SHARP EL-9600
2
2
2ndF
2ndF
A-LOCK
A-LOCK
ENTER
ALPHA
L
ENTER
NAME _____________________________________________________ CLASS __________ DATE __________
7.3
PROGRAMS FOR RECTANGULAR AND
TRAPEZOIDAL APPROXIMATION OF AREA
Entering a program (cont.)
Print W*M
2ndF
PRGM
M
D
I
=
1
2ndF
2ndF
PRGM
2ndF
PRGM
1
ALPHA
Print "RIGHT=" 2ndF
PRGM
1
2ndF
R
I
G
H
T
=
2ndF
PRGM
1
ALPHA
End
2ndF
PRGM
6
ENTER
2
2
×
PRGM
ALPHA
2
PRGM
W
2ndF
A-LOCK
ENTER
W
2ndF
Print W*R
Press 2ndF
PRGM
×
2ndF
2
M
ENTER
A-LOCK
ENTER
ALPHA
R
ENTER
QUIT to save the program and exit the editing mode.
Executing a program
Enter the function you wish to approximate, say f(x) = cos x, by pressing Y=
CL
cos
X/θ/T/n . Press ENTER
CL to clear additional prompts.
To execute the RECTAPP program, press 2ndF
PRGM
A , highlight the
'RECTAPP' program, and press ENTER . Enter 'N' of 5 by pressing 5
ENTER ,
followed by 'A' of 0 and 'B' of 1. You will see a display of left, midpoint and right
rectangular approximations for the area under the cosine function between 0
and 1 with 5 intervals in the partition.
TRAPAPP program.
PROGRAM
KEYSTROKES
Input N
2ndF
PRGM
A
3
ALPHA
N
ENTER
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81
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
Print "MID="
NAME _____________________________________________________ CLASS __________ DATE __________
7.4
TRAPEZOIDAL APPROXIMATION OF AREA
Input A
2ndF
PRGM
Y1(A)⇒L
VARS
A
L
3
ALPHA
ENTER
A
A
ENTER
(
1
ALPHA
)
A
STO
ALPHA
ENTER
Input B
2ndF
PRGM
3
Y1(B)⇒R
VARS
ENTER
ALPHA
B
ENTER
(
ALPHA
Z
ENTER
1
B
)
STO
)
÷
ALPHA
ALPHA
R
ENTER
0→Z
0
STO
ALPHA
(B–A)/N⇒W
(
ALPHA
ALPHA
W
ENTER
A+W⇒X
ALPHA
A
+
Label LOOP
2ndF
2Y1(X)+Z⇒Z
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
–
ALPHA
PRGM
ALPHA
2
ALPHA
B
1
VARS
ENTER
W
STO
2ndF
(
1
Z
ENTER
X+W⇒X
X/θ/T/n
+
ALPHA
If X<BGoto
2ndF
PRGM
3
LOOP
MATH
O
A
W
F
5
ALPHA
O
P
ENTER
B
1
ENTER
Print "TRAP="
2ndF
PRGM
A
1
2ndF
T
A
P
=
2ndF
PRGM
1
(
W
R
End
÷ 2 ) (
) ENTER
2ndF
)
+
ALPHA
X/θ/T/n
ENTER
2ndF
C
(L+Z+R)
O
O
P
PRGM
6
Z
STO
X/θ/T/n
PRGM
2ndF
L
STO
2ndF
Print (W/2)
STO
ENTER
A-LOCK
X/θ/T/n
ClrT
R
X/θ/T/n
N
ENTER
ALPHA
L
82
B
PRGM
PRGM
PRGM
2
2
2
2ndF
2ndF
A-LOCK
A-LOCK
ENTER
ALPHA
ALPHA
L
+
ALPHA
ENTER
Blackline Masters/CALCULUS USING THE SHARP EL-9600
Z
+
ALPHA
NAME _____________________________________________________ CLASS __________ DATE __________
8.1
HYPERBOLIC FUNCTIONS
Steps for graphing the hyperbolic sine function.
1. Turn the calculator on and press Y= CL to access and clear the Y1
prompt. Press ENTER
CL to remove additional expressions.
2. The calculator should be setup in radian measurement with rectangular
coordinates. To complete this setup, press 2ndF
SET UP , touch B DRG,
double touch 2 Rad, touch E COORD, and double touch 1 Rect. Press 2nd
QUIT to exit the SET UP menu and return to the Y prompts.
3. To enter the hyperbolic sine function ( y = sinh x ) for Y1, press MATH ,
➧
touch A CALC, touch
on the screen, double touch 15 sinh, and press
X/θ/T/n .
4. Enter the viewing window by pressing ZOOM , touching F HYP, and double
Steps for graphing the hyperbolic cosine function.
