1

AP Calculus I

Notes 4.1

Extrema on an Interval

Definition of Extrema

Let f be defined on an interval l containing c .

1) )(cf is the minimum of f if )()( xfcf for all x .

2) )(cf is the maximum of f if )()( xfcf for all x .

The minimum and maximum of a function on an interval are the extreme values, or extrema, of the

function on the interval. The overall minimum and maximum of a function are called the absolute

minimum and absolute maximum on the interval.

Ex. 1: Consider the following variations of the function 1)( 2 xxf :

1)( 2 xxf from ]2,1[ 1)(2 xxf from )2,1(

The Extreme Value Theorem:

If f is continuous on a closed interval ],[ ba , then f has both a minimum and a maximum on the interval.

1 2 3 4 5–1 x

1

2

3

4

5

6

–1

–2

y

1 2 3 4 5–1 x

1

2

3

4

5

6

–1

–2

y

1 2 3 4 5–1 x

1

2

3

4

5

6

–1

–2

y

0,2

0,1)(

2

x

xxxg from ]2,1(

2

Relative Extrema:

Consider the graph below of 23 3)( xxxf :

Think of the relative maximum occurring on a __________ of the graph and the relative minimum

occurring in a _______________ of the graph.

Ex. 2: Find the value of the derivative at each of the relative extrema below.

a) xxf sin)(

1 2 3 4 5–1 x

1

2

–1

–2

–3

–4

–5

–6

y

1 2 3 4 5 6 7–1 x

1

2

–1

–2

y

The graph of 23 3)( xxxf has a relative minimum

at the point ____________

The graph of 23 3)( xxxf has a relative maximum

at the point ____________

3

b) 2)( xxf

c)

227

2132)(

2

xx

xxxxg

Note that in each of these examples of relative extrema, the derivative is either _____ or ____________

2 4 6–2 x

2

4

–2

–4

y

1 2 3 4 5 6 7–1–2–3 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

4

1 2 3 4 5 6 7–1–2–3 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

1 2 3 4 5 6 7–1–2–3 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

Definition of Critical Number:

Let f be defined at c . If 0)(' cf or if 'f is undefined at c , then c is a critical number of f .

Theorem – Relative Extrema Occur Only at Critical Numbers

If f has a relative minimum or relative maximum at cx , then c is a critical number of f .

However, if cx is a critical number of f , there is not necessarily a relative extrema at cx .

Counterexample: 3)( xxf

)(xf has a critical value at _____________...

)(xf has a relative extrema at _____________...

5

Guidelines for Finding Absolute Extrema on a Closed Interval:

To find the absolute extrema of a continuous function f on a closed interval ],[ ba , use the following:

1) Find the critical numbers of f in ),( ba

2) Evaluate f at each critical number

3) Evaluate f at each endpoint of ],[ ba

4) The least of these values is the minimum. The greatest is the maximum.

Ex. 3: Find the absolute extrema of 34 43)( xxxf on the interval ]2,1[ .

Ex. 4: Find the absolute extrema of 32

32)( xxxf on the interval ]3,1[

6

Ex. 5: Find the absolute extrema of xxxf sinsin)( 2 on the interval ]2,0[

7

4.1 Practice Sheet

1. List all critical values for the following.

a. x2(x2- 6) b. 2

2

2 1

4

x

x

c. sec x

2. For each of the following find all absolute extrema over the following intervals:

a. x4 + 4x3 – 20x2 + 600 [ - 5, 1 ]

b. 2 6 10x x [ - 4, -1 ]

c. 2

5

x

x [ 0, 2 ]

d. 3sin2x [ 0, 2π ]

3. If a continuous function f has a relative minimum at c over [ a, b ], which of the following must be true?

I. f must have an absolute min on [ a, b ]

II. f ’(c) = 0

III. ( c, f(c) ) is an absolute min

a. I b. II c. III

d. I & II e. II & III

8

AP Calculus I

Notes 4.2

Rolle’s Theorem and the Mean Value Theorem

Exploration

Label the points (1,3) and (5,3) . Draw a graph of a differentiable function that starts at (1,3) and ends

at (5,3) .

What does it mean that the function must be differentiable?

Is there at least one point on the graph for which the derivative is zero?

Would it be possible to draw the graph so that there isn’t a point for which the derivative is zero? Explain your reasoning.

