Prep Session I...Calculus Test Format % of Grade Number of Questions Time allotted Calculator Use Section I 50% Part A 28 multiple choice 55 minutes no calculator Part B 17 multiple

AP Calculus

Prep Session I

February 2, 2008

Houston ISD

Presented by:Advanced Placement Strategies�™, Inc.

Advanced Placement®, AP®, and Advanced Placement Strategies™ are trademarks of the College Entrance Examination Board and used under license by AP Strategies™, Inc. The College Entrance Examination

Board is not involved in the production nor endorses the use of this material.

Calculus Test Format

% of Grade Number of

Questions

Time allotted

Calculator Use

Section I 50% Part A 28 multiple

choice 55 minutes no

calculator

Part B 17 multiple choice

50 minutes graphing calculator required

Section II 50% Part A 3 free

response questions

45 minutes graphing calculator required

Part B

3 free response questions

45 minutes

no calculator

2008 Exams Schedule Week 1

Morning Session 8 a.m.*

Afternoon Session 12 noon*

Monday, May 5 Government and Politics: United States

Government and Politics: Comparative** French Language**

Tuesday, May 6 Computer Science A** Computer Science AB** Spanish Language**

Statistics

Wednesday, May 7 Calculus AB** Calculus BC**

Chinese Language and Culture

Thursday, May 8 English Literature** German Language**

Japanese Language and Culture** French Literature**

Friday, May 9 United States History European History Studio Art (portfolios due)

Week 2 Morning Session 8 a.m.* Afternoon

Session 12 noon*

Afternoon Session 2 p.m.

Monday, May 12 Biology** Music Theory**

Physics B** Physics C: Mechanics**

Physics C: Electricity and Magnetism†

Tuesday, May 13 Environmental Science** Chemistry**

Psychology

Wednesday, May 14 Italian Language and Culture** English Language**

Art History

Thursday, May 15 Macroeconomics** World History**

Microeconomics

Friday, May 16 Human Geography** Spanish Literature**

Latin Literature** Latin: Vergil**

Limits, Continuity,

and the Definition of the Derivative

DEFINITION Derivative of a Function The derivative of the function f with respect to the variable x is the function f whose value at x is

0

( ) (( ) limh

)f x h f xf xh

provided the limit exists. DEFINITION (ALTERNATE) Derivative at a Point The derivative of the function f at the point x a is the limit

( ) ( )( ) limx a

f x f af ax a

provided the limit exists.

DEFINITION Continuity

A function f is continuous at a number a if 1) ( )f a is defined (a is in the domain of f ) 2) lim ( )

x af x exists

3) lim ( ) ( )

x af x f a

Limits, Continuity, and the Definition of the Derivative Page 1 of 12

1. What is ?2cos

2cos

lim0 h

h

h

(A) -1 (B) 2

2 (C) 0 (D) 1 (E) The limit does not exist

2. The graph of the function f is shown in the figure below. Which of the following

statements about f is true? (A) )(lim)(lim xfxf

bxax

(B) 2)(lim xfax

(C) 2)(lim xfbx

(D) 1)(lim xfbx

(E) existnotdoesxfax

)(lim

3. The graph of a function f is shown below. Which of the following statements about f

is false? (A) f is continuous at x = a (B) f has a relative maximum at x = a (C) x = 2 is in the domain of f (D) )(lim)(lim xftoequalisxf

axax

(E) exists )(lim xfax


4. If 2 2

4 40, limx a

x aa thenx a

is

(A) 2

1a

(B) 22

1a

(C) 26

1a

(D) 0 (E) nonexistent

5. 1

limx

1 11 2

1x

x

(A) 14

(B) –1

(C) 14

(D) 0 (E) does not exist

6. 9

limx

93x

x

(A) 6 (B) –6 (C) 0 (D) 1

(E) 13


7. If 3

2 6 , 3( ) , then lim ( )3

5, 3x

x xf x fx

xx

(A) 5 (B) 1 (C) 2 (D) 0 (E) does not exist

8. If 1

1( ) and lim ( ) does not exist, then 2 x k

f x f xx

k

( ) 2 ( ) 1x

f x f x

(A) 2 (B) 3 (C) 1 (D) –2 (E) –1

9. If the graph of f is as shown below, then lim 2

3

X

Y

-2 -1 1 2 3 4 5 6 7 8

-5

-4

-3

-2

-1

1

2

3

4

5

0

(3, -4)

(A) 8 (B) 10 (C) 11

(D) 9 (E) 25


10. 2

1

3 2lim1x

xx

(A) 0 (B) –2

(C) 12

(D) 2 (E) does not exist

11. 0

sinlimx

xx

(A) –1 (B) 0 (C) 1

(D) 12

(E) does not exist 12. Let

2( ) 2 . The lim ( )

xf x x f x

(A) 0 (B) 1 (C) –1 (D) (E)