1. Press Y= CL to remove the hyperbolic sine function.
2. To enter the hyperbolic cosine function ( y = cosh x ) for Y1, press MATH ,
➧
touch
on the screen, double touch 16 cosh x, press X/θ/T/n .
3. Use the same viewing window as before (-6.5 ≤ x ≤ 6.5 by -10 ≤ y ≤ 10) and
press GRAPH to view the graph.
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83
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touching 1 sinh x.
NAME _____________________________________________________ CLASS __________ DATE __________
8.2
HYPERBOLIC FUNCTIONS
Activity 1
Graph the hyperbolic tangent function.
1. Turn the calculator on and press Y=
prompt. Press ENTER
CL to access and clear the Y1
CL to remove additional expressions.
2. The calculator should be setup in radian measurement with rectangular
coordinates. To complete this setup, press 2ndF SET UP , touch B
DRG, double touch 2 Rad, touch E COORD, and double touch 1 Rect. Press
2nd
QUIT to exit the SET UP menu and return to the Y prompts.
3. To enter the hyperbolic tangent function ( y = tanh x ) for Y1, press
➧
MATH , touch A CALC, touch
on the screen, double touch 17 tanh, and
press X/θ/T/n .
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
4. Enter the viewing window by pressing ZOOM , touching F HYP, and
double touching 3 tanh x.
5. Sketch the graph in the box provided.
Activity 2
Graph the hyperbolic cotangent function.
Repeat the steps for Activity 1 for the hyperbolic cotangent function. Enter the
hyperbolic cotangent function (y = coth x) as y = 1/tanh x. Continue to use the
tanh viewing window.
84
Blackline Masters/CALCULUS USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
8.3
HYPERBOLIC FUNCTIONS
Activity 3
Graph the hyperbolic secant function.
Repeat the steps for Activity 1 for the hyperbolic secant function. Enter the
hyperbolic secant function (y = sech x) as y = 1/cosh x. Continue to use the tanh
viewing window.
Graph the hyperbolic cosecant function.
Repeat the steps for Activity 1 for the hyperbolic cosecant function. Enter the
hyperbolic cosecant function (y = csch x) as y = 1/sinh x. Continue to use the
tanh viewing window.
Blackline Masters/CALCULUS USING THE SHARP EL-9600
85
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Activity 4
NAME _____________________________________________________ CLASS __________ DATE __________
9.1
SEQUENCES
1. Turn the calculator on and set the calculator to sequence mode by pressing
2ndF SET UP , touching E COORD, and double touching 4 Seq.
2. Press 2ndF QUIT Y= to access the sequence prompts. Clear
any sequences by presing CL .
3. Enter the sequence generator an = n2 – n for u(n) by pressing X/θ/T/n
x2
–
1
ENTER .
X/θ/T/n ENTER . Enter n1 = 1 for u(nMin) by pressing
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
4. View a table of sequence values by pressing TABLE .
5. Graph the sequence by first setting the format to time and dot modes. Do
this by pressing 2ndF FORMAT ,touching G TYPE, double touching
2 Time, touching E STYLE1, and double touching 2 Dot. Press 2ndF
QUIT to exit the FORMAT menu. Enter the viewing window by pressing
WINDOW EZ , double touching 3, double touching 5 -1<X<10, and double
touching 2 -10<Y<100.
86
Blackline Masters/CALCULUS USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
9.2
SEQUENCES
1. Enter the recursive sequence generator an = an – 1 + 2n for u(n) by pressing
( X/θ/T/n –
)
+ 2 X/θ/T/n ENTER .
Y= CL 2ndF u
1
Enter a1 = 1 by pressing 1
ENTER .
3. Graph the sequence in the same window as 9.1 by pressing
GRAPH .
Blackline Masters/CALCULUS USING THE SHARP EL-9600
87
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
2. View a table of sequence values by pressing TABLE .
NAME _____________________________________________________ CLASS __________ DATE __________
9.3
SEQUENCES
Activity 1
The Fibonnoci sequence is the sequence where the previous two terms are
added together to form the next term. The first two terms are a1 = 1 and a2 = 1.
1. Enter the Fibonnoci sequence generator an = an – 1 + an – 2 for u(n) by pressing
) + 2ndF u ( X/θ/T/n
Y= CL 2ndF u ( X/θ/T/n – 1
–
2
,
1
{
)
ENTER . Enter a1 = 1 and a2 = 1 by pressing 2ndF
2ndF } ENTER .
1
2. View a table of sequence values by pressing TABLE .
3. Graph the sequence in the same window as 9.1 by pressing GRAPH .
1. Return the SET UP to rectangular mode by pressing 2ndF
SET UP ,
touching E COORD, and double touching 1 Rect. Press 2ndF
QUIT
to
exit the SET UP menu.