Rolle’s Theorem:

9

Ex. 1: Determine whether the function xxf sin)( over ]2,0[ satisfies the conditions of Rolle’s

Theorem. If f does, then find the guaranteed points.

Ex. 2: Find the two zeroes of the function and determine whether Rolle’s Theorem can be applied on the

interval of the zeros. If Rolle’s Theorem can be applied, find the values of c such that 0)(' cf .

a) 2)1)(3()( xxxf b) 4)( 32

xxf

10

The Mean Value Theorem:

If f is continuous on the closed interval ],[ ba and differentiable on the open interval ),( ba , then there

exists a number c in ),( ba such that ( ) ( )

'( )f b f a

f cb a

.

In other words, if f is continuous on the closed interval ],[ ba and differentiable on the open interval,

then there is some number c in ),( ba such that the _____________________ rate of change is equal to

the ________________ rate of change.

Ex. 3: Given 4

( ) 5f xx

, find all values of c in the open interval (1,4) such that ( ) ( )

'( )f b f a

f cb a

,

if they exist.

(1, 1)

(4, 4)

1 2 3 4 5 x

1

2

3

4

5

y

11

Ex. 4: Given 2

3( ) 4 6f x x x , find all values of c in the open interval ( 1,1) in which the average rate

of change is equal to the instantaneous rate of change. Also, over the interval (1,8) .

Ex. 5: 1999 BC

12

Section 4.3 Increasing and Decreasing Functions Discovery Worksheet

Use the graph of f(x) above to answer the following questions:

1. Find f ‘ (b)

2. Find f ‘ (c)

3. Find f ‘ (d)

4. Is the graph of f(x) increasing / decreasing on the interval [a,b]

5. Is the graph of f(x) increasing / decreasing on the interval [b,c]

6. Is the graph of f(x) increasing / decreasing on the interval [c,d]

7. Is the graph of f(x) increasing / decreasing on the interval [d,e]

8. Is f ‘ (x) positive / negative on the interval [a,b]

9. Is f ‘ (x) positive / negative on the interval [b,c]

10. Is f ‘ (x) positive / negative on the interval [c,d]

11. Is f ‘ (x) positive / negative on the interval [d,e]

12. Based on the observations above, given f ‘ (x), how can you tell when a function is increasing and when it is decreasing?

13. At which value(s) of x do relative minima occur?

14. Based on what you see above, relative minima occur when f ‘ (x) changes from _______________________________ to ________________________________.

15. At which value(s) of x do relative maxima occur?

16. Based on what you see above, relative maxima occur when f ‘ (x) changes from _______________________________ to ________________________________.

13

AP Calculus I

Notes 4.3

Increasing and Decreasing Functions and the First Derivative Test

Exploration

Examine the graph below.

1. On which intervals is the function increasing?

2. On which intervals is the function decreasing?

3. What separates these intervals?

4. What is true about the slopes of the tangent lines in the

intervals in which the function is increasing?

5. What is true about the slopes of the tangent lines in the

intervals in which the function is decreasing?

Definitions of Increasing and Decreasing Functions

A function is increasing if for two numbers 1x and 2x in the interval, 1x < 2x implies 1 2( ) ( )f x f x .

A function is decreasing if for two numbers 1x and 2x in the interval, 1x < 2x implies 1 2( ) ( )f x f x .

Theorem – Test for Increasing and Decreasing Functions

Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a,

b).

1. If ( ) 0f x for all x in (a, b), then f is increasing on [a, b].

2. If ( ) 0f x for all x in (a, b), then f is decreasing on [a, b].

3. If ( ) 0f x for all x in (a, b), then f is constant on [a, b].

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

14

Guidelines for Finding Intervals on Which a Function is Increasing or Decreasing

Let f be continuous on the interval (a, b). To find the open intervals on which f is increasing or decreasing,

use the following steps:

1. Find the derivative.

2. Solve for critical values ( f’(x) = 0 or undefined )

3. Plot critical numbers on a number line.

4. Test intervals for positive (+) or negative ( – )

A function is strictly monotonic on an interval if it is either increasing on the entire interval or decreasing

on the entire interval.

Ex. 1: Find the intervals on which 3 23

2( )f x x x is increasing or decreasing.

Ex. 2: Find the intervals on which 2( ) sin sinf x x x is increasing or decreasing from [0,2 ) .