13. 24

4 6lim2 5 1x

xx x 2

(A) 0 (B) 1 (C) (D) (E) none of these


14. 2

23

3lim6 9x

x xx x

(A) –3 (B) –1 (C) 1 (D) 3 (E) does not exist

15. For 0

1 1 10,limh

xh x h x

(A) 2

1x

(B) 2x

(C) 2

1x

(D) 2

2x

(E) 0

16. 32

32

1 2lim

1x

x

x

(A) 8 (B) 1 (C) 0

(D) (E) –8

17. 2

2

6 1lim ,200 4 2x

n kn kn

(A) 3 (B) 6 (C) 12

(D) 8 (E) 2


18. 8 5 6 4 4 2

9 6 7 5 5 3

10 10 10lim10 10 10x

x x xx x x

(A) 0 (B) 1 (C) –1

(D) 110

(E) 110

19. The 0

tan3 tan 3limh

x h xh

is

(A) 0 (B) 23sec 3x

(C) 2sec 3x

(D) 3cot 3x (E) nonexistent

20. If lim where L is a real number, which of the following must be true? ( ) L,

x af x

(A) ( )f x exists (B) ( )f x is continuous at x a (C) ( )f x is defined at x a (D) f a L (E) none of the above

21. What is 8 8

0

8 0.5 8 0.5limh

hh

(A) 0

(B) 14

(C) 12

(D) 1 (E) the limit does not exist


22. 0

sin sinlimh

hh

(A) 1 (B) 0 (C) –1

(D) (E)

23. 3 3

limx a

x ax a

(A) 0 (B) 32 a

(C) 3 23 a

(D) 3

3aa

(E) does not exist

24. 0

tan 2 tan 2limh

x h xh

(A) 0 (B) 2cot 2x

(C) 2sec 2x

(D) 22 sec 2x (E) does not exist


25. If which of the following must be true?

3lim ( ) 7,x

f x

I. f is continuous at 3x II. f is differentiable at 3x III. (3) 7f

(A) none (B) II only (C) III only

(D) I and III only (E) I, II, and III

26. If 2 5 7( )

2(2)

x xf xx

f k for x 2 and if f is continuous at 2x then k =

(A) 0

(B) 16

(C) 13

(D) 1

(E) 75

27. For what value of c is 23 2,

( )5, 1

x xf x

cx x1

continuous

(A) 3 (B) –3 (C) 6 (D) none (E) 0


28. Consider the function sin ,

( ), 0

x xf x x

k x

0 in order for ( )f x to be continuous at

, the value of k must be 0x

(A) 0 (B) 1 (C) –1 (D) (E) a number >1

29. Let

2 1,( ) 11 , 1

x 1xf x xx

Which of the following statements is correct?

(A) ( )f x is continuous at 1 since ( )f x is defined at 1x (B) ( )f x is continuous at 1 since

1lim ( )x

f x exists

(C) ( )f x is not continuous at 1 since ( )f x is not defined at 1x (D) ( )f x is not continuous at 1 since

1lim ( )x

f x does not exist

(E) ( )f x is not continuous at 1 since 1

lim ( ) (1)x

f x f

30. Use and find the 22 ,

( )2 ,

x xf x

x x00 0

( )limh

f x h f xh

(A) 0 (B) 1 (C) –1 (D) 2 (E) does not exist


3

31. At x = 3, the function given by 2 , 3

( )6 9,

x xf x

x x is

(A) Undefined (B) Continuous but not differentiable (C) Differentiable but not continuous (D) Neither continuous nor differentiable (E) Both continuous and differentiable

32. If f is a continuous function on [a,b], which of the following is necessarily true?

(A) f exist on (a,b) (B) 0( )f x is a maximum of f, then 0( )f x = 0 (C)

0 00lim ( ) (lim ), ( , )

x x x xf x f x for x a b

(D) ( ) 0 [ , ]f x for some x a b (E) the graph of f is a straight line

33. Which of the following is true about the function f if 2

2

12 5

xf x

x x 3 ?

I. f is continuous at x = 1 II. The graph of f has a vertical asymptote at x = 1.

III. The graph of f has a horizontal asymptote at y = 12

.

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


2

Free Response 1. Let f be the function defined as follows:

, where a and b are constants 2

| 2 | 4,( )

, 2x x

f xax bx x

(a) If a = -1/2 and b = 4, is f continuous for all x? Justify your answer. (This question implies one should name the intervals of continuity and justify with limits) (b) Describe all values of a and b for which f is a continuous function. (c) For what values of a and b is f both continuous and differentiable.


