CALCULUS with the Casio FX-9750G Plus CALCULUS with the Casio FX-9750G Plus Limits Derivatives Continuity Slope Linear Functions Differentiability Polynomials Trigonometric Functions Graphing Models Slope Fields Anti Derivatives Integration Riemann Sums Activities for the Classroom 9750-CALC Casio, Inc. CALCULUS with the Casio FX-9750G Plus Activities for the Classroom All activities in this resource are also compatible with the Casio CFX-9850G Series. CALCULUS with the Casio FX-9750G Plus Kevin Fitzpatrick ® 2005 by CASIO, Inc. 570 Mt. Pleasant Avenue Dover, NJ 07801 www.casio.com 9750-CALC The contents of this book can be used by the classroom teacher to make reproductions for student use. All rights reserved. No part of this publication may be reproduced or utilized in any form by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval system without permission in writing from CASIO. Printed in the United States of America. Design, production, and editing by Pencil Point Studio Contents Activity 1: Looking at Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Teaching Notes Student Activity Calculator Notes and Answers Activity 2: Do Limits Take Sides? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Teaching Notes Student Activity Calculator Notes and Answers Activity 3: A Graphical Look at Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Teaching Notes Student Activity Calculator Notes and Answers Activity 4: Introduction to Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Teaching Notes Student Activity Calculator Notes and Answers Activity 5: Being Locally Linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Teaching Notes Student Activity Calculator Notes and Answers Activity 6: Continuity Meets Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Teaching Notes Student Activity Calculator Notes and Answers Activity 7: Derivative Behavior of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Teaching Notes Student Activity Calculator Notes and Answers Activity 8: Derivative Behavior of Common Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Teaching Notes Student Activity Calculator Notes and Answers Copyright © Casio, Inc. Calculus with the Casio fx-9750G Plus iii Activity 9: Looking at Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Teaching Notes Student Activity Calculator Notes and Answers Activity 10: Looking at Slope Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Teaching Notes Student Activity Calculator Notes and Answers Activity 11: Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Teaching Notes Student Activity Calculator Notes and Answers Appendix: Overview of the Calculator Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 iv Calculus with the Casio fx-9750G Plus Copyright © Casio, Inc. Activity 1 Looking at Limits Teaching Notes Topic Area: Limits Class Time: one 45-50 minute class period Overview This activity will encourage students to use graphical and numerical representations to examine the behavior of a function as it approaches a particular input value. A limit is one of the foundation concepts in any calculus course. The idea behind this activity is to have the student investigate both numerically and graphically the behavior of the output of a function as its input moves closer and closer to some point of interest. The emphasis will be on examining the behavior of the function as its gets near a particular input value. Even though the function may reach that input value, the activity will be centered more on what happens as the input gets closer and closer to the value of interest. Objectives • To develop an understanding of meaning of a "limit” • To be able to estimate the value of a limit using a numerical view from a table and a graphical view Getting Started Using the Casio fx-9750G Plus, have students work in pairs or small groups arranged prior to beginning the activity to maximize student involvement and ownership of the results. Prior to using this activity: • Students should be able to produce and manipulate graphs and tables of values manually and with the graphing utility. • Students should have a basic understanding of the language of functions. • Students should be able to identify rational and exponential functions. Ways students can provide evidence of learning: • If given a function, the student can state and explain what the limit is at a particular value. • If given a graphical representation of a function, the student can state and explain what the limit is at a particular value. • If given a tabular representation of a function, the student can state and explain what the limit is at a particular value. Common mistakes to be on the lookout for: • Students may use viewing windows that appear to show functions being defined when they are not. • Students may use an input or table value with an increment so small that the calculator will display a rounded value that does not actually exist. • Students may use an input or table value with an increment so small that the calculator will return an error message regarding memory overflow. Copyright © Casio, Inc. Activity 1 • Calculus with the Casio fx-9750G Plus 1 Name _____________________________________________ Class ________ Date ________________ Activity 1 • Looking at Limits Introduction This activity will encourage you to use graphical and numerical representations to examine the behavior of a function as it approaches a particular input value. Using the Casio fx-9750G Plus you will be working in pairs or small groups. Problems and Questions 2 Examine the value of the function f(x)= x – 1 as the value of x gets close to 1. x–1 1. Go to the MENU and choose the TABLE option. 2. Enter the function in Y1. 3. Set up the table as shown below. 4. Display the table and record the function values when x = {0,1,2}. x y 0 1 2 5. Explain why the values you recorded either did or did not match up with your expectations. _____________________________________________________________________________ _____________________________________________________________________________ 6. Now have the table start at .5, and change the pitch to .5 as well. 7. Record the values you get for x = {.5, 1, 1.5} x y .5 1 1.5 2 Calculus with the Casio fx-9750G Plus • Activity 1 Copyright © Casio, Inc. Name _____________________________________________ Class ________ Date ________________ Activity 1 • Looking at Limits 8. Repeat the process, this time starting the table at .75 and changing the pitch to .25, then record the function values for x = {.75, 1, 1.25}. x y .75 1 1.25 9. Repeat the process twice more. • the first time starting at .9 with a pitch of .1 Record the values for x = {.9, 1, 1.1} x y .9 1 1.1 • the second time starting at .99 with a pitch of .01. Record the values for x = {.99, 1, 1.01} x y .99 1 1.01 10. What would you expect to see if the pitch was changed to .001, to .0001? _____________________________________________________________________________ 11. What function value does it appear to close in on? _____________________________________________________________________________ Now examine the graph of the same function to see the behavior. 12. Choose GRAPH from the Menu and set the INIT viewing window as shown below. Copyright © Casio, Inc. Activity 1 • Calculus with the Casio fx-9750G Plus 3 Name _____________________________________________ Class ________ Date ________________ Activity 1 • Looking at Limits Then sketch the graph on the axis below. 13. Go to ZOOM and press F2 (zoom factors), set the zoom factors as follows: • Xfact: 4 • Yfact: 2 14. Graph the function again, Trace to the point (.9, 1.9) and Zoom-In. Write a description of what you see and include a sketch to support your statements. _____________________________________________________________________________ _____________________________________________________________________________ 15. Trace to the point (1.025, 2.025) and Zoom-In again. Write a description of what you see and include a sketch to support your statements. _____________________________________________________________________________ _____________________________________________________________________________ 16. Continue to repeat the process, tracing closer and closer to the value x = 1, from values both above and below x = 1, each time Zoom-In, until you are comfortable drawing a conclusion. 17. If the values of a function come closer and closer to a single value, that value is called the limit of the function and is expressed as "as x approaches some value (c), f(x) has a limit of L" Rewrite your conclusion to these examinations using the phrasing shown here. _____________________________________________________________________________ _____________________________________________________________________________ x2 – 1 18. Examine the function: f(x)= x – 2 around the value x = 2 using a table set up starting at x = 1, ending at x = 3, and having a pitch of 1, record the values for x = {1,2,3}. x y 1 2 3 4 Calculus with the Casio fx-9750G Plus • Activity 1 Copyright © Casio, Inc. Name _____________________________________________ Class ________ Date ________________ Activity 1 • Looking at Limits 19. Change the pitch of the table (table increments) as before, first to .5, then to .25, then to .1 and finally to .01. Each time, recording the values directly above and below x = 2 in each case. x 2 y x 2 y x 2 y x y 2 20. Now use the same graphical analysis process with this function and write a conjecture based upon the numerical and graphical evidence. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 21. When an input approaches a single value and the output also approaches a single value the function is said to have a limit, however when the output does not approach a single value, the function is said to have no limit. Using the phrasing from Question 17, express your conclusion using the proper phrasing. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ Further Exploration Find the limit, if it exists, for each of the following. If it does not exist, explain why. 22. 23. 24. 2 lim 2xx – 50 3 + 15 ______________________________________________________________ x lim 3 – 1 x→0 x ______________________________________________________________ lim x + 2 x+4 ______________________________________________________________ x→5 x→0 25. xlim →-3 Copyright © Casio, Inc. 3x + 12 4–x ______________________________________________________________ Activity 1 • Calculus with the Casio fx-9750G Plus 5 Calculator Notes and Answers for Activity 1 To Get to the TABLE screen: • From the Main MENU either press 7, or use the arrow keys to highlight TABLE and press EXE. To get to the TABLE SET UP: • While in the Table Function, press F5 (RANGE) key. To get to the Zoom Factors screen: • After graphing press SHIFT F2 (Zoom). • Press F2 (FACT) key. Answers: 4. 5. Answers will vary, however, most students should recognize that at x = 1 there is division by zero and that is creating the error being displayed. 6. n/a 7. 8. 9. 10. Answers will vary but a good answer should contain the fact that they value is closing in on 2 as x approaches 1. 6 Calculus with the Casio fx-9750G Plus • Activity 1 Copyright © Casio, Inc. Calculator Notes and Answers for Activity 1 11. Here the answers should not vary, a value of 2 is the correct answer. 12. A good sketch will show the hole in the graph and look something like this: Note: The reason for setting the particular viewing window in this activity is to make sure the hole is visible. The calculator will only show the gap if it is a specific pixel it is asked to light up and that pixel does not exist at that point. In a many other viewing windows the point (1,2) would not be one that the 9750 would try to graph, thus in connected mode the hole would not appear and the graph would appear to be continuous. 13. n/a 14. Answers will vary but should contain a statement about the maintenance of the discontinuity (hole) in the graph. 15. The description should include mention of the hole and a better description would include a statement about the value closing in on 2, while still not existing at x = 2. 16. A good conclusion would center around the value getting infinitely close to 2 as x gets closer and closer to 1. 17. "as x approaches 1, f(x) has a limit of 2" 18. 19. Copyright © Casio, Inc. Activity 1 • Calculus with the Casio fx-9750G Plus 7 Calculator Notes and Answers for Activity 1 20. Students should produce some graphs showing the following sketch, and the idea of asymptotes should be mentioned. Note: This is a good time to discuss the window again, here there is not missing pixel but care needs to be taken to show both branches of the graph. If the proper vertical window is not set, only one branch will be found leading to an incorrect answer. 21. "as x approaches 2 f(x) has a no limit" 22. 0 23. 1.099 approximately 24. 1 25. .247 Note: Some students may realize that 24, 25 can be done by direct substitution, this should cement discussion regarding the fact that while it is not necessary for a limit to actually be a value of the function, it certainly can be. This also can be used to foreshadow a discussion of continuity. 