15

Theorem – The First Derivative Test

Let c be a critical number of a function f that is continuous on an open interval I containing c. If f is

differentiable on the interval, except possibly at c, then f(c) can be classified as follows:

1. If ( )f x changes from negative to positive at c, then f(c) is a ___________________ of f.

2. If ( )f x changes from positive to negative at c, then f(c) is a ___________________ of f.

Ex. 3: Find the relative extrema of xxexf )(

.

Ex. 4: Find the relative extrema of the function 3 2( ) ( 2) ( 3)f x x x .

16

Ex. 5: Find the relative extrema of the function 2

2 3( ) 4f x x .

Ex. 6: Given a graph of 'f

a) Where is f increasing/decreasing?

b) Where does f have relative minimums or maximums on the interval ( -5, 4 ) ?

1 2 3–1–2–3–4 x

1

2

3

–1

–2

–3

–4

f'

17

Review of Derivatives

For problems 1-6 find dx

dy.

1) 2

23 346x

xxy 2) 2

3 32

x

xy

3) xxy cot6 4) xx

xy 5

1

2

5) )2(cos 54 xy 6) 34 tan(6 )f x x x

18

7) 3 26

7 5 5f x

x x

, find ' 1f 8) )2sin( xy , find '

4f

For Problem 9 &10 find 2

2

dx

yd.

9) xy csc 10) 2sin 3y x

Find the equation of the tangent line for the given function.

11) ( ) cos 2f x x x at x

19

AP Calculus I

Notes 4.4

Concavity and the Second Derivative Test

Definitions of Concavity

Let f be differentiable, the graph of f is concave upward if f is increasing on the interval (the slopes are

increasing) and concave downward if f is decreasing on the interval (the slopes are decreasing).

Exploration:

f ( x ) f ‘ ( x ) f ‘ ‘ ( x )

1. Let f be differentiable at c. If the graph of f is concave upward at , ( )c f c , the graph of f lies above the

tangent line at , ( )c f c on some open interval containing c.

2. Let f be differentiable at c. If the graph of f is concave downward at , ( )c f c , the graph of f lies below

the tangent line at , ( )c f c on some open interval containing c.

1 2–1–2 x

1

2

–1

–2

y

1 2–1–2 x

1

2

–1

–2

y

1. On which intervals is the graph of f (x) concave up?

2. On which intervals is the graph of f (x) concave down?

Think of concave up as a ____________ and concave down as an ______________.

1 2–1–2 x

1

2

–1

–2

y"

20

Theorem – Test for Concavity

1. Since concave upward means 'f is increasing and increasing means the derivative is positive,

2. Since concave downward means 'f is decreasing and decreasing means the derivative is negative,

Ex. 1: Determine the open intervals on which the graph of 2/2

)( xexf is concave upward or concave

downward.

Ex. 2: Determine the open intervals in which the graph of 2

2

1( )

4

xf x

x

is concave upward or concave

downward.

21

Theorem – Points of Inflection

If , ( )c f c is a point of inflection of a graph of f, then either ( ) 0f c or ( )f c is undefined. A point of inflection occurs when the concavity changes from upward or downward or vice versa. So, the

point of inflection occurs when "f changes from ____________ to _______________ or vice versa.

Ex. 3: Determine the points of inflection and discuss the concavity of the graph of xxxxf 45 5)(

Ex. 4: Find the points of inflection of ( )f x given '( ) 1f x x x .

22

Concavity also allows us to determine relative maxima of a function.

Theorem – The Second Derivative Test

Let f be a function such that ( ) 0f c and the second derivative of f exists on an interval containing c.

1. If ( ) 0f c , then f(c) is a _____________________ of f.

2. If ( ) 0f c , then f(c) is a _____________________ of f.

3. If ( )f c = 0, the test fails. In such cases, use the First Derivative Test.

Ex. 5: Find the relative extrema of the function 5 3( ) 3 5f x x x using the second derivative test.

Ex. 6: For all x in the closed interval 2,5 , the function f has a positive first derivative and a negative second derivative. Which of the following could be the table of values for f ?

x ( )f x

2 7

3 9

4 12

5 16

x ( )f x

2 7

3 11

4 14

5 16

x ( )f x

2 7

3 9

4 11

5 13

x ( )f x

2 16

3 14

4 11

5 7

x ( )f x

2 16

3 13

4 10

5 7

23

First Derivative

Relative Extrema

Increasing /

Decreasing

Second Derivative

Concavity

Point of Inflection

24

AP Calculus I

Graph of f’

Quick Definition Review

A function ( )f x is increasing when )(' xf __________________________________________________.