Free Response 2. Let f be a function defined by

2

2 1, 2( ) 1 , 2

2

x xf x

x k x

(a) For what value of k will f be continuous at x = 2. Justify your answer algebraically. (b) Using the value of k found in part (a), determine whether f is differentiable at x = 2. Justify your answer. (c) Let k = 4. Determine whether f is differentiable at x = 2. Justify your answer algebraically.

Relationships Between , ,f f f

Calculus – 13 Things I Know

1. If a function is increasing, the first derivative is positive (or occasionally zero, think 3x .)

2. If the first derivative is positive, the function is increasing. 3. If a function is decreasing, the first derivative is negative (or occasionally

zero, think 3x .) 4. If the first derivative is negative, the function is decreasing. 5. If a function is concave up, the second derivative is positive. 6. If the second derivative is positive, the function is concave up. 7. If a function is concave down, the second derivative is negative. 8. If the second derivative is negative, the function is concave down. 9. A function has a relative maximum if the first derivative changes sign from

positive to negative. 10. A function has a relative minimum if the first derivative changes sign from

negative to positive. 11. A function has a point of inflection if the second derivative is zero or

undefined and if the concavity changes sign. 12. If the first derivative is increasing, the function is concave up. 13. If the first derivative is decreasing, the function is concave down.

Relationships Between , ,f f f Page 1 of 12

Multiple Choice:

1. At which of the five points on the graph below are 2

2

dxydand

dxdy both negative?

(A) A (B) B (C) C (D) D (E) E 2. The graph of the derivative of f is shown in the figure below. Which of the following could be the graph of f ?


3. The graph of a twice-differentiable function f is shown in the figure below. Which of the following is true?

(A) f f f( ) ( ) ( )1 1 1

1111

(B) f f f( ) ( ) ( )1 1(C) f f f( ) ( ) ( )1 1(D) f f f( ) ( ) ( )1 1(E) f f f( ) ( ) ( )1 1

4.

Let f be a function whose domain is the open interval (1,5). The figure above shows the graph of . Which of the following describes the relative extrema of and the points of inflection of the graph of ?

f ff

(A) 1 relative maximum, 1 relative minimum, and no point of inflection (B) 1 relative maximum, 2 relative minimum, and no point of inflection (C) 1 relative maximum, 1 relative minimum, and 1 point of inflection (D) 1 relative maximum and 2 points of inflection (E) 1 relative minimum and 2 points of inflection


5.

The graphs of the derivatives of the functions f, g, and h are shown above. Which of the functions f, g, or h have a relative maximum on the open interval a < x < b? (A) f only (B) g only (C) h only (D) f and g only (E) f, g, and h 6. Which of the following statements about the function given by

4 3( ) 2f x x x is true?

(A) The function has no relative extremum (B) The graph of the function has one point of inflection, and the function has two relative extrema. (C) The graph of the function has two points of inflection, and the function has one relative extremum. (D) The graph of the function has two points of inflection, and the function has two relative extrema. (E) The graph of the function has two points of inflection, and the function has three relative extrema.

7. The graph of is concave down for 4 3 23 16 24 4y x x x 8

(A) 0x(B) 0x

(C) or 2x 23

x

(D) 23

x or 2x

(E) 2 23

x


8. Let f be a function defined for all real numbers x. If 2

|4|)(2

xxxf , then f is

decreasing on the interval (A) )2,((B) ),((C) ( 2, 4)(D) ),2((E) ),2(

9. If , then the graph of f has inflection points when x = 2)2)(1()( xxxxf

(A) -1 only (B) 2 only (C) –1 and 0 only (D) –1 and 2 only (E) –1, 0, and 2 only

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

f increasing?

(A) ,21

(B) 2

1,21

(C) ,0(D) 0,

(E) 21

,


11. Let g be the function given by ( ) 100 20sin 10cos2 6t tg t . For , g is

decreasing at a decreasing rate when t =

80 t

(A) 0.949 (B) 2.017 (C) 3.106 (D) 5.965 (E) 8.000

12. The derivative of 4 5

( )3 5x xf x attains its maximum value at x =

(A) –1 (B) 0 (C) 1

(D) 43

(E) 53

13. For what value of k will kxx

have a relative maximum at ? 2x

(A) –4 (B) –2 (C) 2 (D) 4 (E) none of these

14. The maximum value of 3 2( ) 2 9 12 1f x x x x on 1, 2 is

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4


15. For what value of k will 2

8x kx

have a relative minimum at 4x ?

(A) –32 (B) –16 (C) 0 (D) 16 (E) 32

16. The function f given by 3( ) 12 24f x x x

(A) is increasing for , decreasing for 2x 2 x 2 , increasing for 2x (B) is decreasing for , increasing for 0x 0x (C) is increasing for all x (D) is decreasing for all x (E) is decreasing for , increasing for 2x 2 x 2 , decreasing for 2x 17. Let f be a polynomial function with degree greater than 2. If

, which of the following must be true for at least one value of x

and ( ) ( ) 1a b f a f bbetween a and b?

