CALCULUS - Mr Waddellmrwaddell.net/tech/docs/calcs/casio/Calculus and the FX-9750G Plus.… · A limit is one of the foundation concepts in any calculus course. The idea behind this

CALCULUSw

ith the Casio FX-9750G Plus Casio, Inc.

CALCULUS with the Casio FX-9750G PlusActivities for the Classroom

9750-CALC

CALCULUSwith the Casio FX-9750G Plus

Activities for the Classroom

Limits

Derivatives

Continuity

Slope

Linear FunctionsDifferentiabilityPolynomialsTrigonometric Functions

Graphing Models

Slope Fields

Anti Derivatives

Integration

Riemann Sums

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 AvenueDover, NJ 07801www.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

Copyright © Casio, Inc. Calculus with the Casio fx-9750G Plus iii

ContentsActivity 1: Looking at Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Teaching NotesStudent ActivityCalculator Notes and Answers

Activity 2: Do Limits Take Sides? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Teaching NotesStudent ActivityCalculator Notes and Answers

Activity 3: A Graphical Look at Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Teaching NotesStudent ActivityCalculator Notes and Answers

Activity 4: Introduction to Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Teaching NotesStudent ActivityCalculator Notes and Answers

Activity 5: Being Locally Linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Teaching NotesStudent ActivityCalculator Notes and Answers

Activity 6: Continuity Meets Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Teaching NotesStudent ActivityCalculator Notes and Answers

Activity 7: Derivative Behavior of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 42Teaching NotesStudent ActivityCalculator Notes and Answers

Activity 8: Derivative Behavior of CommonTrigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Teaching NotesStudent ActivityCalculator Notes and Answers

iv Calculus with the Casio fx-9750G Plus Copyright © Casio, Inc.

Activity 9: Looking at Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Teaching NotesStudent ActivityCalculator Notes and Answers

Activity 10: Looking at Slope Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Teaching NotesStudent ActivityCalculator Notes and Answers

Activity 11: Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Teaching NotesStudent ActivityCalculator Notes and Answers

Appendix:Overview of the Calculator Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Copyright © Casio, Inc. Activity 1 • Calculus with the Casio fx-9750G Plus 1

Activity 1Looking at LimitsTopic Area: Limits

Class Time: one 45-50 minute class period

Overview

This activity will encourage students to use graphical and numerical representationsto 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 thisactivity is to have the student investigate both numerically and graphically the behav-ior of the output of a function as its input moves closer and closer to some point ofinterest. The emphasis will be on examining the behavior of the function as its getsnear 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 groupsarranged 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.

Teaching Notes

2 Calculus with the Casio fx-9750G Plus • Activity 1 Copyright © Casio, Inc.

Introduction

This activity will encourage you to use graphical and numerical representations toexamine 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

Examine the value of the function as the value of x gets close to 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}.

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}

Name _____________________________________________ Class ________ Date ________________

Activity 1 • Looking at Limits

f(x)= x2 – 1x – 1

x

0

1

2

y

x

.5

1

1.5

y


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}.

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}

• the second time starting at .99 with a pitch of .01. Record the values for x = {.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.

Name _____________________________________________ Class ________ Date ________________


x

.75

1

1.25

y

x

.9

1

1.1

y

x

.99

1

1.01

y


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.

_____________________________________________________________________________

_____________________________________________________________________________

18. Examine the function: 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}.

Name _____________________________________________ Class ________ Date ________________


f(x)= x2 – 1x – 2

x

1

2

3

y


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.

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. ______________________________________________________________

25. ______________________________________________________________

Name _____________________________________________ Class ________ Date ________________


x

2

y x

2

y x

2

y x

2

y

limx�5

2x2 – 503x + 15

limx�0

3x – 1x

limx�0

x + 2x + 4

limx�-3

3x + 124 – x


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 TABLEand 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.


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 makesure the hole is visible. The calculator will only show the gap if it is a specific pixel it isasked to light up and that pixel does not exist at that point. In a many other viewingwindows the point (1,2) would not be one that the 9750 would try to graph, thus in con-nected 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.