8 Calculus with the Casio fx-9750G Plus • Activity 1 Copyright © Casio, Inc. Activity 2 Do Limits Take Sides? Teaching Notes Topic Area: Limits Class Time: one 45-50 minute class period Overview This activity will encourage students to use graphical and numerical representations to examine the idea of a limit needing to be the same from both directions of approach. The concept of a limit creates the framework for discussing continuity. Using splitdefined functions, the goal of this activity is to put a face on the idea of one-sided limits. Objectives • To develop an understanding of meaning of one sided limits • To be able to understand and communicate the idea that for a function to have a limit at a point, it must approach the same output value from either direction. Class Time: This activity is designed to be used in one 45-50 minute class period. Getting Started Using the Casio fx-9750G Plus, have students work in pairs or small groups arranged prior to beginning the activity to maximize student involvement and ownership of the results. Prior to using this activity: • Students should be able to produce and manipulate graphs and tables of values manually and with the graphing utility. • Students should be able to produce split defined (or piecewise) functions. • Students should have a basic understanding of the language of functions. • Students should be able to identify rational and exponential functions. Ways students can provide evidence of learning: • If given a split defined function, the student can produce a picture of the function using the calculator. • If given a graphical representation of a function, the student can state and explain what the limit is as it approaches an input value from the left side or the right side. Common mistakes to be on the lookout for: • Students may produce a graph on the calculator and not be able to communicate the concept of a split-defined function as window chosen may produce the appearance of single formula. Copyright © Casio, Inc. Activity 2 • Calculus with the Casio fx-9750G Plus 9 Name _____________________________________________ Class ________ Date ________________ Activity 2 • Do Limits Take Sides? Introduction This activity will have you use graphical and numerical representations to examine the idea of a limit needing to be the same from both directions of approach. Using the Casio fx-9750G Plus you will be working in pairs or small groups. Problems and Questions { Examine the behavior of the function: f(x)= x – 4, x < 2 as the value of x x – 1, x > 2 approaches 2: 1. Choose GRAPH from the MENU, enter the function. 2. Set the initial viewing window to Standard by pressing F3 (STD). 3. Copy the graph on the axis shown and describe what you see: _____________________________________________________________________________ _____________________________________________________________________________ 4. Using the trace function, record your observations as to what happens as you trace along the function moving closer and closer to the value x = 2. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 5. Using any zoom technique you prefer, keep both branches visible and keeping x = 2 toward the center of the window redraw the graph getting a closer and closer look at the output of the function. Explain what you see. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 6. From your knowledge of limits, and based upon what you see in this case, what is the xlim2 f(x) ? Explain your answer. → _____________________________________________________________________________ _____________________________________________________________________________ 10 _____________________________________________________________________________ Calculus with the Casio fx-9750G Plus • Activity 2 Copyright © Casio, Inc. Name _____________________________________________ Class ________ Date ________________ Activity 2 • Do Limits Take Sides? 7. The symbolic notation: lim+ f(x) means to investigate the limit of the function, x→c f(x), as x approaches some value c through values that are greater than c (frequently called "from the right"). In this case, using your trace cursor, copy the graph and show what that means. 8. Describe your results using some ordered pairs to show the respective input and output relationships. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 9. How would you now answer the question: Find lim+ f(x) ? x→2 _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 10. Based upon this investigation so far, how would you describe the notation: lim f(x) ? x 2– → _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 11. How would you answer the question: Find lim– f(x)? Why? x→2 _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 12. How would you now answer the question: Find xlim2 f(x) ? Why? → _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ Copyright © Casio, Inc. Activity 2 • Calculus with the Casio fx-9750G Plus 11 Name _____________________________________________ Class ________ Date ________________ Activity 2 • Do Limits Take Sides? 13. Graph the function h(x)= { 1 , x < –1 x+ 3 x2 + 3, –1< x< 2 –x + 9, x > 2 in the window Sketch what you see on the axes. 14. Find each of the following limits and explain how you arrived at your conclusion lim h(x) ______________________________________________________________ lim h(x) ______________________________________________________________ lim h(x) ______________________________________________________________ lim h(x) ______________________________________________________________ lim h(x) ______________________________________________________________ lim h(x) ______________________________________________________________ lim h(x) ______________________________________________________________ h. lim- h(x) x 3 ______________________________________________________________ a. x→–3 b. x→–3- c. x→–3+ d. x→–1+ e. f. g. x→–1 x→0 x→3 → 12 Calculus with the Casio fx-9750G Plus • Activity 2 Copyright © Casio, Inc. Calculator Notes and Answers for Activity 2 How to graph a split defined function: • • • • Enter each branch in its own Y= slot then create the restrictions by using putting them in [lower, upper] Example to graph f(x)= x – 4, x < 2 you would enter it as follows: x – 1, x > 2 Y1 = x – 4, [lower, 2] { Y2 = x –1 , [2, upper] Note: The lower and upper can usually be just the min and max of the viewing window if you only have two branches. 3. The graph in standard window Note: When students graph it they should be very clear to indicate that there are open circles at the endpoints of the "jump." 4. As the input value gets closer to 2, the lower branch gets closer to –2, while the upper branch gets closer to 1 5. Should have the same results are in #4, but the numbers should be getting closer to –2 and 1 respectively 6. The function does not have a limit as x approaches 2 since the values are different depending upon the direction you approach the input. 7. One view of what happens as the cursor gets closer to 2, answers will vary. Copyright © Casio, Inc. Activity 2 • Calculus with the Casio fx-9750G Plus 13 Calculator Notes and Answers for Activity 2 8. Answers will vary, see above for some possible ordered pairs. 9. limit is 1 10. What is the limit of the function, as the input approaches 2 from values below 2 (or to the left of 2)? 11. The limit is –2. The explanations will vary, but a good explanation should cover the fact that as the value "walks" along the function from values to the left of 2, the input gets increasingly closer to –2. 12. Answer should be the same as 6. 13. This is a representation of what the student should sketch. 14a. xlim h(x) = None, two different one sided limits –3 → b. xlim h(x) = – (Note: while "none" is also acceptable, – is a more complete →–3description of what is actually taking place.) h(x) = (Note: while "none" is also acceptable, is a more complete c. x lim →–3+ description of what is actually taking place.) h(x) = 4 d. x lim →–1+ e. xlim h(x) = None, two different one-sided limits –1 → f. xlim0 h(x) = 3 → g. xlim3 h(x) = None, two different one-sided limits → h. xlim h(x) = 12 3- → 14 Calculus with the Casio fx-9750G Plus • Activity 2 Copyright © Casio, Inc. Activity 3 A Graphical Look at Continuity Teaching Notes Topic Area: Derivatives and Continuity Class Time: an exploratory introduction during the first 30 minutes of a class period on the topic of continuity Overview This activity will have students explore the concept of continuity at a point. It will also allow them to discover that simply having a limit at a point will not guarantee that the function is also continuous. It also explores the idea that a having a limit is a necessary, but not a sufficient condition to determine the continuity of a function at a point, and through all points. Objectives • To develop a visual understanding of how limits and continuity relate • To be able to understand and communicate what it means for a function to be continuous at a point Getting Started Using the Casio fx-9750G Plus, have students work in pairs or small groups arranged prior to beginning the activity to maximize student involvement and ownership of the results. Prior to using this activity: • Students should be able to produce and manipulate graphs of functions manually and with the graphing utility. • Students should be able to produce split defined (or piecewise) functions. • Students should have a basic understanding of the language of limits. Ways students can provide evidence of learning: • Students should be able to produce graphs of functions and communicate symbolically, graphically and verbally the relationship between having a limit and being continuous. Common mistakes to be on the lookout for: • Students may produce a graph on the calculator in such a way that the window chosen may produce the appearance of a continuous function when, in fact, it is not. • Students may confuse the pixel values with the actual function values. Copyright © Casio, Inc. Activity 3 • Calculus with the Casio fx-9750G Plus 15 Name _____________________________________________ Class ________ Date ________________ Activity 3 • A Graphical Look at Continuity Introduction This activity will have have you explore the concept of continuity at a point. It will also allow you to discover that simply having a limit at a point will not guarantee that the function is also continuous. Using the Casio fx-9750G Plus you will be working in pairs or small groups. Problems and Questions Explore the behavior of the function f(x)= x2 – x – 6 around the vertex: 1. Go to the GRAPH menu and, in the viewing window, produce the graph of the function f(x) and copy it to the axes. 2. Find and record the vertex of the function. 3. Making sure your zoom factors are set to 4 for both X and Y, trace to the vertex and zoom in, record what you see. 4. What does it appear the value of xlim5 f(x) is? Explain why you arrived at that → answer. _____________________________________________________________________________ _____________________________________________________________________________ 16 Calculus with the Casio fx-9750G Plus • Activity 3 Copyright © Casio, Inc. Name _____________________________________________ Class ________ Date ________________ Activity 3 • A Graphical Look at Continuity 5. Now explore the behavior of the split-defined function: g(x)= Use the same viewing window as before. { x2 – x – 6, x< .5 –6, x = .5 x2 – x – 6, x>.5 Record what you see below. 6. What does it appear the value of xlim g(x) is? →.5 How does it compare to xlim f(x) ? .5 → _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 7. Now, trace to a value where x = .4, and zoom in, describe and record what you see. _____________________________________________________________________________ _____________________________________________________________________________ g(x) , lim - g(x) , lim g(x) 8. Find: xlim x→.5 x→5 →.5+ 9. Now find g(.5), how does this compare to your answers above? _____________________________________________________________________________ _____________________________________________________________________________ 10. Draw a conclusion about the relationship between limits and continuity. _____________________________________________________________________________ _____________________________________________________________________________ Copyright © Casio, Inc. Activity 3 • Calculus with the Casio fx-9750G Plus 17 Calculator Notes and Answers for Activity 3 1. 2. Vertex is (.5, -6.25) and can be found symbolically or using the MIN function in the G-Solve folder. 3. This is the screen sequence for the zoom. Nothing unusual should be seen. The vertex remains, the function is continuous. 4. The limit is –6.25, the vertical value of the vertex. Answer will vary as to how it was arrived at. Care should be taken to point out that simply tracing to a value is not confirmation enough and can be tricky. Direct substitution is a valid explanation. A good answer might also include a mention of "passing through" or even a mention of continuity. 18 Calculus with the Casio fx-9750G Plus • Activity 3 Copyright © Casio, Inc. Calculator Notes and Answers for Activity 3 5. This provides a good look at the split defined function. The graph produced in the given window will be as shown on the left. The discontinuity will not be immediately apparent from this graph. 6. The limit is –6.25. Answers may vary as students begin to get the idea that the change in the definition of the function may be creating some problems, although not with the limit. This is a good checkpoint for the understanding of what it means to be a "limit." 7. Copyright © Casio, Inc. This is the screen sequence that produces the desired screen. Activity 3 • Calculus with the Casio fx-9750G Plus 19 Calculator Notes and Answers for Activity 3 8. All three limits are –6.25, although some students may try to refine the answers to longer decimals. This provides another good opportunity to stress the idea of "limit" as a value the function approaches. 9. g(.5) = -6, a value different from the limit. 10. A good answer will include the fact that the function has a gap or a hole or a jump (ie, a point of discontinuity at x = .5). The idea is to have them begin to think about the fact that simply having a limit does not guarantee the continuity of a function. 