I. ( ) 0f x II. ( ) 0f xIII. 0f x

(A) None (B) I only (C) II only (D) I and II only (E) I, II and III

18. Given the function defined by 5( ) 3 20 3f x x x , find all values of x for which the

graph of f is concave up.

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


19. If the graph of 3 2 4y x ax bx has a point of inflection at (1, -6), what is the

value of b?

(A) 3 (B) 0 (C) 1 (D) 3 (E) It cannot be determined from the information given.

20. If the graph of 2( ) 2 kf x xx

has a point of inflection at 1x , then the value of

k is

(A) 1 (B) –1 (C) 2 (D) –2 (E) 0

21. The function 4 2 8 1y x bx x has a horizontal tangent and a point of inflection for the same value of x . What must be the value of b ?

(A) 6 (B) 4 (C) 1 (D) -1 (E) -6

22. Let be a relative maximum for 2, 2g 3 2( ) 2 6g x x hx kx . Use the fact

that 1 12, 2

g is an inflection point to find the value of h k .

(A) -15 (B) -9 (C) 9 (D) 15 (E) 24


23. Let h be a continuous and differentiable function defined on 0,2 . Some function values of h and are given in the table below: h

x

0

2

32

2

h x

3

6

4

0

-2

h x 332

-1

2 1

If 2sin 2p x h x , evaluate 2

p .

(A) -2 (B) -1 (C) 0 (D) 1 (E) 2 Use the following table to help you answer questions 24 and 25:

x f x f x 0 7 -1 1 3 1 2 -1 6 3 2 4 4 5 -3

24. What is 2m if ?

3m x f x

(A) -18 (B) -3 (C) 0 (D) 3 (E) 18 25. What is if 0k xk x f e ? (A) -2 (B) -1 (C) 1 (D) 2 (E) 3


4

Free Response 1, No Calculator

x 0 0 1x 1 1 x 2 2 2 3x 3 3 4x ( )f x –1 Negative 0 Positive 2 Positive 0 Negative ( )f x 4 Positive 0 Positive DNE Negative –3 Negative

( )f x –2 Negative 0 Positive DNE Negative 0 Positive Let f be a function that is continuous on the interval [0, 4). The function f is twice differentiable except at . The function f and its derivatives have the properties indicated in the table above, where DNE indicates that the derivatives of f do not exist at

.

2x

2x (a) For 0 , find all values of x at which f has a relative extremum. Determine

whether f has a relative maximum or a relative minimum at each of these values. Justify your answer.

x

(b) On the axes provided, sketch the graph of the function that has all the

characteristics of f.

X

Y

O


4

4

(c) Let g be the function defined by on the open interval .

For , find all values of x at which g has a relative extremum. Determine whether g has a relative maximum or a relative minimum at each of these values. Justify your answer.

1( ) ( )

xg x f t dt (0, 4)

0 x

(d) For the function g defined in part (c), find all values of x, for 0 x , at which

the graph of g has a point of inflection. Justify your answer.


Free Response 2, No Calculator. The figure above shows the graph of f , the derivative of the function f, on the closed interval . The graph of 1 x 5 f has horizontal tangent lines at x = 1 and x = 3. The function f is twice differentiable with f(2) = 6. (a) Find the x-coordinate of each of the points of inflection of the graph of f. Give a

reason for your answer. (b) At what value of x does f attain its absolute minimum value on the closed interval

? At what value of x does f attain its absolute maximum value on the closed interval ? Show the analysis that leads to your answers.

1 x 551 x


(c) Let g be the function defined by ( ) ( )g x x f x . Find an equation for the line

tangent to the graph of g at x = 2.

Related Rates

Strategy for Solving Related Rate Problems (Calculus, Finney, Demana, Waits, Kennedy)

Understand the problem.

Identify the variable whose rate of change you seek and the variable (or variables) whose rate of change you know.

Develop a mathematical model of the problem. Draw a picture (many of these problems involve geometric figures) and label the parts that are important to the problem. Be sure to distinguish constant quantities from variables that change over time. Only constant quantities can be assigned numerical values at the start.

Write an equation relating the variable whose rate of change you seek with the variable(s) whose rate of change you know. The formula is often geometric, but it could come from a scientific application.

Differentiate both sides of the equation implicitly with respect to time t. Be sure to follow all the differentiation rules. The Chain Rule will be especially critical, as you will be differentiating with respect to the parameter t.

Substitute values for any quantities that depend on time. Notice that it is only safe to do this after the differentiation step. Substituting too soon “freezes the picture” and makes changeable variables behave like constants, with zero derivatives.