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 pixelbut care needs to be taken to show both branches of the graph. If the proper vertical win-dow 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, thisshould 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 fore-shadow a discussion of continuity.



Activity 2Do Limits Take Sides? Topic Area: Limits

Class Time: one 45-50 minute class period

Overview

This activity will encourage students to use graphical and numerical representationsto examine the idea of a limit needing to be the same from both directions ofapproach.

The concept of a limit creates the framework for discussing continuity. Using split-defined functions, the goal of this activity is to put a face on the idea of one-sidedlimits.

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


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.

Teaching Notes


Name _____________________________________________ Class ________ Date ________________

Introduction

This activity will have you use graphical and numerical representations to examinethe idea of a limit needing to be the same from both directions of approach.



Examine the behavior of the function: as the value of xapproaches 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 ? Explain your answer.

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

Activity 2 • Do Limits Take Sides?

f(x)= x – 4, x<2x – 1, x>2{

limx�2

f(x)


7. The symbolic notation: means to investigate the limit of the function, 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 ?

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

10. Based upon this investigation so far, how would you describe the notation:?

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

11. How would you answer the question: Find ? Why?

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

12. How would you now answer the question: Find ? Why?

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

Name _____________________________________________ Class ________ Date ________________

Activity 2 • Do Limits Take Sides?lim

x�c+ f(x)

limx�2+ f(x)

limx�2– f(x)

limx�2– f(x)

limx�2

f(x)


13. Graph the function h(x)= 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

a. ______________________________________________________________

b. ______________________________________________________________

c. ______________________________________________________________

d. ______________________________________________________________

e. ______________________________________________________________

f. ______________________________________________________________

g. ______________________________________________________________

h. ______________________________________________________________

Name _____________________________________________ Class ________ Date ________________

Activity 2 • Do Limits Take Sides?

{ x2 + 3, –1< x<2–x + 9, x > 2

1x + 3

, x < –1

limx�–3

h(x)

limx�–3- h(x)

limx�–3+ h(x)

limx�–1+ h(x)

limx�–1

h(x)

limx�0

h(x)

limx�3

h(x)

limx�3- h(x)


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 you would enter it as follows:

• 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.

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.

f(x)= x – 4, x<2x – 1, x>2{

The graph in standard window

Note: When students graph it they shouldbe very clear to indicate that there areopen circles at the endpoints of the "jump."



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.

14a. = None, two different one sided limits

b. = –� (Note: while "none" is also acceptable, –� is a more complete description of what is actually taking place.)

c. = � (Note: while "none" is also acceptable, � is a more complete description of what is actually taking place.)

d. = 4

e. = None, two different one-sided limits

f. = 3

g. = None, two different one-sided limits

h. = 12


This is a representation of what the student should sketch.

limx�–3

h(x)

limx�–3- h(x)

limx�–3+ h(x)

limx�–1+ h(x)

limx�–1

h(x)

limx�0

h(x)

limx�3

h(x)

limx�3- h(x)


Activity 3A Graphical Look at ContinuityTopic Area: Derivatives and Continuity

Class Time: an exploratory introduction during the first 30 minutes of a class periodon the topic of continuity

Overview

This activity will have students explore the concept of continuity at a point. It willalso allow them to discover that simply having a limit at a point will not guaranteethat the function is also continuous.

It also explores the idea that a having a limit is a necessary, but not a sufficient con-dition 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


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.

Teaching Notes


Introduction

This activity will have have you explore the concept of continuity at a point. It willalso allow you to discover that simply having a limit at a point will not guaranteethat the function is also continuous.



Explore the behavior of the function 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 is? Explain why you arrived at that answer.

_____________________________________________________________________________

_____________________________________________________________________________

Name _____________________________________________ Class ________ Date ________________

Activity 3 • A Graphical Look at Continuity

f(x)= x2 – x – 6

limx� 5

f(x)


5. Now explore the behavior of the split-defined function: g(x)=

Use the same viewing window as before.

Record what you see below.

6. What does it appear the value of is?

How does it compare to ?