20 Calculus with the Casio fx-9750G Plus • Activity 3 Copyright © Casio, Inc. Activity 4 Introduction to Derivatives Teaching Notes Topic Area: Derivatives Overview This activity will have students begin to connect the concept of slope and rate of change to the derivative. It also provides an introduction to the concept that the slope of a function extends beyond linear slope, but that using the slope of a line can foster a discussion of average vs. instantaneous rates of change. Objectives • To develop an understanding of the slope of a function that is not just linear • To be able to understand and communicate the visuals connected with the average rate of change and the secant line to a function Getting Started Using the Casio fx-9750G Plus, have students work in pairs or small groups arranged prior to beginning the activity to maximize student involvement and ownership of the results. Prior to using this activity: • Students should be able to produce and manipulate graphs of functions manually and with the graphing utility. • Students should be able to use the statistics Menu to produce linear and quadratic regression models. • Students should have a basic understanding of the language of limits. • Students should have an understanding of what a secant line is. • Students should have an understanding of slope as a rate of change. Ways students can provide evidence of learning: • Students should be able to produce graphs of functions and communicate symbolically, graphically, numerically and verbally the relationship between the slope of a line, a function and an average rate of change Common mistakes to be on the lookout for: • Not being able to relate the slope to a real world rate of change concept • Not being able to communicate the slope as the rate of change of output over input Copyright © Casio, Inc. Activity 4 • Calculus with the Casio fx-9750G Plus 21 Name _____________________________________________ Class ________ Date ________________ Activity 4 • Introduction to Derivatives Introduction This activity provides an introduction to the concept that the slope of a function extends beyond linear slope, but that using the slope of a line can foster a discussion of average vs. instantaneous rates of change. Using the Casio fx-9750G Plus you will be working in pairs or small groups. Problems and Questions 1. Calculate the slope of the line connecting the points (2,5) and (5,2)? ____________________________________ 2. Describe the meaning of the slope you just found in terms of input and output. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 3. Now calculate the slope of the line connecting the points (-1,8) and (11,-4) ____________________________________ 4. What conclusions, if any, can you draw about these 4 points? Explain. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 5. Name two other points that would share the same characteristics as these points? Explain your choices. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 6. If, at the end of his first year of employment, Mike’s annual salary was $42,000 and at the end of his 3rd year of employment with the same company, Mike’s annual salary was $49,000. What conclusion could you draw about the growth of Mike’s salary over that period of time? Explain. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 22 Calculus with the Casio fx-9750G Plus • Activity 4 Copyright © Casio, Inc. Name _____________________________________________ Class ________ Date ________________ Activity 4 • Introduction to Derivatives 7. Given the same data as above, if Mike were to stay with the same company for 10 years, predict what his salary should be at the end of those 10 years. Explain. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 8. What if Mike’s actual salary after 10 years was $100,000? How does that agree with your prediction from above? How does that compare to the rate of growth you used in your prediction in item #7? _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 9. Create a good model using the data at the end of the first, third and tenth year salaries. Record the result here and explain why you chose your model. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 10. Using your model from item 9, what would you say that average change in Mike’s salary was between years 4 and 10? Between years 4 and 9? Between years 4 and 6? Explain how you arrived at your answers. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 11. How might you estimate the rate that Mike’s salary would be growing at the end of the 5th year with the company? _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 12. Now find the equation of the secant line connecting the points (4.9, 58767) and (5.1, 59972) ____________________________________ Copyright © Casio, Inc. Activity 4 • Calculus with the Casio fx-9750G Plus 23 Name _____________________________________________ Class ________ Date ________________ Activity 4 • Introduction to Derivatives 13. Graph the model you created in item 9, and the equation of the line from item 12 in the following viewing window: Copy the graph and explain what you see: _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 14. The derivative of a function at a point (also known as the instantaneous rate of change) is the same as the slope of the line tangent to the function at that point. Based upon your exploration what could you estimate the derivative of your salary model to be at the end of the 5th year? And how does that translate to Mike’s salary growth rate during that same time period? _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ Extension Given the function f(x)= 3x2 – 2x + 1 find a good estimate for the equation of the line tangent to f(x) at x = 2. Explain your process and how accurate you think you are. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 24 Calculus with the Casio fx-9750G Plus • Activity 4 Copyright © Casio, Inc. Calculator Notes and Answers for Activity 4 1. -1 2. Answers will vary: A complete answer should include a mention of the relative change of a decrease in output by 1 for every increase in the input of 1. 3. -1 4. Answers will vary. However all should include mention that they have the same slope. Plotting the points using the STAT mode will also show that they are on the same line. Care should be taken to point out that JUST because they share a slope does not put them on the same line. 5. Answers will vary. Any other points that have slopes of –1 will work, however, if the answer given to #4 includes the co linearity of the points, then the additional points chosen should also be on that same line. 6. Answers will vary, but should include a mention that his salary has raised an average of $3500 per year over the time period in question. 7. $73,500 This answer can be found by either using the slope or creating the equation of the line connecting the points (1,42000) and (3, 49000) and extrapolating. 8. That actual salary would be greater, thus the growth rate will have had to have been greater at some point for that to take place. If a numerical comparison of the growth rates are attempted, it must be made clear by the student what they are using to create that new comparison and they should be prompted to explain why they have made that choice. 9. A good answer should be the creation of the quadratic equation that results from using the three points (1,42000) , (3, 49000) and (10, 100000). 10. Answers will vary. Most students will likely find the values of the model associated with 4, 6, 9, and using the given value at 10 and find the slopes of the respective secant lines. Some students may begin to suggest that because of the function behavior, these secant values are not good predictors. Between 4 and 10: Average increase is $7706 per year Between 4 and 9: Average increase is $7286 per year Between 4 and 6: Average increase is $6024 per year Copyright © Casio, Inc. Activity 4 • Calculus with the Casio fx-9750G Plus 25 Calculator Notes and Answers for Activity 4 11. Answers will vary. Some students might take the growth between 4 and 5 [$5603] and then 5 and 6 [6445] and take the average [$6024] some may begin to estimate closer, perhaps anticipating the question asked in item 12, some may estimate over an even closer slope interval. Care should be taken to make sure that the students continue to use slope and discuss rate of change and not simply plug 5 into some model and use the output for the answer to the question. 12. y = 6025x + 29244.50 13. While answers will vary, a good answer should point out that the parabola is the model of the actual data and the line is the secant line connecting the two given points. Some answers may being to bring up the concept of the tangent line and it’s very close relationship to the curve at the point of tangency. 14. The actual value of the derivative at 5, to the nearest cent is $6023.81 This is close to the secant line slopes as the student gets closer and closer to 5 from either side. Here a discussion of limits as it pertains to the finding of the slope is also a good extension. Extension Answers will vary, the actual answer is y = 10x – 11 . Care should be taken to be sure that students don’t simply use the calculator function to create the line without being able to communicate the connection between the slope of the secant line/tangent line and the value of the function at x = 2. The student estimation of accuracy will depend upon their process. 26 Calculus with the Casio fx-9750G Plus • Activity 4 Copyright © Casio, Inc. Calculator Notes and Answers for Activity 4 How to do a regression on the Casio fx-9750 Plus 1. From the MENU press 2 (STAT) 2. Input the x-values into List 1, and the y-values into List 2. 3. Press F2 (CALC), then F3(REG). 4. Your basic menu choices then become: F1(linear), F2(med-med line), F3(quadratic), F4(cubic), F5(quartic), F6(next page). 5. After choosing the model you want, the next screen will produce the values and the general model. Copyright © Casio, Inc. Activity 4 • Calculus with the Casio fx-9750G Plus 27 Calculator Notes and Answers for Activity 4 You also have the option of graphing the points, creating and copying the model from there. a) Start at the STAT menu, put the values in the lists as you need, this time press F1(GRPH), then choose F1(GPH1). The calculator will set a proper window and plot the points. b) You now have the same model choices along the F1-F6 keys. c) After you make your choice it will create the model and give you the options to draw it, and or copy it to the function grapher. d) Choose F5 (COPY) and it will take you to the Y= screen where you can choose the place you want to put it and press EXE to store the entire function which you can then access at any time by going to the GRAPH section from the main MENU. e) If you choose DRAW it will draw the model through the points you’ve graphed. (b) (c) (d) (e) (Accessing the GRAPH section and the newly stored function) 28 Calculus with the Casio fx-9750G Plus • Activity 4 Copyright © Casio, Inc. Activity 5 Being Locally Linear Teaching Notes Topic Area: Derivatives and Slope Class Time: an exploration during the first part of a class period while connecting the slope of a function to the derivative Overview This activity will begin to bring home the point that as the behavior around a single point on a differentiable function is examined, the function will "flatten out" and very much resemble the behavior of a line drawn through the point of interest. The example given should motivate a discussion of what it means to be locally linear with regard to a differentiable function. Objectives • To connect the much earlier concept of linear slope to the examination of the rate of change of a function and the idea of what a derivative is • To be able to understand and communicate the visual and numerical ideas of linear slope and its relationship to the instantaneous rate of change of any function Getting Started Using the Casio fx-9750G Plus, have students work in pairs or small groups arranged prior to beginning the activity to maximize student involvement and ownership of the results. Prior to using this activity: • Students should be able to produce and manipulate graphs of functions manually and with the graphing utility. • Students should have an understanding of "decimal" and "standard" window and how to easily produce them. • Students should be able to use Zoom features of the graphing utility to examine specific parts of the graph. • Students should have an understanding of slope of a line as a rate of change. Ways students can provide evidence of learning: • Students should be able to produce graphs of functions and communicate changes taking place to the appearance of a function as they zoom in on a particular value. Common mistakes to be on the lookout for: • Not understanding the zoom process and what is taking place Copyright © Casio, Inc. Activity 5 • Calculus with the Casio fx-9750G Plus 29 Name _____________________________________________ Class ________ Date ________________ Activity 5 • Being Locally Linear Introduction This activity will begin to bring home the point that as the behavior around a single point on a differentiable function is examined, the function will "flatten out" and very much resemble the behavior of a line drawn through the point of interest. Using the Casio fx-9750G Plus you will be working in pairs or small groups. Problems and Questions 1. Graph the function y = x2 – 2x – 3 in the viewing window. Record the results below. 2. What can you say about the slope of the function over the viewing window? _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 3. Set your zoom factors to: 30 Calculus with the Casio fx-9750G Plus • Activity 5 Copyright © Casio, Inc. Name _____________________________________________ Class ________ Date ________________ Activity 5 • Being Locally Linear Trace to x = 2 and zoom in at that point. Record your results below. 4. Using the trace function, record both the x and y values immediately above and below x = 2: x y 2 -3 5. Find the equation of the line connecting the first and third points in your table above. ____________________________________ 6. Graph the line along with the original function in the last window you have and record the results below. 7. Zoom in on both at x = 2 and describe what you see. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ Copyright © Casio, Inc. Activity 5 • Calculus with the Casio fx-9750G Plus 31 Name _____________________________________________ Class ________ Date ________________ Activity 5 • Being Locally Linear 8. As the behavior of a function is examined closer and closer to a particular point of interest, in many cases the function begins to "flatten out", ie, become approximately linear over a very small neighborhood around the particular point of interest. This behavior is called being "locally linear" and for this small interval can be very closely approximated by examining the behavior of the line tangent to the graph at the particular point of interest. With this in mind, examine the graph of y = Sin(x), with the settings in radian mode, in the same window used at the beginning of this activity. Record what you see below. Then, change the setting to degree mode and explain why the results change in light of this activity. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 32 Calculus with the Casio fx-9750G Plus • Activity 5 Copyright © Casio, Inc. Calculator Notes and Answers for Activity 5 1. 2. Answers will vary: A good answer should minimally contain comments about the slope changing throughout the behavior of the function. [A more "advanced" answer would contain comments about the slope changing from negative to positive, and perhaps even mentioning where the slope is zero. 3. 4. x y 1.975 -3.049375 2 -3 2.025 -2.949375 Copyright © Casio, Inc. Activity 5 • Calculus with the Casio fx-9750G Plus 33 Calculator Notes and Answers for Activity 5 5. y = 2x – 6.999375 6. 7. This is what the calculator will show. A good answer will include comments that the line and the function begin to be very "close together" around the value of x = 2. Some students with greater insight might begin to discuss the line being very close to tangent (Care should be taken to point out that while it "looks" pretty tangent, the line being discussed is not tangent, but a secant line in a very small neighborhood of x = 2) For some students an extra zoom or two might clarify the idea being presented. 8. in radian mode in degree mode The goal here is for students to realize that if the mode is changed to degree, they are now looking at a graph that is being produced over only a neighborhood +6.3 degrees away from Sin(0) thus creating a graph very close to y = 0 for that interval. Note, students should also be encouraged to zoom around the Sine graph at any point and be asked to communicate the fact that relatively few zooms will produce a very "linear" looking graph. All explanations should be accompanied by a description of the window that is producing the viewed result. 34 Calculus with the Casio fx-9750G Plus • Activity 5 Copyright © Casio, Inc. Activity 6 Continuity Meets Differentiability Teaching Notes Topic Area: Derivatives and Continuity Class Time: an exploration during the first half of a class period to point out visually that continuity is a necessary but not sufficient condition for differentiability. Overview This activity will begin to extend the idea of local linearity and derivative. It will also connect those concepts to continuity and point out that continuity is a necessary but not sufficient condition for differentiability. The connections will be made visually using the idea of local linearity (or what happens when it’s missing). Symbolic derivatives will, where appropriate, be used to support these findings. Objectives • To connect the ideas of slope, local linearity and differentiability • To be able to understand and communicate the idea that continuity alone does not guarantee that a function has a derivative Getting Started Using the Casio fx-9750G Plus, have students work in pairs or small groups arranged prior to beginning the activity to maximize student involvement and ownership of the results. Prior to using this activity: • Students should be able to produce and manipulate graphs of functions manually and with the graphing utility. • Students should have had an introduction to basic symbolic derivatives to make an easier connection to the visuals. • Students should be able to use Zoom features of the graphing utility to examine specific parts of the graph, including setting the zoom factors. • Students should have an understanding of slope of a function at a point as the visual presentation of the derivative. Ways students can provide evidence of learning: • Students should be able to produce graphs of functions and communicate why a certain function may not have a derivative at a certain point. • Students should be able to, where appropriate, back up their graphical presentation with symbolic analysis. Common mistakes to be on the lookout for: • Not understanding the zoom process and what is taking place • Not being able to communicate the concept of derivative verbally • Entering the rational exponents incorrectly resulting in the calculator producing a graph different that the one desired Copyright © Casio, Inc. Activity 6 • Calculus with the Casio fx-9750G Plus 35 Name _____________________________________________ Class ________ Date ________________ Activity 6 • Continuity Meets Differentiability Introduction This activity will begin to extend the idea of local linearity and derivative. It will also connect those concepts to continuity. The connections will be made visually using the idea of local linearity (or what happens when it’s missing). Problems and Questions Explore the behavior of the function: y = x2/3 around the value x = 2. 1. Sketch the graph of the function y = x2/3 in the INIT default viewing window. Record the graph below and describe what you see. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 2. Trace to the value of x = 2 and with your zoom factors set to 4 for X and Y, zoom in twice. Record what you see and explain what is going on. 3 2 1 1 2 3 _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 36 Calculus with the Casio fx-9750g Plus • Activity 6 Copyright © Casio, Inc. Name _____________________________________________ Class ________ Date ________________ Activity 6 • Continuity Meets Differentiability 3. Using trace, fill in the following values for the function accurate to 5 decimal places. x y 2.0125 Pt 1 2.00625 Pt 2 2 Pt 3 1.99375 Pt 4 1.9875 Pt 5 4. Calculate slopes of Pt 1 & Pt 2, then Pt 2 & Pt 3, then Pt 3 & Pt 4, Then Pt 4 & Pt 5 and record them as Slope 1, Slope 2, Slope 3, and Slope 4: Slope 1 Slope 2 Slope 3 Slope 4 5. What do your results indicate? Explain how the graph you saw either agrees or disagrees with those results. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 6. Now let’s examine the same function around the point x = 0 by graphing the function in the INIT window, tracing to x = 0, and with the zoom factors still set at 4, zoom in twice. Record the graph below. Copyright © Casio, Inc. Activity 6 • Calculus with the Casio fx-9750G Plus 37 Name _____________________________________________ Class ________ Date ________________ Activity 6 • Continuity Meets Differentiability Describe what you see. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 7. Using trace, fill in the following values for the function accurate to 5 decimal places. x y -0.0125 Pt 1 -0.00625 Pt 2 0 Pt 3 0.00625 Pt 4 0.0125 Pt 5 8. Repeat the same slope procedure as before: Calculate slopes of Pt 1 & Pt 2, then Pt 2 & Pt 3, then Pt 3 & Pt 4, Then Pt 4& Pt 5 and record them as Slope 1, Slope 2, Slope 3, and Slope 4: Slope 1 Slope 2 Slope 3 Slope 4 9. What do these results indicate? Compare them to the results from the exploration of the graph around x = 2. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 38 Calculus with the Casio fx-9750g Plus • Activity 6 Copyright © Casio, Inc. Name _____________________________________________ Class ________ Date ________________ Activity 6 • Continuity Meets Differentiability 10. While continuity is a necessary condition for a function to have a derivative at that same point, it is not a sufficient condition as these two examples indicate. The function explored is both continuous and differentiable at x = 2, however, it is continuous but NOT differentiable at x = 0. Use symbolic derivatives to support the visual evidence found in these explorations. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 11. Can you come up with some other simple functions that might provide places where the function is continuous and differentiable at one point in its domain, and continuous but NOT differentiable at another point? Copyright © Casio, Inc. Activity 6 • Calculus with the Casio fx-9750G Plus 39 Calculator Notes and Answers for Activity 6 1. Descriptions will vary. A good answer will include a statement about there being a hard corner at x = 0. 2. The graph should be virtually linear, while descriptions will vary, there should be a comment about the "straightening out" of the function. Answers may include comments about seeing a good "linear approximation" of the function at x = 2. There should also be comments regarding the continuity around x = 2. 3. Using trace, fill in the following values for the function accurate to 5 decimal places. x y 2.0125 1.59401 Pt 1 2.00625 1.59071 Pt 2 2 1.58740 Pt 3 1.99375 1.58409 Pt 4 1.9875 1.58078 Pt 5 4. 40 Slope 1 0.528 Slope 2 0.5296 Slope 3 0.5296 Slope 4 0.5296 Calculus with the Casio fx-9750g Plus • Activity 6 Copyright © Casio, Inc. Calculator Notes and Answers for Activity 6 5. A good answer will include statements about the slopes being the same and the graph becoming linear around the point x = 2. The graph should show a picture that is highly linear in the small neighborhood of x = 2. 6. In stark contrast to the prior exploration, a good answer should include comments about the graph NOT straightening out, (becoming locally linear). There should also be some comments about the continuity being maintained. 7. x y -0.0125 .05386 Pt 1 -0.00625 .03393 Pt 2 0 0 Pt 3 0.00625 .03393 Pt 4 0.0125 .05386 Pt 5 8. Slope 1 -3.188 Slope 2 -5.4288 Slope 3 5.4288 Slope 4 3.188 9. Answers will vary. A good answer should include a direct comparison indicating that the graph is not becoming locally linear around x = 0, while it did "straighten out" around x = 2. The idea that the graph is continuous at both x = 0 and x = 2 should be discussed. 2 10. dy 3 x dx ( ) = 23 x -1 3 A good answer will point out that the derivative at x = 2 exists (and = .52913, very close to the value found in the exploration). However, the derivative at x = 0 does not exist (division by 0). In fact, repeated zooming around x = 0 will continue to provide the same slope with different signs on either side of x = 0. Copyright © Casio, Inc. Activity 6 • Calculus with the Casio fx-9750G Plus 41 Activity 7 Derivative Behavior of Polynomials Teaching Notes Topic Area: Derivatives Overview This activity will lead students to make connections between the behavior of some well known polynomial functions and their derivatives. They will be asked to plot the functions, confirm the expected behavior using the grapher and then overlay the derivative, confirming again using the grapher. Objectives • To be able to express verbally and graphically the behavior of some well known functions • To be able to understand and communicate the behavior of the derivative of these well known functions to the function itself • To make sure that students can express the behavior of the derivative as producing output values relative to the SLOPE of the original function, and not simply compare output values to output values Getting Started Using the Casio fx-9750G Plus, have students work in pairs or small groups arranged prior to beginning the activity will provide students with opportunity to exchange ideas. Prior to using this activity: • Should be able to produce and manipulate graphs of functions manually and with the graphing utility • Should have had an introduction to basic symbolic derivatives to make an easier connection to the visuals • Should have a basic understanding of the transformations of polynomial functions • Should be able to use the Casio fx-9750G Plus to graph a derivative How to graph a derivative on the Casio fx-9750G Plus. • In the GRAPH Menu, in the Y= (entry) screen 42 • Press OPTN. • Press F2 (CALC). • Press F1 (d/dx) • Entry syntax: d/dx (function, x) Example: Calculus with the Casio fx-9750g Plus • Activity 7 Copyright © Casio, Inc. Name _____________________________________________ Class ________ Date ________________ Activity 7 • Derivative Behavior of Polynomials Introduction This activity will begin to extend the idea of local linearity and derivative. It will also connect those concepts to continuity. The connections will be made visually using the idea of local linearity (or what happens when it’s missing). The derivative of a function represents the behavior of the slope of the function at each point along its domain. The goal of this activity is to have you able to make the connections to the picture of the function and the picture of the behavior of the slope of the function. Problems and Questions 1. Draw the graphs of the following functions in the window. y = 2x 2. y = 2x + 5 y = 2x – 3 Describe the behavior of the slope of each function. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 3. Using your calculator, draw the function y = 2x – 3 and the graph of its slope on the same axes. Copy it below. 4. Does this agree with what you expected to see? Explain. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ Copyright © Casio, Inc. Activity 7 • Calculus with the Casio fx-9750G Plus 43 Name _____________________________________________ Class ________ Date ________________ Activity 7 • Derivative Behavior of Polynomials 5. Given the general equation of a linear function: ax + by = c , generalize the relationship between the linear function and its derivative. Provide some examples to support your hypotheses. 6. Using the same window as before, draw the graph of: y = x2 on the axes shown below. Confirm the behavior on your calculator. 7. Describe the behavior of the slope of the function over the following intervals: • (– 0) _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ • (0, ) _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 44 Calculus with the Casio fx-9750g Plus • Activity 7 Copyright © Casio, Inc. Name _____________________________________________ Class ________ Date ________________ Activity 7 • Derivative Behavior of Polynomials 8. Based upon your knowledge of what a derivative is, what would you say the derivative of the function is when x = 0? Explain. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 9. Sketch the function over again on the axes provided below and then overlay what you think the behavior of the derivative would look like. 10. Use your grapher to produce the picture of the actual derivative, does it agree with the graph you produced manually? 11. Now try the same procedure with the following function: y = –3(x–2)2 +2 The function and derivative manually: Copyright © Casio, Inc. The function and derivative by calculator: Activity 7 • Calculus with the Casio fx-9750G Plus 45 Name _____________________________________________ Class ________ Date ________________ Activity 7 • Derivative Behavior of Polynomials 12. Given the general equation of a quadratic function: y = –ax2 + bx +c, generalize the relationship between the quadratic function and its derivative. Provide some examples to support your hypotheses. 13. Given that the general form of a polynomial is y = –anxn + an–1xn–1 + ... + a0 make a general statement about any polynomial function and its derivative. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 14. Provide one fourth degree example to support your conclusion. 46 Calculus with the Casio fx-9750g Plus • Activity 7 Copyright © Casio, Inc. Calculator Notes and Answers for Activity 7 1. 2. Answers will vary, but the goal is to have students discuss that the slopes are all the same (lines are parallel). A likely answer will also include a comment that the slope = 2 for each line. 3. 4. Answers will vary. A complete answer should contain a statement regarding the fact that the slope is constant therefore the graph of the derivative should be a horizontal line. 5. Answers will vary. A complete answer should contain a statement that the slope of the line will always be a horizontal line. y = –a/b All provided examples should contain linear functions and horizontal lines as the derivative sketches. Students thinking farther ahead may start with a horizontal line as an example and then show the line y = 0 as the derivative. 6. 7. (– 0) A complete answer should cover the fact that in this entire interval the slope is negative but changing. Some answers may include statements about the slope "slowing down" or being smaller or less as the interval approaches 0 [alternately may include statements about the slope "speeding up" as the interval moves away from zero]. Care should be taken that the students are talking about the behavior of the slope relative to the values of x. (0, ) A complete answer should cover the fact that in this entire interval the slope is positive but changing. Some answers may include statements about the slope "speeding up" or being larger or more as the interval moves away from 0. Copyright © Casio, Inc. Activity 7 • Calculus with the Casio fx-9750G Plus 47 Calculator Notes and Answers for Activity 7 8. A complete answer should include a statement that the derivative = 0 @ x = 0. The explanation could use the difference quotient/limit approach using small values from the graph, or could make the connection that the tangent is horizontal at x = 0. 9. 10. Same graph as above. They should agree. 11. Both graphs should agree, if not, further discussion needs to take place about the derivative representing the picture of the slope. 12. A complete answer should contain statements that the derivative of a quadratic function will always be linear. Students should be very clear that the line exits above the x-axis when the slope of the function is positive, has a root at the vertex of the parabola, and exits below the x-axis when the slope is negative. Examples should be consistent. Require them to verbalize their support choices. 13. The goal is to have students recognize that the derivative of any polynomial will be another polynomial of one degree less. Good answers will also contain state ments consistent with the fact that the derivative graph is above the x-axis when the slope of the function is positive, has a root at any vertex, and exits below the x-axis when the slope is negative. This might require further investigation. This is also a good lead into the power rule for derivatives. 14. Answers will vary, one example provided here: 48 Calculus with the Casio fx-9750g Plus • Activity 7 Copyright © Casio, Inc. Activity 8 Derivative Behavior of Common Trigonometric Functions Teaching Notes Topic Area: Derivatives Overview This activity will lead students to making connections between the behaviors of some well known trigonometric functions and their derivatives. They will be asked to plot the functions, confirm the expected behavior using the grapher and then overlay the derivative, confirming again using the grapher. The derivative of a function represents the behavior of the slope of the function at each point along its domain. The goal of this activity is to have students make the connections to the picture of the function and the picture of the behavior of the slope of the function. Objectives • To be able to express verbally and graphically the behavior of some well known trigonometric functions • To be able to understand and communicate the behavior of the derivative of these well known functions to the function itself • To make sure that students can express the behavior of the derivative as producing output values relative to the SLOPE of the original function, and not simply compare output values to output values Getting Started Using the Casio fx-9750G Plus, have students work in pairs or small groups arranged prior to beginning the activity to share ideas. Prior to using this activity: • Should be able to produce and manipulate graphs of functions manually and with the graphing utility • • Should have a basic understanding of the behavior and appearance of basic trigonometric functions Should be able to use the Casio fx-9750G Plus to graph a derivative How to graph a derivative on the Casio fx-9750G Plus. • In the GRAPH Menu, in the Y= (entry) screen • Press OPTN. • Press F2 (CALC). • Press F1 (d/dx) • Entry syntax: d/dx (function, x) Copyright © Casio, Inc. Example: Activity 8 • Calculus with the Casio fx-9750G Plus 49 Name _____________________________________________ Class ________ Date ________________ Activity 8 • Derivative Behavior of Common Trigonometric Functions Introduction This activity will have you making connections between the behavior of some well known trigonometric functions and their derivatives. You will be asked to plot the functions, confirm the expected behavior using the grapher and then overlay the derivative, confirming again using the grapher. The derivative of a function represents the behavior of the slope of the function at each point along its domain. The goal of this activity is to have you make the connections to the picture of the function and the picture of the behavior of the slope of the function. Problems and Questions Make sure the calculator is in radian mode. 1. Draw the graph of y = Sin(x) in the default initial window and record it here. 2. Describe the slope of the function over the interval [0, 2π] . _____________________________________________________________________________ _____________________________________________________________________________ 3. Using your understanding of derivative as slope, sketch the function, y = Sin(x) and its derivative over the interval [0, 2π] . 4. Using your calculator, produce the same graphs as above. Do the graphs produced agree with what you expected to see? Explain. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 50 Calculus with the Casio fx-9750g Plus • Activity 8 Copyright © Casio, Inc. Name _____________________________________________ Class ________ Date ________________ Activity 8 • Derivative Behavior of Common Trigonometric Functions 5. Draw the graph of y = 2Sin(x), in the interval [0, 2π] and record it here. Then using your knowledge of slope, overlay the graph of the slope function (the derivative). 6. Have your calculator produce the graph of the derivative. Does it agree with your sketch? Explain. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 7. Draw the graph of y = Sin(2x), in the interval [0, 2π] and record it here. Then using your knowledge of slope, overlay the graph of the slope function (the derivative). 8. Have your calculator produce the graph of the derivative. Does it agree with your sketch? Explain. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ Copyright © Casio, Inc. Activity 8 • Calculus with the Casio fx-9750G Plus 51 Name _____________________________________________ Class ________ Date ________________ Activity 8 • Derivative Behavior of Common Trigonometric Functions 9. Draw the graph of y = Cos(x) in the default initial window and record it here. 10. Describe the slope of the function over the interval [0, 2π] . _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 11. Using your understanding of derivative as slope, sketch the function, y = Cos(x) and its derivative over the interval [0, 2π] . 12. Using your calculator, produce the same graphs as above. Do the graphs produced agree with what you expected to see? Explain. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 13. Draw the graph of y = 2Cos(x), in the interval [0, 2π], record it here. Then using your knowledge of slope, overlay the graph of the slope function (the derivative). 52 Calculus with the Casio fx-9750g Plus • Activity 8 Copyright © Casio, Inc. Name _____________________________________________ Class ________ Date ________________ Activity 8 • Derivative Behavior of Common Trigonometric Functions 14. Have your calculator produce the graph of the derivative. Does it agree with your sketch? Explain. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 15. Draw the graph of y =Cos(2x), in the interval [0, 2π] , record it here. Then using your knowledge of slope, overlay the graph of the slope function (the derivative). 16. Have your calculator produce the graph of the derivative. Does it agree with your sketch? Explain. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 17. Compare and contrast the behaviors of the derivatives of the Sine and Cosine functions. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 18. Given the general functions y = A • Sin (x) and y = Sin (Bx), and using the calculator, explore their derivative behaviors for additional values of A and B. Do the same for the Cosine functions and draw a general set of conclusions of the effects of A and B on the derivative behavior. Can you come up with a general symbolic rule using these results? If so, what is it? _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ Copyright © Casio, Inc. Activity 8 • Calculus with the Casio fx-9750G Plus 53 Calculator Notes and Answers for Activity 8 1. 2. Answers will vary. A Good answer will include statements that the slope is π 3π positive (increasing) over the intervals 0, 2 and 2 , 2π and negative (decreasing) over the interval π , 3π . ( 2 2 ) [ ) ( ] A well thought out answer should also include statements that the slope is = 0 at the vertices. 3. Answers may vary but should look like the graph the calculator produces for question #4. 4. If the graphs do not agree in 3 & 4, discussion should take place regarding the differences. 5. The drawn in derivative should look like the result from #6. 6. 54 Calculus with the Casio fx-9750g Plus • Activity 8 Copyright © Casio, Inc. Calculator Notes and Answers for Activity 8 7. 8. Here there is the first real difference that might cause some confusion. The amplitude of the slope is different than the amplitude of the original function. It is difficult to arrive at this just from a graph, a student whose hand sketched graph includes this, has likely already used the symbolic rules or has used some function values to get the actual slopes. 9. . 10. Good answers will be similar to the response to the Sin function indicating that the slope is positive (increasing) over the interval (π, 2π)and negative (decreasing) over the interval (0, π) . A well thought out answer should also include statements that the slope is = 0 at the vertices. 11. 12. Should see same as the answer shown to #11 above. Copyright © Casio, Inc. Activity 8 • Calculus with the Casio fx-9750G Plus 55 Calculator Notes and Answers for Activity 8 13. The graph of y=2Cos(x) 14. The function and its derivative 15. The Graph of y = Cos(2x) 16. The function and its derivative 56 Calculus with the Casio fx-9750g Plus • Activity 8 Copyright © Casio, Inc. Calculator Notes and Answers for Activity 8 17. Answers will vary. Complete answers should include statements that the Sine function derivative produces graphs that look behave like the Cosine function, while the Cosine function derivative produces graphs that seem to be the opposite (negation) of the Sine function. There should also be mention that changing the amplitude of the function is consistent with the amplitude of the derivative, but changing the period of the function is consistent with the period of the derivative, but also changes the amplitude of the derivative. 