Interpret the solution. Translate your mathematical result into the problem setting (with appropriate units) and decide whether the result makes sense. Check to make sure you answered the question asked.

Related Rates P. 1 of 17

Circles: 1. (calculator) The radius of a circle is increasing at a constant rate of 0.2 meters per second.

What is the rate of increase in the area of the circle at the instant when the circumference of the circle is 20 meters?

(A) (B) (C) 20.04 / secm 20.4 / secm 24 / sem c (D) (E) 220 / secm 2100 / secm

2. (calculator) The radius of a circle is decreasing at a constant rate of 0.1 centimeter per second. In terms of the circumference C, what is the rate of change of the area of the circle, in square centimeters per second?

(A) 0.2 C (B) 0.1 C (C) 0.12

C

(D) (E) 20.1 C 20.1 C


3. The radius of a circle is increasing at a nonzero rate, and at a certain instant, the rate of

increase in the area of the circle is numerically equal to the rate of increase in its circumference. At this instant, the radius of the circle is numerically equal to the rate of increase in its circumference. At this instant, the radius of the circle is

(A) 1 (B) 12

(C) 2 (D) 1 (E) 2

4. When the area in square units of an expanding circle is increasing twice as fast as its

radius in linear units, the radius is

(A) 14

(B) 14

(C) 1 (D) 1 (E)


Triangle Area: 5. (calculator) If the base b of a triangle is increasing at a rate of 3 inches per minute while

its height h is decreasing at a rate of 3 inches per minute, which of the following must be true about the area A of the triangle?

(A) A is always increasing. (B) A is always decreasing. (C) A is decreasing only when b h . (D) A is decreasing only when . b h (E) A remains constant.

6. The rate of change of the area of an equilateral triangle with respect to its side s at 2s is approximately:

(A) 34

(B) 3 (C) 32

(D) 7 (E) 12


7. Let BAC in triangle ABC with AB and ACc b , where b and c are constants

and c . Side BC changes length as the measure of b changes. Find the instantaneous

rate of change of the area of triangle ABC when 3

. Assume the instantaneous rate

of change of is 2.

(A) 2cb (B) 3cb (C) 32

cd (D) 2

cd (E) 2cb

Triangles—Pythagorean Theorem 8. (calculator) A railroad track and a road cross at right angles. An observer stands on the

road 70 meters south of the crossing and watches an eastbound train traveling at 60 meters per second. At how many meters per second is the train moving away from the observer 4 seconds after it passes through the intersection?

(A) 57.60 (B) 57.88 (C) 59.20

(D) 60.00 (E) 67.40


9. The top of a 25-foot ladder is sliding down a vertical wall at a constant rate of 3 feet per minute. When the top of the ladder is 7 feet from the ground, what is the rate of change of the distance between the bottom of the ladder and the wall?

(A) 78

feet per minute (B) 724

feet per minute (C) 724

feet per minute

(D) 78

feet per minute (E) 2125

feet per minute

z y

x

10. The sides of the rectangle above increase in such a way that 1 and 3dz dx dydt dt dt

. At

the instant when and 4x 3y , what is the value of dxdt

?

(A) 13

(B) 1 (C) 2 (D) 5 (E) 5


11. A missile rises vertically from a point on the ground 75000 feet from a radar station. If

the missile is rising at the rate of 16500 feet per minute at the instant when it is 38000 feet high, what is the rate of change, in radians per minute, of the missile’s angle of elevation from the radar station at this instant?

(A) 0.175 (B) 0.219 (C) 0.227 (D) 0.469 (E) 0.507

Volume--Cones

12. The volume of a cone of radius r and height h is given by 213

V r h . If the radius and

the height both increase at a constant rate of 12

centimeter per second, at what rate, in

cubic centimeters per second, is the volume increasing when the height is 9 centimeters and the radius is 6 centimeters?

(A) 2

(B) 10 (C) 24 (D) 54 (E) 108


t

Volume—Cubes 13. The volume V measured in cubic inches of un-melted ice remaining from a melting ice

cube after t seconds is . How fast is the volume changing when seconds?

22000 40 0.2V t40t

(A) 3

26 insec

(B) 243in

sec (C)

3

120 insec

(D) 3

0 insec

(E) 243in

sec

14. If the volume of a cube is increasing at a rate of 3

300 inmin

at the instant when the edge is

20 inches, then the rate at which the edge is changing is

(A) 14

inmin

(B) 12

inmin

(C) 13

inmin

(D) 1 inmin

(E) 34

inmin


15. The volume of a cube in increasing at the rate of 20 cubic centimeters per second. How

fast, in square centimeters per second, is the surface area of the cube increasing at the instant when each edge of the cube is 10 centimeters long?