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

7. Now, trace to a value where x = .4, and zoom in, describe and record what you see.

_____________________________________________________________________________

_____________________________________________________________________________

8. Find: , ,

9. Now find g(.5), how does this compare to your answers above?

_____________________________________________________________________________

_____________________________________________________________________________

10. Draw a conclusion about the relationship between limits and continuity.

_____________________________________________________________________________

_____________________________________________________________________________

Name _____________________________________________ Class ________ Date ________________

Activity 3 • A Graphical Look at Continuity

{ x2 – x – 6, x< .5–6, x = .5x2 – x – 6, x>.5

limx�.5

g(x)limx�.5

f(x)

limx�.5+ g(x) lim

x�.5- g(x) limx�5

g(x)


1.

2. Vertex is (.5, -6.25) and can be found symbolically or using the MIN function in the G-Solve folder.

3.

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.


This is the screen sequence for the zoom.Nothing unusual should be seen.

The vertex remains, the function is continuous.


5.

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.


This provides a good look at the splitdefined function.

The graph produced in the given windowwill be as shown on the left.

The discontinuity will not be immediatelyapparent from this graph.

This is the screen sequence that producesthe desired screen.


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.



Activity 4Introduction to DerivativesTopic Area: Derivatives

Overview

This activity will have students begin to connect the concept of slope and rate ofchange to the derivative.

It also provides an introduction to the concept that the slope of a function extendsbeyond linear slope, but that using the slope of a line can foster a discussion of aver-age 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




• 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.


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

Teaching Notes


Introduction

This activity provides an introduction to the concept that the slope of a functionextends beyond linear slope, but that using the slope of a line can foster a discus-sion of average vs. instantaneous rates of change.



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.

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

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)

____________________________________

Name _____________________________________________ Class ________ Date ________________



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 find a good estimate for the equation of theline tangent to f(x) at x = 2. Explain your process and how accurate you think youare.

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

Name _____________________________________________ Class ________ Date ________________


f(x)= 3x2 – 2x + 1


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





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 functionto create the line without being able to communicate the connection between theslope of the secant line/tangent line and the value of the function at x = 2. The studentestimation of accuracy will depend upon their process.



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.



You also have the option of graphing the points, creating and copying the model fromthere.

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)



Activity 5Being Locally LinearTopic Area: Derivatives and Slope

Class Time: an exploration during the first part of a class period while connecting theslope of a function to the derivative

Overview

This activity will begin to bring home the point that as the behavior around a singlepoint on a differentiable function is examined, the function will "flatten out" and verymuch resemble the behavior of a line drawn through the point of interest. The exam-ple given should motivate a discussion of what it means to be locally linear withregard 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




• 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.


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

Teaching Notes


Introduction

This activity will begin to bring home the point that as the behavior around a sin-gle 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.



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:

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:

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.

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

Name _____________________________________________ Class ________ Date ________________


x y

2 -3


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.

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

Name _____________________________________________ Class ________ Date ________________



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

2 -3

1.975 -3.049375

2.025 -2.949375


5. y = 2x – 6.999375

6.

7.

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.


This is what the calculator will show.


Activity 6Continuity Meets DifferentiabilityTopic Area: Derivatives and Continuity

Class Time: an exploration during the first half of a class period to point out visuallythat 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 alsoconnect those concepts to continuity and point out that continuity is a necessary butnot sufficient condition for differentiability. The connections will be made visuallyusing the idea of local linearity (or what happens when it’s missing). Symbolic deriv-atives 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




• 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.


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

Teaching Notes

36 Calculus with the Casio fx-9750g Plus • Activity 6 Copyright © Casio, Inc.

Name _____________________________________________ Class ________ Date ________________

Introduction

This activity will begin to extend the idea of local linearity and derivative. It willalso connect those concepts to continuity. The connections will be made visuallyusing the idea of local linearity (or what happens when it’s missing).


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.

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

Activity 6 • Continuity Meets Differentiability

3

2

1

1 2 3


3. Using trace, fill in the following values for the function accurate to 5 decimal places.

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:

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.