18. y = A • Sin(x)→ dy = A • Cos(x) dx dy y = Sin(Bx)→ = B • Cos(Bx) dx y = A • Cos(x)→ dy = -A • Sin(x) dx dy y = Cos(Bx)→ = -B • Sin(Bx) dx Copyright © Casio, Inc. Activity 8 • Calculus with the Casio fx-9750G Plus 57 Activity 9 Looking at Relationships Teaching Notes Topic Area: Derivatives and Graphs Overview A great deal of information about a function can be found by analyzing the behavior of the first and second derivatives. This activity will provide a graphical examination of the relationships between the function and its derivatives. Objectives • Be able to explain information about the graph of a function based on the first and second derivatives • Know that the derivative of a function is positive when the function increases, and negative when the function decreases • Know that a positive second derivative means the function is concave upward and a negative second derivative means the function is concave downward Getting Started Using the Casio fx-9750G Plus, students can work this activity independently or in pairs.. Prior to using this activity: • Students should be able to take basic symbolic derivatives. • Students should know the terms relative minimum and relative maximum. • Students should be able to produce the graph of a derivative and second derivative from the calculator. Ways students can provide evidence of learning: • Students should be able to explain how the first derivative yields information about the increasing/decreasing nature of the function. • Students should be able to explain how the second derivative yields information about the concavity of the graph. Common mistakes to be on the lookout for: • Students may understand where a function is increasing or decreasing but they may misinterpret that on the graph as thinking the function is always above or below the x-axis instead of the graph of the derivative being positive/negative. • 58 The speed at which the calculator shows a second derivative graph is relatively slow. Some students may conclude there is no graph being produced. Calculus with the Casio fx-9750g Plus • Activity 9 Copyright © Casio, Inc. Name _____________________________________________ Class ________ Date ________________ Activity 9 • Looking at Relationships Introduction This activity will provide a graphical examination of the relationships between the function and its first and second derivatives. The increasing/decreasing nature of a function can be examined by the positive/negative behavior of its derivative. Similarly, the upward/downward concavity can be examined by the positive/negative behavior of its second derivative. Problems and Questions 1. Graph the function y = 2x3 – 3x2 – 12x + 4 in the window. 2. Record the results here: 3. At what x-values does it appear the function reaches its relative minimum and maximum values? ____________________________________ 4. Using the G-Solve functions, confirm those values and find the minimum and maximum function values. 5. Record the domain interval/intervals where the function increases. ____________________________________ 6. Record the domain interval/intervals where the function decreases. ____________________________________ 7. Explain what kind of values would you expect the derivative to have over the interval where the function increases. ____________________________________ Copyright © Casio, Inc. Activity 9 • Calculus with the Casio fx-9750G Plus 59 Name _____________________________________________ Class ________ Date ________________ Activity 9 • Looking at Relationships 8. Using the d/dx from the OPTN menu, set Y2 to produce the graph of the derivative, graph both the function and the derivative together and record here. (Be sure to LABEL on your sketch which is the function and which is the derivative.) 9. From the graph, what are the y-values of the derivative where the original function has a relative maximum or minimum? ____________________________________ 10. Explain the nature of the y-values of the derivative over the interval(s) where the original function increases. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 11. Explain the nature of the y-values of the derivative over the interval(s) where the original function decreases. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 12. When the derivative crosses the x-axis explain what happens to the graph of the original function? _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 13. Over what x interval(s) does the derivative increase? ____________________________________ 60 Calculus with the Casio fx-9750g Plus • Activity 9 Copyright © Casio, Inc. Name _____________________________________________ Class ________ Date ________________ Activity 9 • Looking at Relationships 14. Over what x interval(s) does the derivative decrease? ____________________________________ 15. If the first derivative is increasing, do you expect the second derivative to be positive or negative? Explain. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 16. If the first derivative is decreasing, do you expect the second derivative to be positive or negative? Explain. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 17. Using the d2/dx2 from the OPTN menu, set Y3 to produce the graph of the second derivative, graph both the first and second derivative together and record here. (Be sure to LABEL on your sketch which is the first and which is the second derivative). 18. Do the graphs produced match your expectations? If not, explain any differences you see. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ Copyright © Casio, Inc. Activity 9 • Calculus with the Casio fx-9750G Plus 61 Name _____________________________________________ Class ________ Date ________________ Activity 9 • Looking at Relationships 19. The graph of a function is concave down when the graph of the first derivative is decreasing. Sketch the portion of the original function that is concave down and record it here. Explain what is true about both the first and second derivatives over the interval you just sketched. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 20. A point of inflection is a point where the concavity changes. Based upon your exploration what is the point of inflection for the original graph? Explain how you know. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 62 Calculus with the Casio fx-9750g Plus • Activity 9 Copyright © Casio, Inc. Calculator Notes and Answers for Activity 9 2. 3. Relative Max at x =-1, Relative Min at x = 2 4. Using the G-Solve functions, confirm those values and find the minimum and maximum function values. 5. Increases: (–, –1) (2, ) 6. Decreases: (–1, 2) 7. Positive values 8. Der Function 9. Y = 0 in both cases 10. Where the function is increasing, the y-values of the derivative are positive 11. Where the function is decreasing, the y-values of the derivative are negative 12. The function reaches a relative extreme point 13. (.5, ) 14. (–, .5) 15. Positive. It should follow the same behavior as the relationship between the original function and its 1st derivative 16. Negative, same reason as above. Copyright © Casio, Inc. Activity 9 • Calculus with the Casio fx-9750G Plus 63 Calculator Notes and Answers for Activity 9 17. 1st Der. 2nd Der. 18. Answers may vary, but they should match. 19. The first derivative is decreasing, the second derivative is negative. 20. The point of inflection is (.5, -2.5). The concavity will change at the root of the second derivative which is x = .5, that produces a y-value of –2.5 in the original function. 64 Calculus with the Casio fx-9750g Plus • Activity 9 Copyright © Casio, Inc. Activity 10 Looking at Slope Fields Teaching Notes Topic Area: Slope Fields Overview A Slope Field is a visual representation of the solution to a differential equation created by creating a series of small linear approximations to the slope at certain points. In this activity students will sketch some slope fields and then confirm their appearance using the calculator. This can be used as a first introduction to the idea of Slope Fields and also as a general introduction to antidifferentiation. Objectives • Understand what a slope field represents in terms of dy/dx • Create and explain a slope field from a given differential equation Getting Started Using the Casio fx-9750G Plus and the Slope Field Program, have students begin the activity independently and then share and discuss their results with another student. Prior to using this activity: • Students should have a working knowledge of differentiation and be conversant with the language of differential equations. • It is not necessary for students to know any symbolic antidifferentiation methods. Ways students can provide evidence of learning: • Students should be able to sketch their own slope fields for a given differential equation over specific grid points using pencil and paper. Common mistakes to be on the lookout for: • Students may misunderstand that the graph being produced is the graph of the solution to dy/dx (the graph of the antiderivative) and be confused when the given equation does not seem to fit the slope field. • For example, the slope field of the expression dy/dx = x, correctly drawn will produce a parabolic fit, students may incorrectly expect a linear fit. Copyright © Casio, Inc. Activity 10 • Calculus with the Casio fx-9750G Plus 65 Activity 10 Slope Field Program This program can be found in the download section of the Casio Education website: http://www.casioeducation.com Notes: • The program stores the slope field in Pic 1 automatically. • The differential equation must be stored in Y1 in the GRAPH menu prior to executing the program. • The optimal window is the INIT window, set while in the GRAPH menu, prior to executing the program. • When the slope field has been created on the calculator press AC/on key to break out of the program. To overlay the slope field on the graph of a proposed solution graph: • Execute the program, press AC/ON when done. • Return to GRAPH menu. • Put the proposed solution graph in Y1. • Draw the graph. • Press OPTN. • Press F1 (Pict). • Press F2 (Rcl). • Press F1 (Pic 1). The picture of the slope field will be placed on the graph of the proposed solution for comparison. 66 Calculus with the Casio fx-9750G Plus • Activity 10 Copyright © Casio, Inc. Name _____________________________________________ Class ________ Date ________________ Activity 10 • Looking at Slope Fields Introduction A Slope Field is a visual representation of the solution to a differential equation created by a series of small linear approximations to the slope at certain points. In this activity students will sketch some slope fields and then confirm their appearance using the calculator. An equation like dy/dx = 2x which contains a derivative is called a "differential equation". The problem becomes finding a function, y in terms of x, when we are given its derivative. Note the phrasing: we are looking for "a solution" and not "the solution". This is due to the fact that the slope will allow us to arrive at a family of curves and missing some initial condition value we will not be able to arrive at a single solution but should be able to make some conclusions as to the appearance of the family of solutions. If we examine the plot of the slope over a series of grid points we end up with a Slope Field. If given the differential equation dy/dx = f(x,y) a plot of short line segments with slopes f(x,y) over specific grip points produces a slope field. This slope field will give you a look at the behavior of the solution to the original differential equation. Problems and Questions 1. Fill in the accompanying table representing the slope of dy/dx = 2x for the grid points shown, [for example at the point (0,0) you should a slope of 0, at (1,2) a slope of 2 etc]. (x,y) dy/dx (x,y) dy/dx (x,y) dy/dx (x,y) dy/d (x,y) -6, -2 -6, -1 -6, -0 -6, 1 -6, 2 -5,-2 -5,-1 -5,-0 -5,1 -5,2 -4,-2 -4,-1 -4,-0 -4,1 -4,2 -3,-2 -3,-1 -3,-0 -3,1 -3,2 -2,-2 -2,-1 -2,-0 -2,1 -2,2 -1,-2 -1,-1 -1,-0 -1,1 -1,2 0,-2 0,-1 0,-0 0,1 0,2 1,-2 1,-1 1,-0 1,1 1,2 2,-2 2,-1 2,-0 2,1 2,2 3,-2 3,-1 3,-0 3,1 3,2 4,-2 4,-1 4,-0 4,1 4,2 5,-2 5,-1 5,-0 5,1 5,2 6,-2 6,-1 6,-0 6,1 6,2 Copyright © Casio, Inc. dy/dx Activity 10 • Calculus with the Casio fx-9750G Plus 67 Name _____________________________________________ Class ________ Date ________________ Activity 10 • Looking at Slope Fields 2. Sketch a small line segment with the slopes calculated, centered at each grid point. For example at the point (1,2) which will have a slope of 2, this is what you should sketch: And for the points (-1,2) (0,2) and (1,2) (with slopes of –2, 0, 2 respectively) you should see this: 3. What familiar family of curves does this slope field seem to indicate is the solution to the given differential equation? Explain. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 68 Calculus with the Casio fx-9750G Plus • Activity 10 Copyright © Casio, Inc. Name _____________________________________________ Class ________ Date ________________ Activity 10 • Looking at Slope Fields 4. Using the Slope Field program, have the calculator produce the slope field for dy/dx = 2x. How does it compare with what you hand sketched? _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 5. Have the calculator draw one member of the family of curves you think is best represented by the slope field, then have it overlay the slope field on the graph produced. Record what you see here: 6. Look at the graph the calculator produced. How does it compare to what you expected to see? _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ Extension: • Use the following differential equations and generate a slope field for each. • Have the calculator generate the slope field • Draw a conclusion about the general solution to the differential equation. a) dy/dx = x2 Copyright © Casio, Inc. Activity 10 • Calculus with the Casio fx-9750G Plus 69 Name _____________________________________________ Class ________ Date ________________ Activity 10 • Looking at Slope Fields b) dy/dx = Sin(x) c) dy/dx = ex 70 Calculus with the Casio fx-9750G Plus • Activity 10 Copyright © Casio, Inc. Calculator Notes and Answers for Activity 10 1. (x,y) dy/dx (x,y) dy/dx (x,y) dy/dx (x,y) dy/d (x,y) dy/dx- -6, -2 -12 -6, -1 -12 -6, -0 -12 -6, 1 -12 -6, 2 12 -5,-2 -10 -5,-1 -10 -5,-0 -10 -5,1 -10 -5,2 -10 -4,-2 -8 -4,-1 -8 -4,-0 -8 -4,1 -8 -4,2 -8 -3,-2 -6 -3,-1 -6 -3,-0 -6 -3,1 -6 -3,2 -6 -2,-2 -4 -2,-1 -4 -2,-0 -4 -2,1 -4 -2,2 -4 -1,-2 -2 -1,-1 -2 -1,-0 -2 -1,1 -2 -1,2 -2 0,-2 0 0,-1 0 0,-0 0 0,1 0 0,2 0 1,-2 2 1,-1 2 1,-0 2 1,1 2 1,2 2 2,-2 4 2,-1 4 2,-0 4 2,1 4 2,2 4 3,-2 6 3,-1 6 3,-0 6 3,1 6 3,2 6 4,-2 8 4,-1 8 4,-0 8 4,1 8 4,2 8 5,-2 10 5,-1 10 5,-0 10 5,1 10 5,2 10 6,-2 12 6,-1 12 6,-0 12 6,1 12 6,2 12 2. The slope field should reflect a parabolic shape. 