(A) 43

(B) 2 (C) 4

(D) 6 (E) 8

Volume/Surface Area—Spheres

16. If the radius of a sphere is increasing at the rate of 2 inches per second, how fast, in cubic inches per second, is the volume increasing when the radius is 10 inches?

(A) 800 (B) 800 (C) 3200 (D) 40 (E) 80


17. The radius r of a sphere is increasing at the uniform rate of 0.3 inches per second. At the

instant when the surface area S becomes 100 square inches, what is the rate of increase, in cubic inches per second, in the volume V?

2 344 and 3

S r V r

(A) 10 (B) 12 (C) 22.5 (D) 25 (E) 30

18. The volume of an expanding sphere is increasing at a rate of 12 cubic feet per second. When the volume of the sphere is 36 cubic feet, how fast, in square feet per second, is the surface area increasing?

(A) 8 (B) 6 (C) 8

(D) 83

(E) 10


2002 AB 6 Form B

Ship A is traveling due west toward Lighthouse Rock at a speed of 15 kilometers per hour (km/hr). Ship B is traveling due north away from Lighthouse Rock at a speed of 10 km/hr. Let x be the distance between Ship A and Lighthouse Rock at time t, and let y be the distance between Ship B and Lighthouse Rock at time t, as shown in the figure above. (a) Find the distance, in kilometers, between Ship A and Ship B when x 4 km and y 3 km. (b) Find the rate of change, in km/hr, of the distance between the two ships when x 4 km and

km. y 3


(c) Let be the angle shown in the figure. Find the rate of change of , in radians per hour,

when x 4 km and y 3 km.


1999 AB 6

In the figure above, line is tangent to the graph of 2

1yx

at point P, with coordinates

2

1,ww

, where w > 0. Point Q has coordinates ( , . Line crosses the x-axis at point R,

with coordinates .

0)w

( ,0)k (a) Find the value of k when w = 3. (b) For all w > 0, find k in terms of w.


(c) Suppose that w is increasing at the constant rate of 7 units per second. When w = 5, what is

the rate of change of k with respect to time? (d) Suppose that w is increasing at the constant rate of 7 units per second. When w = 5, what is

the rate of change of the area of PQR with respect to time? Determine whether the area is increasing or decreasing at this instant.


1995 AB5 and BC3

As shown in the figure above, water is draining from a conical tank with height 12 feet and diameter 8 feet into a cylindrical tank that has a base with area 400 square feet. The depth h, in feet, of the water in the conical tank is changing at the rate of ( 12)h feet per minute. (The

volume V of a cone with radius r and height h is 21 .3

V r h )

(a) Write an expression for the volume of water in the conical tank as a function of h. (b) At what rate is the volume of water in the conical tank changing when ? Indicate

units of measure. 3h


(c) Let y be the depth, in feet, of the water in the cylindrical tank. At what rate is y changing

when ? Indicate the units of measure. 3h


2003 AB 5 and BC 5 A coffeepot has the shape of a cylinder with radius 5 inches, as shown in the figure above. Let h be the depth of the coffee in the pot, measured in inches, where h is a function of time t, measured in seconds. The volume V of coffee in the pot is changing at the rate of 5 h cubic inches per second. (The volume V of a cylinder with radius r and height h is .) 2V r h

(a) Show that 5

dh hdt

.


(b) Given that h = 17 at time t = 0, solve the differential equation

5dh hdt

for h as a

function of t. (c) At what time t is the coffeepot empty?

Derivatives and

Rate of Change

Theorems

THEOREM: Intermediate Value Theorem for Continuous Functions (IVT) A function that is continuous on a closed interval ( )y f x ,a b takes on every value between ( )f a and ( )f b . In other words, if 0y is between ( )f a and ( )f b , then

0 ( )y f c for some c in ,a b . Example:

The function above is continuous over the interval 0 x 3 . Therefore, every y-value between 1 and 4 is guaranteed somewhere on the interval, at least one time.

THEOREM: Mean Value Theorem for Derivatives (MVT) If is continuous at every point of the closed interval ( )y f x ,a b and differentiable at every point of its interior , then there is at least one point c in at which ( , )a b ( , )a b

( ) ( )( ) f b f af cb a

Example:

For the given function, the slope of the segment connecting (0 and (3 is 1. If the , 1) , 4)function is continuous on the closed interval and differentiable on the open interval, there is at least one place where the slope of the tangent line will be 1. This occurs when

32

x .

Page 1 of 11

Derivatives and Rate of Change

Graphing Calculator allowed.