Name _____________________________________________ Class ________ Date ________________


2.0125

2.00625

2

1.99375

1.9875

x y

Pt 1

Pt 2

Pt 3

Pt 4

Pt 5

Slope 1

Slope 2

Slope 3

Slope 4


Describe what you see.

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________


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:

9. What do these results indicate? Compare them to the results from the exploration of the graph around x = 2.

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

Name _____________________________________________ Class ________ Date ________________


-0.0125

-0.00625

0

0.00625

0.0125

x y

Pt 1

Pt 2

Pt 3

Pt 4

Pt 5

Slope 1

Slope 2

Slope 3

Slope 4


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?

Name _____________________________________________ Class ________ Date ________________



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.


4.


2.0125

2.00625

2

1.99375

1.9875

x y

Pt 1

Pt 2

Pt 3

Pt 4

Pt 5

Slope 1

Slope 2

Slope 3

Slope 4

1.59401

1.59071

1.58740

1.58409

1.58078

0.528

0.5296

0.5296

0.5296


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.

8.

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.

10.

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.


-0.0125

-0.00625

0

0.00625

0.0125

x y

Pt 1

Pt 2

Pt 3

Pt 4

Pt 5

Slope 1

Slope 2

Slope 3

Slope 4

x = xdydx ( )2

323

-13

.05386

.03393

0

.03393

.05386

-3.188

-5.4288

5.4288

3.188


Activity 7Derivative Behavior of PolynomialsTopic Area: Derivatives

Overview

This activity will lead students to make connections between the behavior of somewell known polynomial functions and their derivatives. They will be asked to plot thefunctions, confirm the expected behavior using the grapher and then overlay thederivative, 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 groupsarranged prior to beginning the activity will provide students with opportunity toexchange 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

• Press OPTN.

• Press F2 (CALC). Example:

• Press F1 (d/dx)

• Entry syntax: d/dx (function, x)

Teaching Notes


Name _____________________________________________ Class ________ Date ________________

Introduction

This activity will begin to extend the idea of local linearity and derivative. It willalso connect those concepts to continuity. The connections will be made visuallyusing 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 ateach point along its domain. The goal of this activity is to have you able to makethe connections to the picture of the function and the picture of the behavior of theslope of the function.


1. Draw the graphs of the following functions in the window.

y = 2x y = 2x + 5 y = 2x – 3

2. 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.

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

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, �)

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

Name _____________________________________________ Class ________ Date ________________



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: The function and derivative by calculator:

Name _____________________________________________ Class ________ Date ________________



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.

Name _____________________________________________ Class ________ Date ________________



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.



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 statements 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:



Activity 8Derivative Behavior of CommonTrigonometric FunctionsTopic Area: Derivatives

Overview

This activity will lead students to making connections between the behaviors of somewell known trigonometric functions and their derivatives. They will be asked to plotthe functions, confirm the expected behavior using the grapher and then overlay thederivative, confirming again using the grapher.

The derivative of a function represents the behavior of the slope of the function ateach point along its domain. The goal of this activity is to have students make theconnections to the picture of the function and the picture of the behavior of the slopeof 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 groupsarranged 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. Example:

• Press F2 (CALC).

• Press F1 (d/dx)

• Entry syntax: d/dx (function, x)

Teaching Notes


Introduction

This activity will have you making connections between the behavior of some wellknown trigonometric functions and their derivatives. You will be asked to plot thefunctions, confirm the expected behavior using the grapher and then overlay thederivative, confirming again using the grapher.

The derivative of a function represents the behavior of the slope of the function ateach point along its domain. The goal of this activity is to have you make the con-nections to the picture of the function and the picture of the behavior of the slopeof the function.


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.

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

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).


_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

Name _____________________________________________ Class ________ Date ________________



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).

Name _____________________________________________ Class ________ Date ________________




_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

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).


_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

17. Compare and contrast the behaviors of the derivatives of the Sine and Cosinefunctions.

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

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?

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

Name _____________________________________________ Class ________ Date ________________



1.

2. Answers will vary. A Good answer will include statements that the slope is

positive (increasing) over the intervals and and negative

(decreasing) over the interval .

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.