3. Students should recognize this as a family of parabolas. However, the apparent vertical nature of the horizontal extremities could throw some students off. This presents a good opportunity to discuss the nature of graphical approximations. 4. If the original field was drawn correctly, they should see a graph very similar to what they hand sketched. 5. Answers will vary based upon the chosen graph. Shown here are the graphs for y = x2 and y = x2 – 2. Copyright © Casio, Inc. Activity 10 • Calculus with the Casio fx-9750G Plus 71 Calculator Notes and Answers for Activity 10 6. If done correctly they should be very similar. If not, this presents a good opportunity to discuss why their graphs are not accurate. Extension: a) a cubic family of curves 1 (actual family: y = 3 x2 + c) b) also a trigonometric family (actual family: y = -Cos(x) + c) 72 Calculus with the Casio fx-9750G Plus • Activity 10 Copyright © Casio, Inc. Calculator Notes and Answers for Activity 10 c) an exponential family (actual family y = ex + c) Copyright © Casio, Inc. Activity 10 • Calculus with the Casio fx-9750G Plus 73 Activity 11 Riemann Sums Teaching Notes Topic Area: Anti Derivatives, Integration Overview This activity will present students with the tools to calculate and analyze Riemann sums. They will hand sketch rectangles and use that to approximate the area under a curve. They will then use the calculator to perform increasing numbers of calculations to observe the convergence of upper and lower Riemann Sums with regular partitions as the size of the partitions decrease and the number of rectangles increase. Objectives • Calculate Riemann Sums • Develop an understanding of when Riemann sum approximation will be over or under the actual value of a definite integral • Observe the convergence of the upper and lower Riemann sum values as the number of rectangles increases Getting Started Using the Casio fx-9750G Plus, have students work in pairs or small groups arranged prior to beginning the activity. Students should have a working knowledge of the derivative function as a model for representing average and instantaneous change and should be able to use the Sum and Sequence commands. Prior to using this activity: • Students should have a working knowledge of the derivative function as a model for representing average and instantaneous change. • Students should be able to use the Sum and Sequence commands. • It is not necessary for students to know any symbolic antidifferentiation methods. Ways students can provide evidence of learning: • Students should be able to sketch their own upper and lower rectangles for a given function over specified intervals. • Students should be able to explain the difference in the upper and lower Riemann Sums. Common mistakes to be on the lookout for: • Students may, in calculating the sums by hand, forget to multiply the series sum by the base width. • 74 Students should be given the programs only after they have demonstrated ability to construct a Riemann Sum on their own. Calculus with the Casio fx-9750G Plus • Activity 11 Copyright © Casio, Inc. Activity 11 Riemann Sum Programs Riemann Sum Drawing Program: Teaching Notes These programs can be found in the download section of the Casio Education website: http://www.casioeducation.com Riemann Sum Calculation program: (Calculates both the left and right RSum) Copyright © Casio, Inc. Activity 11 • Calculus with the Casio fx-9750G Plus 75 Name _____________________________________________ Class ________ Date ________________ Activity 11 • Riemann Sums Introduction A Riemann Sum is a method of approximation for calculating the area under a curve. When a function represents change, the area under the curve represents the accumulation of that change. For instance, if you have curve measuring velocity over time, the sums of those velocities over specific time intervals represents the distance traveled during that time period. This activity will present you with the tools to calculate and analyze Riemann sums. You will hand sketch rectangles and use that to approximate the area under a curve. You will then use the calculator to perform increasing numbers of calculations to observe the convergence of upper and lower Riemann Sums with regular partitions as the size of the partitions decrease and the number of rectangles increase. Problems and Questions 1. Assume you are on cruise control driving down a clear highway at a constant rate of 60 miles per hour. Record the graph of your velocity over the first 5 hours on the graph below. 70 60 50 40 30 20 10 1 2 3 4 5 2. Given these conditions, how far would you have gone over the first hour, two hours, and three hours? 1 hour: ____________________________________ 2 hours: ____________________________________ 3 hours: ____________________________________ How did you calculate these results? _____________________________________________________________________________ 76 Calculus with the Casio fx-9750G Plus • Activity 11 Copyright © Casio, Inc. Name _____________________________________________ Class ________ Date ________________ Activity 11 • Riemann Sums 3. What is the area under the curve over the first hour? Two hours? Three Hours? 1 hour: ____________________________________ 2 hours: ____________________________________ 3 hours: ____________________________________ 4. Explain why the answers to item #2 and item #3 agree. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 5. Now assume you have a rate of change that is being modeled by the function y = x2 over the interval [0,5] Sketch the function on these axes: 6. Now, it is not as easy to find the area under the curve over the first three hours as the rate of change itself is changing at each point on the interval. We can, however, approximate the area using rectangles. What if we were to put three rectangles, each one unit long, under the curve, with measuring the height of the rectangles at the RIGHT or upper endpoint of the interval. Sketch those rectangles and record here: 7. Calculate the area represented by those three rectangles. Does it seem that approximation will be more or less than the actual area? Why? _____________________________________________________________________________ _____________________________________________________________________________ Copyright © Casio, Inc. Activity 11 • Calculus with the Casio fx-9750G Plus 77 Name _____________________________________________ Class ________ Date ________________ Activity 11 • Riemann Sums 8. Now sketch the same rectangles, this time using the LEFT or lower endpoint to mark the height of the triangles. 9. Calculate the total area represented by these lower rectangles. Does it seem that this approximation will be more or less than the actual area? Why? _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 10. Leaving the interval the same, now change from 3 to 6 equally spaced rectangles. (What will happen to the width of the rectangle in this case?), sketch the rectangles and record the areas first for the RIGHT endpoint, then the LEFT endpoint rectangles. 11. Right Endpoint Left Endpoint Area: _________________________ Area: _________________________ Using the RSUMCALC program, find the areas over the same intervals for 12, 18, and 36 equally spaced rectangles. (Remember to be aware of what it does to the width of each rectangle.) Number of partitions 78 Upper Riemann Sum Calculus with the Casio fx-9750G Plus • Activity 11 Lower Riemann Sum Copyright © Casio, Inc. Name _____________________________________________ Class ________ Date ________________ Activity 11 • Riemann Sums 12. What happens to the width of each rectangle as the number of rectangles increase? _____________________________________________________________________________ 13. What happens to the areas as the number of rectangles increase? Why do you suppose this happens? _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ 14. In your own words explain what these Upper and Lower Riemann Sums represent. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ Extension: Calculate the Upper and Lower Riemann Sums for each of the following functions over the interval indicated and for the number of partitions indicated: Function 1 x Interval Number of partitions [1,4] 4 Upper Riemann Sum Lower Riemann Sum 8 12 16 3x+5 [0, 6] 3 6 10 20 Sin(x) (radian mode) [0, π] 2 4 8 16 Copyright © Casio, Inc. Activity 11 • Calculus with the Casio fx-9750G Plus 79 Calculator Notes and Answers for Activity 11 1. 70 60 50 40 30 20 10 1 2 3 4 5 2. 60 Miles, 120 Miles, 180 Miles. Rate x time = distance 3. 60 units2, 120 units2, 180 units2 4. Answers will vary, a complete answer should contain statements about the area being an accumulation of the change. 5. 6. 7. (1)2 • 1 + (2)2 • 1 + (3)2 • 1 =14 Answers should clearly state the area is more than what is covered, by the function, as the rectangles are above the function in each case. 80 Calculus with the Casio fx-9750G Plus • Activity 11 Copyright © Casio, Inc. Calculator Notes and Answers for Activity 11 8. (Note: Some students may be confused in not seeing the three rectangles, they need to be reminded that the left endpoint here is at x = 0 so there will be no rectangle shown.) 9. (0)2 • 1 + (1)2 • 1 + (2)2 • 1 = 5 Answers should clearly state the area is less than what is covered by the function, as the rectangles are below the function in each case. 10. Right endpoint Area: 11. Left Endpoint 11.375 Area: 6.875 Number of partitions Upper Riemann Sum Lower Riemann Sum 12 10.16 7.91 18 9.76 8.26 36 8.62 7.92 12. The widths get smaller 13. The areas begin to converge toward a common value. The "why" answers will vary, but a good answer should contain a statement about the error decreasing as the amount of rectangles increase. 14. Answers will vary, but should indicate statements about increasingly accurate approximations of the area under a curve. Some students who may have seen the process of integration might connect the Riemann Sum to the definite integral over that same interval. Copyright © Casio, Inc. Activity 11 • Calculus with the Casio fx-9750G Plus 81 Calculator Notes and Answers for Activity 11 Extension: Function 1 x 3x+5 Sin(x) (radian mode) Interval Number of partitions Upper Riemann Sum Lower Riemann Sum [1,4] 4 1.71 1.15 8 1.54 1.26 12 1.48 1.30 16 1.46 1.32 3 66 102 6 75 93 10 78.6 89.4 20 81.3 86.7 2 1.57 1.57 4 1.90 1.90 8 1.82 1.97 16 1.99 1.99 [0, 6] [0, π] Note: Care should be taken to remind students that there is no firm "rule" for whether the left or right endpoint rule will give the best approximation, it depends upon the function. The goal of this activity is simply to bring out the tools for the use of Riemann Sums. 82 Calculus with the Casio fx-9750G Plus • Activity 11 Copyright © Casio, Inc. Appendix Overview of the Calculator Functions This is not meant to be an exhaustive tutorial. This overview is to provide a starting point when beginning to use the Casio fx-9750G Plus graphing calculator. Run Function The Casio fx-9750G Plus is just like any other calculator when in the run function. There are a few extra functions it offers that other calculators do not. Fraction calculations: Using the ab/c key, you can enter numbers as fractions and do any normal mathematical operations. The EXE will give you answers in fraction or mixed number format. You can change from fraction format to a decimal format by pressing the F↔D key and vice versa. You may also change the mixed number to an improper fraction by pressing SHIFT ab/c for d/c. This function will only work if the expression is entered in fraction format originally. d/c ab/c Editing: If you need to make a change in a previous calculation, use the deep recall to retrieve the equation. In the Run function, press the AC/ON to clear the screen. Then use the Up Arrow key to scroll through the previous equations until you come to the one you want to edit. Press the left arrow key to make the changes desired. *Note: The previous calculations will be lost as soon as you exit the Run function. Probability: In the Run function, press the OPTN (option) menu. Press the F6 (arrow right) for more options. Then press the F3(PROB) function. This will allow you to do factorials, permutations, combinations, and random numbers. Copyright © Casio, Inc. Appendix • Calculus with the Casio fx-9750G Plus 83 Statistics Function Entering Data: If there are statistics in the lists, press F6 for more options, then press F4(DEL-A) and F1(YES). Enter the data in List 1 by entering the number and pressing EXE. When you press EXE the number will show up in the list. After the list is complete, you can begin calculating the statistics. Statistics: Press F2 to show the CALC options. Check the F6(SET) for the setup of the calculations. Make sure that the 1Var XList is set at List 1 by highlighting it and pressing F1. Press EXIT to get back to the previous screen and press F1(1VAR). Arrow down to see other statistics. Exit out: Press EXIT twice to get back to the statistics lists. Box and Whiskers: Using the same data as in a histogram, you can make a box and whiskers graph. From the graph option menu, press F6(SET). Arrow down to graph type and set to Box. Press EXIT to get back to the previous screen. Check to see if graph 1 is selected in the SEL option, and press F6(DRAW). Statistics can be seen from here by pressing F1(1VAR). SHIFT F1(Trace) will allow you to trace the graph for quartile ranges using the right arrow key. Press EXIT twice to return to the STAT window. Scatter Plot: Enter data in lists 1 and 2 as done above. Press F1(GRPH) options then F6(SET). Arrow down to graph type and set at SCATter. Make sure that XList and YList are set at List1 and List2 respectively. Press EXIT and then press F1 to see Grph1. If the graph does not appear, EXIT and press F4(SEL) to make sure only Graph 1 is turned on. Line of Best Fit: On the screen with the graph are options for different kinds of regressions. Press the regression you think will fit the data the best and the calculator will give you a regression analysis. Notice the choices at the bottom of the screen. If you press the F5(COPY), it will copy the line to the GRAPH function so that you can see the graph later, if needed. After you press F5(COPY), press EXE to store the graph and it will take you back to the regression information. Press F6(DRAW) function to see the graph. 