2. First find the sequence of the first 10 terms of series ∑ 1/X by first pressing
+ –
× ÷
2ndF
LIST , touching A OPE, and double touching 5 seq( .
÷ X/θ/T/n , . Enter the lower
Enter the generator 1/X by pressing 1
and upper bounds for the sequence by pressing 1
the sequence by pressing ENTER . Press
ANS
1
0
) . Find
to see more of the sequence.
3. Find the partial sum of the first 10 terms by pressing
touching 6 cumul, and pressing 2ndF
,
2ndF
LIST , double
ENTER . Press
▼
CL
▼
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
Activity 2
Find the partial sum of a series, ∑ 1/X, for the first 10 terms.
to
move right in the sequence of partial sums until the last term is seen. This
is the partial sum of the first 10 terms.
88
Blackline Masters/CALCULUS USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
10.1
GRAPHING POLAR EQUATIONS
Steps for graphing a polar function:
1. Turn the calculator on and press 2ndF SET UP , touch E COORD, and
double touch 3 Polar to change to polar mode. While in the SET UP menu,
the calculator should be setup in radian mode. To complete this setup,
touch B DRG, and double touch 2 Rad. Press 2ndF QUIT to exit the SET
UP menu.
2. Make sure calculator is in connected mode by pressing 2ndF FORMAT ,
touching E STYLE1, and double touching 1 Connect. While in the FORMAT
menu, set the calculator to display polar coordinates when tracing, by
touching B CURSOR and double touching 2 PolarCoord. Also, set the
calculator to display the expression during tracing by touching C EXPRESS
and double touching 1 ON. Press 2ndF QUIT to exit the FORMAT menu.
expression. Press ENTER CL to clear additional R prompts.
4. Enter the polar function r = 2(1 – cos θ) for R1, by pressing 2 ( 1
cos X/θ/T/n ) . Notice, when in polar mode the X/θ/T/n key
–
provides a θ for equation entry.
5. Now, graph the polar function in the Decimal viewing window by pressing
➧
ZOOM , touching A ZOOM, touching
on the screen, and touching 7 Dec.
6. This particular shape of curve is called a cardoid. Trace the curve by
pressing TRACE . Notice the expression is displayed at the top of the
screen.
Blackline Masters/CALCULUS USING THE SHARP EL-9600
89
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
3. Press Y= to access the R1 prompt. Press CL to remove an old R1
NAME _____________________________________________________ CLASS __________ DATE __________
10.2
GRAPHING PARAMETRIC EQUATIONS
Steps for graphing a parametric function:
1. Turn the calculator on and press 2ndF SET UP , touch E COORD, and
double touch 2 Param to change to parametric mode. Press 2ndF QUIT
to exit the SET UP menu.
2. Make sure calculator is set to display rectangular coordinates when tracing
by pressing 2ndF
FORMAT , touching B CURSOR and double touching
1 RectCoord. Press 2ndF QUIT to exit the FORMAT menu.
3. To enter the parametric function X1T = 2(cos T)3, Y1T = 2(sin T)3, press
Y= CL 2 ( cos X/θ/T/n ) ab 3 ENTER CL 2 ( sin
X/θ/T/n
)
ab 3
ENTER . Notice, when in parametric mode the
X/θ/T/n key provides a T for equation entry.
pressing ZOOM , touching A ZOOM, touching
➧
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
4. Now, graph the parametric function in the Decimal viewing window by
on the screen, and
touching 7 Dec.
5. Press TRACE and notice the expression and T values now appear on the
range screen.
90
Blackline Masters/CALCULUS USING THE SHARP EL-9600
NAME _____________________________________________________ CLASS __________ DATE __________
10.3
GRAPHING POLAR EQUATIONS
Activity 1
Graph the following polar functions. Sketch what you see in the
box provided.
1. r = 1 + 2 cos θ (limacon)
2. r = (4sin (2θ)) (leminscate)
Decimal viewing window.
Blackline Masters/CALCULUS USING THE SHARP EL-9600
91
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
Decimal viewing window.
NAME _____________________________________________________ CLASS __________ DATE __________
10.4
GRAPHING PARAMETRIC EQUATIONS
Activity 2
Graph the following parametric functions.
