400 Points - Kenton County template.pdf · Jeopardy template Daily Double (wager) Daily Double!!!...

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Attachments

Theme Jeopardy.mp3

CURLZ___.TTF

TEMPSITC.TTF

sosmart.mp3

First Derivative Test ﴾questions﴿.doc

Parrot.bmp

Sample Excel Sheet.xls

37694.72

SMART Notebook

SMART Notebook

SMART Notebook

sosmart

homer

simpson

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SMART Notebook

Name: ________________________

Date: ___________________

The First Derivative Test For Local Extrema

As we have seen, any function f(x) can be graphed by generating a table of values and then manually plotting each point individually. This process, while fairly reliable, is often long and tedious, especially if f(x) is equal to say, x4 – 4x3 – 8x2 + 48x –2 !!! Wouldn’t it be nice if there were some easy way to sketch this function, without going through a million and one calculations?

Well, search no more!!! Calculus and in particular, knowledge of functions and their derivatives, has provided us with a method of doing just that: quickly sketching complicated functions without generating a table of values!!! As you will see, these graphs may not be exactly “to scale” but they will possess all the essential features of the function and will allow the reader to quickly extract any required information. Let’s get started…

1. For each of the following functions, use two DIFFERENT colors to sketch the graphs of f(x) and f’(x) on the SAME coordinate system. In order to save yourself some time, you may use your graphing calculator.

a) f(x) = x3 – 3x + 2

f’(x) = ____________________

2. Identify the turning points of the function f(x) and classify each as either a local maximum or a local minimum.

Turning points of f(x)

Local Max or Min?

3. Divide your graph into INTERVALS by drawing a vertical line through each turning point. Complete the following table.

x < -1

x = -1

(max)

-1< x < 1

x = 1

(min)

x > 1

Is the slope of the tangent to f(x) positive(+), zero, or negative(-) on this interval?

Is f’(x) positive (+) or negative (-) on this interval?

4. At each of the turning points, what do you notice about the slope of the tangent to the function f(x)?

5. At each of the turning points, what do you notice about the value of the function f’(x)?

6. Suggest a way of finding the turning points of a function f(x), if you are given its equation and nothing else (without using your graphing calculator).

b) f(x) = x4 – 8x2 + 8

f’(x) = ____________________



Local Max or Min?




Examine BOTH graphs carefully before answering questions 9-12.

9. What do you notice about the values of the function f’(x) immediately before a local max?

10. What do you notice about the values of the function f’(x) immediately after a local max?

11. What do you notice about the values of the function f’(x) immediately before a local min?

12. What do you notice about the values of the function f’(x) immediately after a local min?

13. Using your own words, write a rule that will allow you to determine whether a turning point is a local maximum or a local minimum based on the values of the derivative f’(x).

14. Your new rule needs a name. What are you going to call your new rule?

15. Repeat for the following function.

c) f(x) = 2x3 + 3x2 – 12x

f’(x) = ____________________



Local Max or Min?




18. Does your rule apply to this function? If not, return to question 13 and modify your rule.

19. Now that you know how to find and classify turning points, you can work backwards (starting with the turning points) to graph the function. Try it! Given the following information and without using your graphing calculator, sketch the graph of f(x).

d) f(x) = 4x3 - 27x + 10

f’(x) = ____________________

x < -1.5

x = -1.5

-1.5 < x < 1.5

x = 1.5

x > 1.5



20. In the table above, where do the values x = -1.5 and x = 1.5 come from? How would you get these values?

21. Start by identify the turning points of the function f(x) and classify each as either a local maximum or a local minimum.


Local Max or Min?

22. Use your graphing calculator to check your answer.

23. Were you able to correctly apply your rule to come up with the graph of f(x)? If yes, try to explain why your rule works. If not, then try to explain why your rule does not work.

24. Given a function f(x), write down the steps that you would follow in order to quickly sketch the graph of f(x).

25. Write a short paragraph summarizing the information contained in this investigation. Please not that you may be required to share your summary with the class.

26. Try graphing the following functions using your new rule and the steps outlined in question 24. Only use your graphing calculator to check your answers. Does your rule work for all functions? If not, return to question 13 and modify your rule.

i) f(x) = -2x5 + 5

ii) f(x) = 3x4 – 4x3

iii) f(x) = 2x4 - 8x3 + 9x2

iv) f(x) = x3

v) f(x) = x3 – 12x + 4

SMART Notebook

SMART Notebook

Sheet1

Centurion Construction Expansion Cost Analysis

Costs

Eastern Region LocationFacilities (build or renovate)Legal FeesMoving CostsTotal Cost

New York City, New York$890,789.00$699,355.00$145,698.001,735,842.00

Philadelphia, Pennsylvania$742,687.00$476,861.00$95,572.001,735,842.00

Washington, D.C.$234,000.00$34,989.00$100,985.00$369,974.00

Orlando, Florida$34,560.00$854,988.00$124,342.001,013,890.00

Total Product Costs1,902,036.002,066,193.00$4,855,548.00

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New York City, New YorkNew York City, New YorkNew York City, New York

Philadelphia, PennsylvaniaPhiladelphia, PennsylvaniaPhiladelphia, Pennsylvania

Washington, D.C.Washington, D.C.Washington, D.C.

Orlando, FloridaOrlando, FloridaOrlando, Florida

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Facilities (build or renovate)

Legal Fees

Moving Costs

Products

Percentage of Total Costs

Product Costs

890789

699355

145698

342687

476861

95572

234000

34989

100985

3456

854988

124342

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SMART Notebook

Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11Page 12Page 13Page 14Attachments Page 1

400 Points - Kenton County template.pdf · Jeopardy template Daily Double (wager) Daily Double!!!...

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