If the graph shown is ( )f x , ________________________________________________________.

If the graph shown is )(' xf , ________________________________________________________.

A function ( )f x is decreasing when )(' xf __________________________________________________.

If the graph shown is ( )f x , ________________________________________________________.

If the graph shown is )(' xf , ________________________________________________________.

A function has a local maximum when )(' xf _________________________________________________.

If the graph shown is ( )f x , ________________________________________________________.

If the graph shown is )(' xf , ________________________________________________________.

A function has a local minimum when )(' xf _________________________________________________.

If the graph shown is ( )f x , ________________________________________________________.

If the graph shown is )(' xf , ________________________________________________________.

A function has a point of inflection when )('' xf ______________________________________________.

If the graph shown is ( )f x , ________________________________________________________.

If the graph shown is )(' xf , ________________________________________________________.

If the graph shown is )('' xf , _______________________________________________________.

25

A function is concave up when )('' xf ______________________________________________________.

If the graph shown is ( )f x , ________________________________________________________.

If the graph shown is )(' xf , ________________________________________________________.

If the graph shown is )('' xf , _______________________________________________________.

A function is concave down when )('' xf _______________________________________________.

If the graph shown is ( )f x , ____________________________________________________.

If the graph shown is )(' xf , ___________________________________________________.

If the graph shown is )('' xf , __________________________________________________.

Ex. 1: Label the graph of )(xf :

26

Ex. 2: Now, let’s make the same graph a graph of )(' xf , assuming that )(xf is a differentiable function.

a) Find the interval(s) in which )(xf is increasing. Justify your answer.

b) Find the interval(s) in which )(xf is decreasing. Justify your answer.

c) Find the interval(s) in which )(xf is concave up. Justify your answer.

d) Find the interval(s) in which )(xf is concave down. Justify your answer.

e) Find all relative extrema and points of inflection of )(xf . Justify your answer.

27

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y'

Ex. 3: Given the function xxxf 43

1)( 3 , find the intervals of increasing, decreasing, concave up,

concave down, relative maxima, relative minima, and points of inflection. Then, use the graph to

verify all your answers.

28

Ex. 4:

29

Ex. 5:

30

Ex. 6: The function f is continuous on the closed interval 3,3 such that ( 3) 4f and (3) 1f . The

functions 'f and ''f have the properties given in the table below.

x 3 1x 1x 1 1x 1x 1 3x

'( )f x Positive Fails to

Exist Negative 0 Negative

''( )f x Positive Fails to

Exist Positive 0 Negative

(a) At what value of x does f attain its relative minimum and maximum value, if any?

(b) At what value of x does f attain its absolute minimum value and absolute maximum value on

the closed interval 3 3x ? Show the analysis that leads to your answer.

(c) Find the coordinatex of each point of inflection of the graph of f . Justify your answer.

(d) Sketch a possible graph of f .

31

Ex. 7:

(d) Given )1(f and )(2)( xfexg , write the equation of the tangent line to g at 1x .

(e) Given g as defined above, for what value(s) of x does g have a relative maximum? Justify your answer.

32

AP Calculus I

Notes 4.7

Optimization

One of the most common applications of calculus involves determining maximum and minimum values

Problem-Solving Strategy for Applied Minimum and Maximum Problems

1) Assign symbols to all given quantities and quantities to be determined. When feasible, make a sketch.

2) Write a primary equation for the quantity that is to be maximized (or minimized).

3) Reduce the primary equation to one having a single independent variable. This may involve the use of

secondary equations relating the independent variables of the primary equation.

4) Differentiate the equation and then solve for the critical values

5) Determine the desired maximum or minimum value depending on the situation.

Ex. 1: A manufacturer wants to design an open box having a square base and a surface area of 108 square

inches. What dimensions will produce a box with maximum volume?

2 4S x xh

33

Ex. 2: A rectangular field, bounded on one side by a building, is to be fenced in on the other three sides. If

3000 feet of fence is to be used, find the dimensions of the largest field that can be fenced in.

Ex. 3: A rectangular page is to contain 24 square inches of print. The margins at the top and bottom of the

page are to be 1.5 inches, and the margins on the left and right are to be 1 inch. What should the

dimensions of the page be so that the least amount of paper is used?