1. Let f be the function given by 2( ) sinf x x . The first positive root for f is

A) 0 B) 2

C) 1.772 D) 2.507 E)

2. For , how many points of intersection are there for the graphs of

10 10xxy e and ? siny x

A) 0 B) 1 C) 2 D) 3 E) 4

3. The root for 3( ) ( 9) 1010xf x x is

A) –124.271 B) –11.283 C) 11.072 D) 11.283 E) 13.142 4. For x > 10, find the root for . 2( ) 289 1420f x x x A) 279 B) 284 C) 289 D) 1420 E) none of these

5. If ( )x x

x xe ef xe e

, how many horizontal asymptotes does f have?

A) 0 B) 1 C) 2 D) 3 E) 4 6. Let 4 2( ) 8 1f x x x x . The equation of the line tangent to f(x) at x = 1 is y = 14x – 5. Calculate the difference between the zero of the function f that occurs in the interval [0, 2] and the zero of the line tangent to f at x = 1. A) 0.123 B) 0.234 C) 0.357 D) 0.679 E) 1.523

Page 2 of 11 7. How many relative extrema are there for the function g if 4 3 2( ) 23 60 20g x x x x A) 1 B) 2 C) 3 D) 4 E) 5 8. If ( ) cos ln xf x x x x e , then (1)f A) 0 B) 1.000 C) 1.321 D) 1.406 E) 1.469 9. The slope of the line tangent to the curve lny x x at x = 2 is closest to A) 0.76 B) 0.80 C) 0.84 D) 0.90 E) 0.94 10. The graph of the function f given 2( ) x 3f x e x changes concavity at x = A) –.58 B) –.52 C) –.46 D) –.40 E) –.34

11. The function f is defined on the closed interval [1, 4] by 2

2

9( )2

xf xx

. For

what values of x in the interval [1, 4] is the instantaneous rate of change equal to the average rate of change of f over the interval [1, 4]. A) x = 2.00 B) x = 2.11 C) x = 2.22 D) x = 2.33 E) x = 2.44 12. If c satisfies the conclusion of the Mean Value Theorem for ( ) secf x x on the

interval 1 12

x , then c is

A) 0.394 B) 0.751 C) 0.787 D) 0.799 E) 1.574

Page 3 of 11 13. A particle moves along the x-axis so that at time t its position is given by . For what values of t is the velocity of the particle increasing?

3( ) ( 1)( 3)x t t t

A) B) 0 C) 13t 3 3t t D) 1 or 3t t E) all t

14. For 2( ) sinf x x and on the interval 2( ) 0.5g x x ,2 2

, the

instantaneous rate of change of f is greater than the instantaneous rate of change of g for which values of x? A) –0.8 B) 0 C) 0.9 D) 1.2 E) 1.5

15. 3

sec 2lim2x

xx

A) –1.010 B) –.990 C) 0 D) 1.041 E) Does not exit 16. Let f be the function given by 23 xf x e and let g be the function given by

36 .g x x At what value of x do the graphs of f and g have parallel tangent lines?

(A) –0.701 (B) –0.567 (C) –0.391 (D) –0.302 (E) –0.258

17. The first derivative of the function f is given by 2cos 1

5xf 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

Page 4 of 11

18.The function f given by 3( ) 12 24f x x x is (A) increasing for x , decreasing for 2 2 2x 2x

2 2x

, increasing for (B) decreasing for , increasing for (C) increasing for all x (D) decreasing for all x (E) decreasing for , increasing for

0x 0x

2x

2

, decreasing for 2x 19. Which of the following is an equation of the line tangent to the graph

4( ) 2f x x x at the point where ( ) 1f x (A) (B) 8 5y x 7y x (C) 0.763y x (D) (E) 0.122y x 2.146y x 20. The graph of the function 3 26 7 2cosx x x x

8

changes concavity at x = (A) –1.581 (B) –1.632 (C) –1.675 (D) –1.894 (E) –2.337 21. If , what is the minimum value of the product xy? 2y x

(A) –16 (B) –8 (C) –4

(D) 0 (E) 2

Page 5 of 11 22. Let ( )f x x . If the rate of change of f at x = c is twice its rate of change at x = 1, then c =

(A) 14

(B) 1 (C) 4

(D) 12

(E) 12 2

23. An equation of the line tangent to the graph of 23 2

xyx

3 at the point (1, 5) is

(A) (B) 1313 8x y 18x y 13 64

13

(C) x y (D) (E) 13 66x y 2 3x y

24. The slope of the line normal to the graph of 2ln(sec )y x at 4

x is

(A) –2 (B) 12

(C) 12

(D) 2 (E) nonexistent

25. If ( ) sin2xf x , then there exists a number c in the interval 3

2 2x that

satisfies the conclusion of the Mean Value Theorem. Which of the following could be c?