0,π2[ ) ( 3π

2, 2π ]π

23π2

, ( )


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.



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



17. Answers will vary. Complete answers should include statements that the Sinefunction 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)� = A • Cos(x)

y = Sin(Bx)� = B • Cos(Bx)

y = A • Cos(x)� = -A • Sin(x)

y = Cos(Bx)� = -B • Sin(Bx)


dydx

dydx

dydx

dydx


Activity 9Looking at RelationshipsTopic Area: Derivatives and Graphs

Overview

A great deal of information about a function can be found by analyzing the behaviorof the first and second derivatives. This activity will provide a graphical examinationof 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 inpairs..

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.

• The speed at which the calculator shows a second derivative graph is relatively slow. Some students may conclude there is no graph being produced.

Teaching Notes


Introduction

This activity will provide a graphical examination of the relationships between thefunction and its first and second derivatives.

The increasing/decreasing nature of a function can be examined by thepositive/negative behavior of its derivative. Similarly, the upward/downward concavity can be examined by the positive/negative behavior of its second derivative.


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.

____________________________________

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?

____________________________________

Name _____________________________________________ Class ________ Date ________________



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.

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

Name _____________________________________________ Class ________ Date ________________



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.

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

Name _____________________________________________ Class ________ Date ________________



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.

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.


FunctionDer


17.

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.


1st Der.

2nd Der.


Activity 10Looking at Slope FieldsTopic Area: Slope Fields

Overview

A Slope Field is a visual representation of the solution to a differential equation cre-ated 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 appear-ance using the calculator. This can be used as a first introduction to the idea of SlopeFields 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 anotherstudent.

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.

Teaching Notes


Activity 10Slope Field Program

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 forcomparison.

This program can be found in the download section ofthe Casio Education website:

http://www.casioeducation.com


Name _____________________________________________ Class ________ Date ________________

Introduction

A Slope Field is a visual representation of the solution to a differential equationcreated 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 theirappearance using the calculator.

An equation like dy/dx = 2x which contains a derivative is called a "differentialequation". The problem becomes finding a function, y in terms of x, when we aregiven its derivative. Note the phrasing: we are looking for "a solution" and not "thesolution". This is due to the fact that the slope will allow us to arrive at a family ofcurves and missing some initial condition value we will not be able to arrive at asingle solution but should be able to make some conclusions as to the appearanceof the family of solutions. If we examine the plot of the slope over a series of gridpoints we end up with a Slope Field.

If given the differential equation dy/dx = f(x,y) a plot of short line segments withslopes f(x,y) over specific grip points produces a slope field. This slope field willgive you a look at the behavior of the solution to the original differential equation.


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].

Activity 10 • Looking at Slope Fields

(x,y)

-6, -2

-5,-2

-4,-2

-3,-2

-2,-2

-1,-2

0,-2

1,-2

2,-2

3,-2

4,-2

5,-2

6,-2

dy/dx (x,y)

-6, -1

-5,-1

-4,-1

-3,-1

-2,-1

-1,-1

0,-1

1,-1

2,-1

3,-1

4,-1

5,-1

6,-1

dy/dx (x,y)

-6, -0

-5,-0

-4,-0

-3,-0

-2,-0

-1,-0

0,-0

1,-0

2,-0

3,-0

4,-0

5,-0

6,-0

dy/dx (x,y)

-6, 1

-5,1

-4,1

-3,1

-2,1

-1,1

0,1

1,1

2,1

3,1

4,1

5,1

6,1

dy/d (x,y)

-6, 2

-5,2

-4,2

-3,2

-2,2

-1,2

0,2

1,2

2,2

3,2

4,2

5,2

6,2

dy/dx


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.

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

Name _____________________________________________ Class ________ Date ________________



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

Name _____________________________________________ Class ________ Date ________________



b) dy/dx = Sin(x)

c) dy/dx = ex

Name _____________________________________________ Class ________ Date ________________



1.

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.