34

Ex. 4: The product of two positive numbers is 200. Minimize the sum of the first and twice the second.

Ex. 5: What is the radius of a cylindrical soda can with volume of 512 cubic inches that will use the

minimum material? (Formulas for a cylinder: 2V r h and 22 2SA r rh )

35

Ex. 6: Max wants to make a box with no lid from a rectangular sheet of cardboard that is 18 inches by 24

inches. The box is to be made by cutting a square of side x from each corner of the sheet and

folding up the sides. Find the value of x that maximizes the volume of the box.

Ex. 7: A Norman window has the shape of a rectangle with a semicircle on top. If the perimeter of the

entire window is 30 feet, find the dimensions of the rectangular portion of the window to allow in the

greatest possible amount of light.

36

Ex. 8: Which points on the graph of 24y x are closest to the point (0,2) ?

Ex. 9: Suppose that the revenue of a company can be represented with the function ( ) 48r x x and the

company’s cost function is 3 2( ) 12 60c x x x x , where x represents thousands of units and

revenue and cost are represented in thousands of dollars. What production level maximizes profit,

and what is the maximum profit to the nearest thousand dollars?

1 2 3–1–2–3 x

1

2

3

4

–1

–2

y

37

1. The product of two positive numbers is 200. Minimize the sum of the first and three times the second.

2. A printed page must contain 60 square centimeters of printed material. There are to be margins of 5 centimeters on either side, and margins of 3 centimeters each on the top and the bottom. How long should

the printed lines be in order to minimize the amount of paper used?

3. A closed box with a square base is to contain 252 cubic feet. The bottom costs $5 per square foot, the top costs $2 per square foot, and the sides cost $3 per square foot. Find the dimensions that will minimize the

cost.

4. A cylindrical can, open at the top, is to hold 500 cm3 of liquid. Find the height and radius that minimize the amount of material needed to manufacture the can.

5. A farmer with 750 feet of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible area of the four pens?

6. An open box is formed by cutting squares of equal size from the corners of a 24 by 15-inch piece of sheet

metal and folding up the sides. Determine the size of the cutout that maximizes the volume of the box.

7. A computer company determines that its profit equation (in millions of dollars) is given by 3 248 720 1000P x x x , where x is the number of thousands of units of software sold and 0 20x .

Optimize the manufacturer’s profit.

8. A window consists of an open rectangle topped by a semicircle and is to have a perimeter of 288 inches.

Find the radius of the semicircle that will maximize the area of the window.

9. You have 30” of wire and need to create an equilateral triangle and a square (two separate figures- not

connected). What dimensions will generate the least area? (Hint: 2 23

0.4334

equilateral triangleA x x )

38

Chapter 4 Practice Problems

Non-calculator.

_____ 1. The figure shows the graph of the derivative of a function. The derivative is never negative. Which

statement is false?

(A) The function has no relative maximum value. (B) The function has no relative minimum value. (C) The function is always concave up. (D) The function has exactly three points of inflection. (E) The function is increasing for all x.

_____ 2. A function f is differentiable for all values of x, and ( ) 0 f x at 3, 1,x x and 2x . If

2( ) 2 7f x x x , where does a relative minimum occur?

(A) 3 x (B) 1x (C) 2x (D) 3, 2 x x (E) 3, 1 x x

_____ 3. Let f be a function with a derivative 234 54273)(' xxxxf . What are the x-coordinates of the

relative maxima of the graph of f ?

(A) 0 only (B) 3 only (C) 0 and 6 only (D) 3 and 6 only (E) 0, 3, and 6

_____ 4. A particle moves along the x-axis so that at time 0t its position is given by 3 2( ) 2 21 72 53 x t t t t . At what time t is the particle at rest?

(A) 1t only (B) 3t only (C) 7

2t only (D) 3t and

7

2t (E) 3t and 4t

_____5. The product of two positive numbers is 150. Minimize the sum of twice the first number and three

times the second number. What is the sum?

(A 10 (B) 15 (C) 60 (D) 65 (E) 75

_____ 6. Let f be the function given by xxexf x 32)( . The graph of f is concave up when

(A) 2 x (B) 2 x (C) 1 x (D) 1 x (E) 0x

39

_____ 7. The function f is continuous and differentiable on the closed interval [3, 7] . The table above

gives selected values of f on this interval. Which of the following statements must be true?