84 Calculus with the Casio fx-9750G Plus • Appendix Copyright © Casio, Inc. Matrix Function Entering Matrices: In the matrix function, you will see a list of matrices you can enter. With the first available matrix highlighted, enter the number of rows needed first and then the number of columns. Press EXE after each entry. After the second EXE, the actual matrix will show up on the screen. When you enter the matrix, the numbers will go across first and then down to the next row. From this screen, you can do basic matrix operations by pressing F1(R-OP). You can add and switch rows here. Matrix Operations: Most other basic matrix operations can be done in the Run function. Press OPTN in the Run function and then press F2(MAT) for matrix operations. You can add and subtract matrices, find the determinant, transpose, augment, send a matrix to lists, and finding the identity of a matrix. After the function has been entered, press EXE to see the resulting matrix. List Function Sorting data: The data entered in the statistics lists will show up here as well. If you would like to erase the lists, highlight the list and press F4(DEL-A). Then enter the new data as before. You may sort the data by pressing F1. It will prompt you by asking how many lists, then it will prompt you to select a list. When sorting more than one list, the calculator will ask for a base list and a second or third list. You can sort the data in ascending (SRT-A) order or descending order (SRT-D). Other Operations: Press the OPTN key for more options in the List function. These options include moving matrices, finding the minimum, maximum, mean, median, mode, and product of lists. Copyright © Casio, Inc. Appendix • Calculus with the Casio fx-9750G Plus 85 Graph Function Entering Equations: When you enter the graph function, the graph edit screen is up. Type in an equation using the XT or the letter X button for the variable X. You can change the Y= to an inequality by pressing F3(TYPE) before entering in the equation. Press F6 for more options for the equation. Press EXIT to get back out to the original options. Press F6(DRAW) to see the graph. View of the Graph: If the graph cannot be seen, press SHIFT F3(V-window). Press F3(STD) for a standard 20-interval view, press F1(INIT) for initial settings, and F2(TRIG) for a standard trigonometry function. You may also set the setting manually and store that setting by pressing F4(STO). Press EXIT to get back to the graph-edit screen. You may view more than one graph at a time by pressing F1 on each graph you want to show. This is how you can solve systems of equations. Dual Screen: In the graph function, press SHIFT(SETUP). Arrow down to Dual Screen and switch it to Graph by pressing F1. Press EXIT to get to the previous screen. Notice when you press DRAW, whatever picture of the graph was last seen will be on the left, and the frame on the right will be blank. Zoom in on the graph and that part of the graph will show up on the right. You may also use the dual screen for a graph and a table. Go into the setup again and change the dual screen to G to T for Graph to Table. Press EXIT to get back to the edit screen. The graph will appear with a blank table. Press SHIFT F1(Trace) to trace values on the graph. To put those values in the table press EXE. Put multiple values in the table by pressing EXE repeatedly. Solving for a Specific Point: After you graph two inequalities you can find the point of intersection of those graphs by using the G-solve function. Press F5(G-Solv) (you do not have to press the SHIFT key if the graphs are already on the screen). Press F5(ISCT). The calculator will think for a few seconds and then trace the graph to the intersection point for you. You can also find the roots, maximum, minimum, y-intercept, y and x coordinates for given values, and definite integrals. 86 Calculus with the Casio fx-9750G Plus • Appendix Copyright © Casio, Inc. Dynamic Function Built in List of Equations: The dynamic function is used to demonstrate the effect of changing certain variables in an equation. There are seven built in functions that are common in basic algebra and trigonometry. To get to this list, press F5(B-IN) after you are in the Dynamic function. To choose one of the built-in functions, highlight and press F1(SEL). Setting the Variable and Speed: After choosing the equation in question, press F4(VAR) to choose the active variable. Highlight the variable you want to change and enter in values for the other variables. Notice the arrow to the right of the Dynamic Variable. That is the speed indication. Press F3(SPEED) to change the speed of the dynamic function. Highlight the desired speed and press F1(SEL). Each speed has a different symbol. This symbol is shown next to the variable on the previous screen. Press EXIT to return to that screen. Setting the range: In the dynamic variable screen, press F2(RANG). Set the start and end of the range you want to show. The pitch is the interval between numbers in range. Press EXIT to get to the previous screen. Viewing the dynamic graph: In the Dynamic Variable screen, press F6(DYNA). When you do this, the calculator will say "One Moment Please" while loading the graph. Once the graph is on the screen, press EXE to see the change in the variable. When you press the EXE, notice the active variable will change. Entering equations manually: If the equation you need is not in the built-in list you can enter the equation in manually. Use the alpha key and letters to enter variables. The variables entered in manually will be available in the dynamic function. Copyright © Casio, Inc. Appendix • Calculus with the Casio fx-9750G Plus 87 Table Function Entering Equations: Equations that have been entered into the graph function or dynamic function will show up here. To delete the equation, press F2, F1. Enter an equation. Set the type and range. Press F6(TABL) to see the table. The x-values are located in the first column and the y-values in the second. In the table you can type any value for x and press EXE and the calculator will calculate the y-value. Viewing a Graph of the function: The two choices at the bottom of the table screen are G-CON and G-PLT. Press F5(G-CON), a connected graph, and press F6(G-PLT), a scatter graph of the function. Recursion Function Inputting Recursion Formulas: Go into the Recursion function by pressing 8. Once in the function, you must select what type of recursion function you would like. Press F3(TYPE). Select the type of recursion by pressing the corresponding F1, F2, or F3. This will choose between sequences of one, two, and three terms respectively. Two enter in a equation press F4 for variable options. After the equation is entered, press EXE to store. Creating a Table: You must first set the range of the table. Set the range by pressing F5(RANG). The range specifies the start and ending value for the variable n, where a and b should start, and where the pointer starting point is on the graph. Press F2(a1), then enter in your values for start, end, and a1. The variable n will go in increments of 1. EXIT back out to the equation and press F6(TABL) to see the table. The options for the table appear across the bottom of the screen. You have four options: to delete the recursion formula table, to draw a connected line graph of the formula, to draw a plot type graph of the formula, or to draw a graph and analysis of the convergence/divergence of the graph (WEB). FORM takes you back to the formula. 88 Calculus with the Casio fx-9750G Plus • Appendix Copyright © Casio, Inc. Conics Function Graphing a Conic: In the Main Menu, press 9 for Conics. The first screen you come to will be a choice of conic equations already input into the calculator memory. Choose one by arrowing and highlighting the chosen equation and press EXE. Then enter the values in for the variables in the equation as listed. After each value entered, press EXE to store and move to the next value. Press F6(DRAW) to see the graph of the conic equation. Note, the graph of a circle may not necessarily show up as a circle because the view window needs to be set manually (Choose a X-value that is twice as large as the Y-value to get a perfect circle). Equation Function Solving Systems of Equations: To solve for a system of equations, select the equation function by pressing the XT button. Choose F1 for Simultaneous equations. Choose the number of unknowns you have by pressing the corresponding function key. Enter in the values for each of variables in the equations. After the values are entered in, press F1(SOLV). The calculator will solve for the unknown values. REPT at the bottom of the screen will take you back to the previous screen. Solving for a Variable in Polynomials: You can solve for a variable in polynomials up to the third degree. Press F2 for Polynomial when you enter the equation function. The calculator will ask you to specify the degree of the polynomial by pressing either F1 for 2 or F2 for 3. Enter in the values for the polynomial in the matrix shown and then press F1(SOLV). The two values for X will show up in a matrix. Solving Equations: Enter into the equation solver by pressing F3 for Solver. To enter the equation in question, you may enter numbers, alpha-characters, and symbols. If you do not put an equals sign in the equation the calculator will assume that the equation is to the left of the equals sign and that a zero follows it. To specify a letter or number other than zero, type SHIFT, = and then type the value. After the equation is entered, press EXE to store. The variables in the equation will Copyright © Casio, Inc. Appendix • Calculus with the Casio fx-9750G Plus 89 show up on the screen. Enter the known values and press EXE to store each one. Highlight the unknown value and then press F6(SOLV). The value for the unknown will be shown as well as the value for the left-hand side and right-hand side of the equation to show how accurate the answer is. Program Function Running Programs: After you are in the program function, highlight the program you would like to run and press F1(EXE). Other Options: In this function you can also EDIT a program, create a NEW program, DEL a program, or DEL-All programs. If you press F6 for more options, you can also find (SRC) a program or rename (REN) a program. Time Value of Money Function Doing Financial Calculations: In the financial function you have the ability to calculate several variables using simple interest, compound interest, cash flow, amortization, conversion, cost, selling, price, margin, and day and date calculation. There are many abbreviations in the different modes of the function: APR: annual percentage rate BAL: balance of principal after installment C/Y: compounding periods per year Csh: list for cash flow Cst: cost D: number of days d1: date 1 d2: date 2 EFF: effective interest rate FV: future value I%: periodic/ annual interest rate INT: interest portion of installment IRR: internal rate of return Mrg: margin n: number of compound periods NFV: net future value 90 NPV: net present value P/Y: installment periods per year PBP: pay back period PM1: first installment PM2: second installment PMT: payment PRN: principal portion of installment PV: present value Sel: selling price SFV: simple future value SI: Simple Interest SINT: total interest from installment PM1 to installment PM2 SPRN: total principal from installment PM1 to installment PM2 Calculus with the Casio fx-9750G Plus • Appendix Copyright © Casio, Inc. Link Function Transmitting: The link function is used to transmit and receive data from other calculators. Calculators can share information from the program list, tables, graphs, lists, and statistics. By pressing F1 you can transmit data to another calculator. The calculator will ask for the type of transmission you are making. At this point, F1 will allow you to select what you want to transmit. The calculator will give you a list that consists of lists, matrices, files, graphs, pictures, variables, and receive 1 and receive 2. The receive options are for receiving 1 list or 2 lists simultaneously. Select from this list what to transmit and press F6 for the transmission. Receiving: The only thing required for receiving data is the press F2(REC). The transmitting calculator must do all the work! Image Set Mode: In the Link function menu, F6 is the image set mode. The images are sent by pressing the F↔D key. *Note that the F÷D key will not change a fraction to a decimal or vice versa if the image set mode is set to monochrome. The shift between decimals to fraction can occur only if the image set mode is turned off. Contrast Function Setting the Contrast: You can adjust the contrast of the screen by using the left and right arrow keys. Press the right arrow key to darken the contrast and the left arrow key to lighten the contrast. Copyright © Casio, Inc. Appendix • Calculus with the Casio fx-9750G Plus 91 Memory Function Memory Usage: To check memory usage, select it by highlighting it and pressing EXE. You can delete entire sections of the memory as listed only. You cannot delete individual lists or programs from here. This tool is useful to see where you have memory used and how much memory you have left on the calculator. The calculator will give you the option of backing out before you erase any section. Resetting Memory: Highlight the Reset option and press EXE. This option will reset the entire memory of the calculator. This will clear all programs and any statistics, graphs, matrices, lists, tables, and equations you have entered. The calculator will again let you back out if you accidentally press Reset. 92 Calculus with the Casio fx-9750G Plus • Appendix Copyright © Casio, Inc.

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