(x,y)

-6, -2

-5,-2

-4,-2

-3,-2

-2,-2

-1,-2

0,-2

1,-2

2,-2

3,-2

4,-2

5,-2

6,-2

dy/dx

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

(x,y)

-6, -1

-5,-1

-4,-1

-3,-1

-2,-1

-1,-1

0,-1

1,-1

2,-1

3,-1

4,-1

5,-1

6,-1

dy/dx

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

(x,y)

-6, -0

-5,-0

-4,-0

-3,-0

-2,-0

-1,-0

0,-0

1,-0

2,-0

3,-0

4,-0

5,-0

6,-0

dy/dx

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

(x,y)

-6, 1

-5,1

-4,1

-3,1

-2,1

-1,1

0,1

1,1

2,1

3,1

4,1

5,1

6,1

dy/d

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

(x,y)

-6, 2

-5,2

-4,2

-3,2

-2,2

-1,2

0,2

1,2

2,2

3,2

4,2

5,2

6,2

dy/dx-

12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12


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

(actual family: y = x2 + c)

b) also a trigonometric family

(actual family: y = -Cos(x) + c)


13


c) an exponential family

(actual family y = ex + c)



Activity 11Riemann SumsTopic Area: Anti Derivatives, Integration

Overview

This activity will present students with the tools to calculate and analyze Riemannsums. They will hand sketch rectangles and use that to approximate the area undera curve. They will then use the calculator to perform increasing numbers of calcula-tions to observe the convergence of upper and lower Riemann Sums with regularpartitions as the size of the partitions decrease and the number of rectanglesincrease.

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 groupsarranged prior to beginning the activity. Students should have a working knowl-edge of the derivative function as a model for representing average and instanta-neous 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.

• Students should be given the programs only after they have demonstrated ability to construct a Riemann Sum on their own.

Teaching Notes


Activity 11Riemann Sum Programs

Riemann Sum Drawing Program:

Riemann Sum Calculation program: (Calculates both the left and right RSum)

Teaching Notes

These programs can be found in the download sectionof the Casio Education website:

http://www.casioeducation.com


Introduction

A Riemann Sum is a method of approximation for calculating the area under acurve. When a function represents change, the area under the curve represents theaccumulation of that change. For instance, if you have curve measuring velocityover time, the sums of those velocities over specific time intervals represents thedistance traveled during that time period.

This activity will present you with the tools to calculate and analyze Riemannsums. You will hand sketch rectangles and use that to approximate the area undera curve. You will then use the calculator to perform increasing numbers of calcula-tions to observe the convergence of upper and lower Riemann Sums with regularpartitions as the size of the partitions decrease and the number of rectanglesincrease.


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.

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?

_____________________________________________________________________________

Name _____________________________________________ Class ________ Date ________________

Activity 11 • Riemann Sums

70

60

50

40

30

20

10

1 2 3 4 5


Name _____________________________________________ Class ________ Date ________________

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?

_____________________________________________________________________________

_____________________________________________________________________________



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.

Right Endpoint Left Endpoint

Area: _________________________ Area: _________________________

11. 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.)

Name _____________________________________________ Class ________ Date ________________


Number of partitions

UpperRiemann Sum

LowerRiemann Sum


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 functionsover the interval indicated and for the number of partitions indicated:

Name _____________________________________________ Class ________ Date ________________


Function

3x+5

Sin(x)(radianmode)

Interval

[1,4]

[0, 6]

[0, π]


4

8

12

16

3

6

10

20

2

4

8

16

UpperRiemann Sum

LowerRiemann Sum

1x


1.

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 =14Answers should clearly state the area is more than what is covered, by the function, as the rectangles are above the function in each case.


70

60

50

40

30

20

10

1 2 3 4 5


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 Left Endpoint

Area: 11.375 Area: 6.875

11.

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.



12

18

36

UpperRiemann Sum

10.16

9.76

8.62

LowerRiemann Sum

7.91

8.26

7.92


Extension:

Note: Care should be taken to remind students that there is no firm "rule" for whetherthe left or right endpoint rule will give the best approximation, it depends upon thefunction. The goal of this activity is simply to bring out the tools for the use ofRiemann Sums.