I. The maximum value of f on [3, 7] is 20.

II. There exists c, for 3 7 c , such that ( ) 0 f c .

III. 0)(' xf for 54 x .

(A) I only (B) II only (C) III only (D) I and III only (E) I, II, and III

_____ 8. A test plane flies in a straight line with positive velocity v(t), in miles per minute, where v is a

differentiable function of t. Selected values of v(t) for [0,40] are shown in the table.

t(minutes) 0 5 10 15 20 25 30 35 40

v(t) (mpm) 7.0 9.2 9.5 7.0 4.5 2.4 2.4 4.3 7.3

Based on the values in the table, what is the smallest number of instances at which the acceleration

of the plane could equal zero on the open interval (0,40)?

(A) 1 (B) 2 (C) 3 (D) 4 (E) Not enough information

_____ 9. The figure shows the graph of )(' xf , the derivative of a function, f . The derivative is continuous

and 0)2(' f . Which of the following is true?

I. The function has at least one relative minimum point. II. The function has at least one relative maximum point. III. The function has no relative maximum point.

(A) I only (B) II only (C) III only (D) I and II (E) I and III

_____ 10. Let f be the function given by 4ln)( 2 xxf . At what x-value is the relative minimum found?

(A) None (B) 2 (C) 0 (D) 2 (E) 2 and 2

_____ 11. Let f be defined as 2cos4)( xxxf . When is the graph of f concave down from 2,0 ?

(A)

3

4,

3

2 (B)

6

7,

6

5 (C)

3

2,0

(D)

3

5,

3

(E)

2,

3

4,

3

2,0

x 3 4 5 6 7

( )f x 20 17 12 16 20

40

_____ 12. Find the maximum value of the function 1642)( 23 xxxxf on ]4,1[ .

(A) 3

2 (B) 1 (C) 2 (D) 4 (E) 32

13. A function f is continuous on the interval [-5, 5] and its first and second derivatives have the values given in

the following table:

a. Find the x-coordinates of the relative maxima and minima of f on [-5, 5], if any. Justify your answer.

b. Find the x-coordinates of all points of inflection of f on the interval [-5, 5, if any]? Justify your answer.

c. Given 1)3( f , sketch a possible graph for f which satisfies all of the given properties.

x (-5, -1) -1 (-1, 2) 2 (2, 4) 4 (4, 5)

f’(x) Negative 0 Positive Positive Positive 0 Positive

f”(x) Positive Positive Positive 0 Negative 0 Positive

41

Calculator Active.

_____ 14. A particle moves along the x-axis so that at any time 0t its velocity is given by 2( ) ln( 2) v t t t .

What is the acceleration of the particle at time 6t ?

(A) 1.500 (B) 20.453 (C) 29.453 (D) 74.860 (E) 133.417

_____ 15. Find the value c guaranteed by Rolle’s Theorem for 42

arctan2)(

xxxf on [0, 1] .

(A) 0.068 (B) 0.273 (C) 0.363 (D) 0.523 (E) It does not apply.

_____ 16.

_____ 17. The function f has the first derivative given by 3

( )1

xf x

x x

. What is the x-coordinate of the

inflection point of the graph of f ?

(A) 1.008 (B) 0.473 (C) 0 (D) -0.278 (E) f has no inflection point.

_____ 18. Let f be the function with the derivative given by )16.1cos()(' 143.0 xexf x . How many relative

extrema does f have on the interval 61 x ?

(A) One (B) Two (C) Three (D) Four (E) Five

_____ 19. Let f be the function with the derivative given by )16.1cos()(' 143.0 xexf x . How many points of

inflection does f have on the interval 61 x ?

(A) One (B) Two (C) Three (D) Four (E) Five

_____ 20. Which of the following functions satisfy the conditions of Rolle’s Theorem on the interval [0, 2] ?

I. 2( ) 2f x x x II. 1

( )1

f xx

III. ( ) 1f x x

(A) I only (B) II only (C) III only (D) I and III (E) I, II, and III

42

_____ 21. A can of mountain dew has a volume of 21.65 cubic inches. What is the radius of the cylindrical can

that will use the minimum material? (Formulas for a cylinder: 2V r h and 22 2SA r rh )

(A) 1.199in (B) 1.273in (C) 1.362in (D) 1.510in (E) 1.620in

_____ 22.

_____ 23. Find the value of c that satisfies the Mean Value Theorem for )sin(5.1)( xxf from 2.1,3.0 .

(A) 0.4712 (B) 0.664 (C) 0.702 (D) 0.727 (E) 0.750

43

Chapter 4 - AP Materials Name: _____________________ * Unless noted with a “*” a calculator is NOT ALLOWED.