1. X1T = T – sin T (cycloid)
Y1T = 1 – cos T
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
Decimal viewing window
2. X1T = 2 cos T (ellipse)
Y1T = sin T
Decimal viewing window
92
Blackline Masters/CALCULUS USING THE SHARP EL-9600
KEYPAD FOR
THE SHARP EL-9600 GRAPHING CALCULATOR
EL-9600
Equation Editor
STAT PLOT
SPLIT
TBL SET
SUB
FORMAT
CALC
Y=
GRAPH
TABLE
WINDOW
ZOOM
TRACE
SOLVER
CLIP
SHIFT/CHANGE
+ –
× ÷
SLIDE
SHOW
EZ
OFF
PRGM
DRAW
ON
MATRIX
STAT
A-LOCK
TOOL =
INS
SET UP
ALPHA
MATH
DEL
BS
sin A
cos B tan C
cos
sin
,
a b
L
v
Q
V
L1
2
1
<
O
0
i
▼
log
SPACE
L3
K OPTION
VARS
X/θ/T/n
{
(–)
O
S
LIST
X
e
}
P
)
T
FINANCE U
Y
EXE
Z
+
3
ENTRY
F
In
π
(
6
W
L2
N
L6
5
4
2
9
R
L5
x
CL
ex
D 10x E
STO
w
8
-1
RCL J
I
ab
M
7
L4
tan
H a
G
u
x
-1
:
ANS
ENTER
Keypad/CALCULUS USING THE SHARP EL-9600
93
Copyright © 1998, Sharp Electronics Corporation. Permission is granted to photocopy for educational use only.
-1
QUIT
+
-1
Exp
▼
▼
2nd F
▼
SOLUTIONS TO SELECTED ACTIVITIES
1. EVALUATING LIMITS
ADDITIONAL PROBLEMS
1. 12; continuous for all values of x except x = -2
2. does not exist; continuous for all values of x
except x = 2.5
3. e and 1; continuous for all values of x>-1 except x=0
4. 4.6; continuous for all values of x except x = 500
5. 71 wpm; No, 95
BLACKLINE MASTER 1.3
1. f(x) = -1, f(x) = 1, Yes
2. 1, 2/3, does not exist, 1
3. p(20)=9082, 10000
2. DERIVATIVES
ADDITIONAL PROBLEMS
1. 2
2. does not exist
3. 2
4. 3.14159
5. -e-x, x, 1/x
BLACKLINE MASTER 2.1
ACTIVITY 2
2. 6
ACTIVITY 3
1. 54
BLACKLINE MASTER 2.2
ACTIVITY 2
2. 6
ACTIVITY 3
1. does not exist
BLACKLINE MASTER
2.3 ACTIVITY 2
2. 2e2x, 5x
3. TANGENT LINES
ADDITIONAL PROBLEMS
1. y = 4x – 4
2. y = .54x + .301
3. y = 1.298x + .103
4. y = .5x + .5
5. y = 1
4. GRAPHS OF DERIVATIVES
ADDITIONAL PROBLEMS
1. x < .361 dec, concave up relative min at x = -.361
-.361<x<1.667 inc, concave up inflection point at x
= 1.667
1.667<x<3.694 inc, concave down relative max at
x = 3.694
x>3.69 dec, concave down
2. x < -.707 inc, concave up inflection point at x = -.707
-.707<x<0 inc, concave down relative min at x = 0
0<x<.707 dec, concave down inflection point at x
= .707 x>.707 dec, concave up
94
3.
4.
x>0 inc, concave down no max or min
x<0 dec, concave up relative min at x=0
x<0 inc, concave up
BLACKLINE MASTER 4.4
ACTIVITY 1
2. decreasing, negative, 0
increasing, positive, 0
minimum, does not exist, does not exist
ACTIVITY 2
1. increasing, concave down, positive, negative
decreasing, concave down, positive, negative
maximum, x-intercept, maximum
5. OPTIMIZATION
ADDITIONAL PROBLEMS
1. 1352
2. 4000.562 m
3. 14.063 ft, .375 seconds
4. 1.132 v
5. x ≈ 2.678, y ≈ 2.231
BLACKLINE MASTER 5.1
8. 3.873, .684
BLACKLINE MASTER 5.2
6. 3.799, 21.23
BLACKLINE MASTER 5.3
ACTIVITY 1
1. 0, 0, 1
2. 0, 25, 5, 0, 1, 1
3. .924
4. 3.464 days
ACTIVITY 2
1. amount of fence = 2w + l = 2w + 675/w
2. w ≈ 18.371, l ≈ 36.74
6.
SHADING AND CALCULATING AREAS
REPRESENTED BY AN INTEGRAL
ADDITIONAL PROBLEMS
1. 246.738
2. 7.021
3. .828
4. .402
5. 4
BLACKLINE MASTER 6.3
ACTIVITY 1
1. 5.333
3. 1.333
ACTIVITY 2
2. 70, 2.805
90, 14.888
3. 80.051
Answer Key/CALCULUS USING THE SHARP EL-9600
SHARP CALC COVER
02.2.19 10:40 AM
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