Function

3x+5

Sin(x)(radianmode)

Interval

[1,4]

[0, 6]

[0, π]


4

8

12

16

3

6

10

20

2

4

8

16

UpperRiemann Sum

1.71

1.54

1.48

1.46

66

75

78.6

81.3

1.57

1.90

1.82

1.99

LowerRiemann Sum

1.15

1.26

1.30

1.32

102

93

89.4

86.7

1.57

1.90

1.97

1.99

1x

Copyright © Casio, Inc. Appendix • Calculus with the Casio fx-9750G Plus 83

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 asfractions and do any normal mathematicaloperations. The EXE will give you answersin fraction or mixed number format. Youcan change from fraction format to a decimal format by pressing the F↔D keyand vice versa. You may also change themixed number to an improper fraction bypressing SHIFT ab/c for d/c. This functionwill only work if the expression is entered infraction format originally.

Editing:If you need to make a change in a previouscalculation, use the deep recall to retrievethe equation. In the Run function, press theAC/ON to clear the screen. Then use the UpArrow key to scroll through the previousequations until you come to the one youwant 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 dofactorials, permutations, combinations, and random numbers.

d/c

ab/c

84 Calculus with the Casio fx-9750G Plus • Appendix Copyright © Casio, Inc.

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. Afterthe 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 andpressing 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 setto Box. Press EXIT to get back to the previous screen. Check to see if graph 1 isselected in the SEL option, and press F6(DRAW). Statistics can be seen from hereby pressing F1(1VAR). SHIFT F1(Trace) will allow you to trace the graph forquartile ranges using the right arrow key. Press EXIT twice to return to the STATwindow.

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 YListare set at List1 and List2 respectively. Press EXIT and then press F1 to seeGrph1. If the graph does not appear, EXIT and press F4(SEL) to make sure onlyGraph 1 is turned on.

Line of Best Fit:On the screen with the graph are options for different kinds of regressions. Pressthe regression you think will fit the data the best and the calculator will give youa regression analysis. Notice the choices at the bottom of the screen. If you pressthe F5(COPY), it will copy the line to the GRAPH function so that you can seethe graph later, if needed. After you press F5(COPY), press EXE to store thegraph and it will take you back to the regression information. Press F6(DRAW)function to see the graph.


Matrix Function

Entering Matrices:In the matrix function, you will see a list of matrices you can enter. With the firstavailable matrix highlighted, enter the number of rows needed first and then thenumber of columns. Press EXE after each entry. After the second EXE, the actualmatrix will show up on the screen. When you enter the matrix, the numbers will goacross first and then down to the next row. From this screen, you can do basicmatrix 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. PressOPTN in the Run function and then press F2(MAT) for matrix operations. Youcan add and subtract matrices, find the determinant, transpose, augment, send amatrix to lists, and finding the identity of a matrix. After the function has beenentered, 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 liketo erase the lists, highlight the list and press F4(DEL-A). Then enter the new dataas before. You may sort the data by pressing F1. It will prompt you by asking howmany lists, then it will prompt you to select a list. When sorting more than onelist, the calculator will ask for a base list and a second or third list. You can sortthe 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 includemoving matrices, finding the minimum, maximum, mean, median, mode, andproduct of lists.


Graph Function

Entering Equations:When you enter the graph function, the graph edit screen is up. Type in an equation using the X�T or the letter X button for the variable X. You can changethe 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 theoriginal 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 astandard 20-interval view, press F1(INIT) for initial settings, and F2(TRIG) for astandard trigonometry function. You may also set the setting manually and storethat setting by pressing F4(STO). Press EXIT to get back to the graph-editscreen. You may view more than one graph at a time by pressing F1 on eachgraph 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 andswitch 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 willbe on the left, and the frame on the right will be blank. Zoom in on the graphand that part of the graph will show up on the right. You may also use the dualscreen for a graph and a table. Go into the setup again and change the dualscreen 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 valueson the graph. To put those values in the table press EXE. Put multiple values inthe table by pressing EXE repeatedly.

Solving for a Specific Point:After you graph two inequalities you can find the point of intersection of thosegraphs by using the G-solve function. Press F5(G-Solv) (you do not have to pressthe 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.