1) What is the x-coordinate of the point of inflection on the graph 3 21

5 243

y x x ?

A. 5 B. 0 C. 10

3 D. –5 E. –10

2) A particle moves along the x-axis so that its position at time t is given by:

2( ) 6 5x t t t . For what value of t is the velocity of the particle zero?

A. 1 B. 2 C. 3 D. 4 E. 5

3) If 2' '( ) ( 1)( 2)f x x x x then the graph of f has inflection points when x =

A. –1 only B. 2 only C. –1 and 0 only D. –1 and 2 only E. –1, 0, and 2 only

4) The function f is given by 4 2( ) 2f x x x . On which of the following intervals is f increasing?

A. 1

,2

B. 1 1

,2 2

C. 0, D. ,0 E. 1

,2

5) The maximum acceleration attained on the interval 0 3t by the particle whose

velocity is given by 3 2( ) 3 12 4v t t t t is

A. 9 B. 12 C. 4 D. 21 E. 40

6)* The first derivative of the function f is given by 2cos 1

'( )5

xf x

x .

How many critical values does f have on the open interval (0, 10)?

A. One B. Three C. Four D. Five E. Seven

44

7) Let f be the function given by ( )f x x . Which of the following statements about f are true?

I. f is continuous at x = 0. II. f is differentiable at x = 0. III. f has an absolute minimum at x = 0.

A. I only B. II only C. III only D. I and III only E. II and III only

8) If g is a differentiable function such that ( ) 0g x for all real numbers x and if

2'( ) ( 4) ( )f x x g x , which of the following is true?

A. f has a relative maximum at 2x and a relative minimum at 2x . B. f has a relative minimum at 2x and a relative maximum at 2x . C. f has relative minima at 2x and 2x . D. f has relative maxima at 2x and 2x . E. It cannot be determined if f has any relative extrema.

9) Let f be the function with derivative given by 22

'( )f x xx

.

On which of the following intervals is f decreasing?

A. ( , 1 only B. ,0 C. 1,0) only D. 3(0, 2 E. 3 2 , )

10) Let f be the function given by ( ) 2 xf x xe The graph of f is concave down when

A. 2x B. 2x C. 1x D. 1x E. 0x

11)

The derivative 'g of a function g is continuous and has exactly two zeros. Selected values of 'g are

given in the table above. If the domain of g is the set of all real number, then g is decreasing on

which of the following intervals?

A. 2 2x only B. 1 1x only C. 2x D. 2x only E. 2x or 2x

12) Let g be a twice-differentiable function with '( ) 0g x and ' '( ) 0g x for all real numbers x , such

that (4) 12g and (5) 18g . Of the following, which is a possible value for (6)g ?

A. 15 B. 18 C. 21 D. 24 E. 27

x -4 -3 -2 -1 0 1 2 3 4 '( )g x 2 3 0 -3 -2 -1 0 3 2

45

13)* A particle moves along the x-axis so that at any time 0t , its velocity is given by

( ) 3 4.1cos(0.9 )v t t . What is the acceleration of the particle at time 4t ?

A. –2.016 B. –0.677 C. 1.633 D. 1.814 E. 2.978

14)* Let f be the function with derivative given by 2'( ) sin 1f x x . How many relative extrema does f have on the interval 2 4x ?

A. One B. Two C. Three D. Four E. Five

15)* The function f has first derivative given by 3

'( )1

xf x

x x

. What is the x-coordinate of the

inflection point of the graph of f ?

A. 1.008 B. 0.473 C. 0 D. –0.278 E. the graph has no inflection point

16) For all x in the closed interval 2,5 , the function f has a positive first derivative and a negative second derivative. Which of the following could be a table of values for f ?

A. B. C. B. C. D. E.

x ( )f x

2 7

3 9

4 12

5 16

x ( )f x

2 7

3 11

4 14

5 16

x ( )f x

2 16

3 12

4 9

5 7

x ( )f x

2 16

3 14

4 11

5 7

x ( )f x

2 16

3 13

4 10

5 7