calculus and the fx-9750g plus

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

OverviewThis 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.

IntroductionThis 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 QuestionsExamine 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 1

x 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 1

x 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

limx5

2x2 503x + 15

limx0

3x 1x

limx0

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

OverviewThis 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.


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 ________________

IntroductionThis 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.


Problems and QuestionsExamine 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, x2{

limx2

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

xc+f(x)

limx2+

f(x)

limx2

f(x)

limx2

f(x)

limx2

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 21

x + 3, x < 1

limx3

h(x)

limx3-

h(x)

limx3+

h(x)

limx1+

h(x)

limx1

h(x)

limx0

h(x)

limx3

h(x)

limx3-

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, x2{

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.

limx3

h(x)

limx3-

h(x)

limx3+

h(x)

limx1+

h(x)

limx1

h(x)

limx0

h(x)

limx3

h(x)

limx3-

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

OverviewThis 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


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


IntroductionThis 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.


Problems and QuestionsExplore 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< .56, x = .5x2 x 6, x>.5

limx.5

g(x)limx.5

f(x)

limx.5+

g(x) limx.5-

g(x) limx5

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

OverviewThis 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




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


IntroductionThis 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.


Problems and Questions1. 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, Mikes annual salary was $42,000 and at the end of his 3rd year of employment with the same company, Mikes annual salary was $49,000.

What conclusion could you draw about the growth of Mikes 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 Mikes 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 Mikes 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 Mikes 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 Mikes salary growth rate during that same time period?

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

ExtensionGiven 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 its 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 dont 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 youve 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

OverviewThis 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




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


IntroductionThis 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.


Problems and Questions1. 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.

OverviewThis 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 its 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




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 ________________

IntroductionThis 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 its missing).

Problems and QuestionsExplore 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 lets 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 ( )

23

23

-13

.05386

.03393

0

.03393

.05386

-3.188

-5.4288

5.4288

3.188


Activity 7Derivative Behavior of PolynomialsTopic Area: Derivatives

OverviewThis 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 ________________

IntroductionThis 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 its 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.

Problems and Questions1. 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(x2)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 + an1xn1 + ... + 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

OverviewThis 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


IntroductionThis 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.

Problems and QuestionsMake 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, 2] .

_____________________________________________________________________________

_____________________________________________________________________________

3. Using your understanding of derivative as slope, sketch the function, y = Sin(x) and its derivative over the interval [0, 2] .

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, 2] 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, 2] 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, 2] .

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

11. Using your understanding of derivative as slope, sketch the function, y = Cos(x)and its derivative over the interval [0, 2] .

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, 2], 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, 2] , 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.


0,2[ ) ( 32 , 2 ]

2

32

, ( )


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 (, 2)and negative (decreasing) over the interval (0, ) . 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

OverviewA 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


IntroductionThis 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.

Problems and Questions1. 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

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