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 inbasic algebra and trigonometry. To get to this list, press F5(B-IN) after you arein the Dynamic function. To choose one of the built-in functions, highlight andpress 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 theother variables. Notice the arrow to the right of the Dynamic Variable. That isthe 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 previousscreen. Press EXIT to return to that screen.

Setting the range:In the dynamic variable screen, press F2(RANG). Set the start and end of therange 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 graphis on the screen, press EXE to see the change in the variable. When you press theEXE, 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 inmanually. Use the alpha key and letters to enter variables. The variables enteredin manually will be available in the dynamic function.


Table Function

Entering Equations:Equations that have been entered into the graph function or dynamic functionwill show up here. To delete the equation, press F2, F1. Enter an equation. Setthe type and range. Press F6(TABL) to see the table. The x-values are located inthe first column and the y-values in the second. In the table you can type anyvalue 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. PressF5(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 mustselect what type of recursion function you would like. Press F3(TYPE). Selectthe type of recursion by pressing the corresponding F1, F2, or F3. This willchoose between sequences of one, two, and three terms respectively. Two enterin a equation press F4 for variable options. After the equation is entered, pressEXE 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 bshould 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 incre-ments 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 connectedline graph of the formula, to draw a plot type graph of the formula, or to draw agraph and analysis of the convergence/divergence of the graph (WEB). FORMtakes you back to the formula.


Conics Function

Graphing a Conic:In the Main Menu, press 9 for Conics. The first screen you come to will be achoice of conic equations already input into the calculator memory. Choose oneby arrowing and highlighting the chosen equation and press EXE. Then enterthe 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 graphof the conic equation. Note, the graph of a circle may not necessarily show up asa circle because the view window needs to be set manually (Choose a X-value thatis 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 theX�T button. Choose F1 for Simultaneous equations. Choose the number ofunknowns 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, pressF1(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 forPolynomial when you enter the equation function. The calculator will ask you tospecify the degree of the polynomial by pressing either F1 for 2 or F2 for 3. Enterin 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 equationin question, you may enter numbers, alpha-characters, and symbols. If you donot 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 aletter or number other than zero, type SHIFT, = and then type the value. Afterthe equation is entered, press EXE to store. The variables in the equation will


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 theunknown will be shown as well as the value for the left-hand side and right-handside 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 torun and press F1(EXE).

Other Options:In this function you can also EDIT a program, create a NEW program, DEL aprogram, or DEL-All programs. If you press F6 for more options, you can alsofind (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 rateBAL: balance of principal after installmentC/Y: compounding periods per yearCsh: list for cash flowCst: costD: number of daysd1: date 1d2: date 2EFF: effective interest rateFV: future valueI%: periodic/ annual interest rateINT: interest portion of installmentIRR: internal rate of returnMrg: marginn: number of compound periodsNFV: net future value

NPV: net present valueP/Y: installment periods per yearPBP: pay back periodPM1: first installmentPM2: second installmentPMT: paymentPRN: principal portion of installmentPV: present valueSel: selling priceSFV: simple future valueSI: Simple InterestSINT: total interest from installment PM1 toinstallment PM2SPRN: total principal from installment PM1 toinstallment PM2


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, andstatistics. 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, F1will allow you to select what you want to transmit. The calculator will give you alist that consists of lists, matrices, files, graphs, pictures, variables, and receive 1and 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 transmittingcalculator must do all the work!

Image Set Mode:In the Link function menu, F6 is the image set mode. The images are sent bypressing 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 betweendecimals 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 lightenthe contrast.


Memory Function

Memory Usage:To check memory usage, select it by highlighting it and pressing EXE. You candelete entire sections of the memory as listed only. You cannot delete individuallists or programs from here. This tool is useful to see where you have memoryused and how much memory you have left on the calculator. The calculator willgive 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 againlet you back out if you accidentally press Reset.

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CALCULUS - Mr Waddellmrwaddell.net/tech/docs/calcs/casio/Calculus and the FX-9750G Plus.… · A limit is one of the foundation concepts in any calculus course. The idea behind this