(A) 23

(B) 34

(C) 56

(D) (E) 32

Page 6 of 11

26. A particle moves along a line so that at time t, where 0 t , its position is given

by 2

( ) 4cos 102ts t t . What is the velocity of the particle when its

acceleration is zero? (A) –5.19 (B) 0.74 (C) 1.32 (D) 2.55 (E) 8.13

No calculator.

27. If 32( )f x x , then (4)f

(A) - 6 (B) - 3 (C) 3 (D) 6 (E) 8

28. If , then in terms of x and y, 3 33 2 1x xy y 7 dydx

(A) 2

22x y

x y

(B) 2

2

x yx y

(C) 2

2x yx y

(D) 2

22x y

y

(E) 2

21 2x

y

Page 7 of 11

29. If , then tan coty x x dydx

(A) sec cscx x (B) sec cscx x (C) sec cscx x (D) 2 2sec cscx x (E) 2 2sec cscx x 30. If ( ) 2 cosf x x x , which of the following could be ( )f x ?

(A) 3

cos 13x x x

(B) 3

cos 13x x x

(C) 3 cos 1x x x (D) 2 sin 1x x (E) 2 sin 1x x

31. If 2

2 3( ) 2 1f x x x , then (0)f

(A) 43

(B) 0 (C) 23

(D) 43

(E) 2

32. If ( ) sin xf x e , then ( )f x (A) cos xe

(B) cos x xe e

(C) cos x xe e

(D) cosx xe e

(E) cosx xe e

Page 8 of 11 33. The table below gives values for , , , andf f g g at selected values of x. If , then ( ) ( ( ))h x f g x (2)h x ( )f x ( )f x ( )g x ( )g x 2 6 4 2 1

2 4 -3 4 3 4 2 2 1 2

(A) 3 (B) 6 (C) 8 (D) 9 (E) 12 34. The function g is continuous for 3 x 5 and differentiable for . If

and , which of the following statements are true? 3 5x

( 3) 2g (5) 3g There exists c, where , such that: 3 c 5

4

I. ( ) 0g cII. ( ) 0g cIII. ( ) 1g c (A) I only (B) II only (C) III only (D) I and II only (E) I and III only 35. The function g is continuous for 2 x and differentiable for . If

and , which of the following statements are true? 2 4x

( 2) 5g (4) 1g There exists c , where , such that: 2 c 4 I. ( ) 0g cII. ( ) 1g cIII. ( ) 1g c (A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III

Page 9 of 11 36. If the line tangent to the graph of the function f at the point (1 passes through the point , then

, 5)( 3, 3) (1)f

(A) (B) (C) 2 1 12

(D) 1 (E) 2

37. Let f be a twice-differentiable function with a positive first derivative and a negative second derivative. If and (1) 2g (2) 10g , which of the following could be ? (3)g (A) 6 (B) 16 (C) 18 (D) 20 (E) 22 38. Which of the following functions could be increasing at an increasing rate? (A) (D)

x ( )f x 1 10 2 12 3 14 4 16

x ( )f x1 19 2 15 3 12 4 10

(B) (E)

x ( )f x 1 10 2 20 3 28 4 34

x ( )f x1 16 2 14 3 12 4 10

(C)

x ( )f x 1 10 2 12 3 15 4 19

Page 10 of 11 Free Response 1, No Calculator. The twice-differentiable function f is defined for all real numbers and satisfies the following conditions: , (0) 2f (0) 4f , and (0) 3f . (a) The function g is given by for all real numbers, where a is a

constant. Find and ( ) ( )a xg x e f x

(0)g (0)g in terms of a. Show the work that leads to your answers.

(b) The function h is given by ( ) cos ( )h x kx f x for all real numbers, where k is a

constant. Find and write an equation for the line tangent to the graph of h at .

( )h x0x

Page 11 of 11 Free Response 2, No Calculator Consider the curve given by 2 24 7 3x y xy .

(a) Show that 3 28 3

dy y xdx y x

.

(b) Show that there is a point P with x-coordinate 3 at which the line tangent to the curve

at P is horizontal. Find the y-coordinate of P.

(c) Find the value of 2

2

d ydx

at the point P found in part (b). Does the curve have a local

maximum, a local minimum, or neither at the point P? Justify your answer.

Position, Velocity, and Acceleration (Motion)

AP Calculus AB

Position, Velocity, and Acceleration (Motion along a line)

Suppose an object is moving along a straight line (say x-axis) so that it s position x, as a function of time t, on that line is given by ( )x f t . Average velocity of the object over the time interval t to t dt is given by

( ) (dx f t dt f tdt dt

) , or change in positionchange in time

.

Instantaneous velocity of the object is the derivative of the position function ( )x f t with respect to time. ( ) ( )v t x t

Speed is the absolute value of the velocity. ( ) .dxSpeed v tdt

Acceleration is the derivative of velocity with respect to time. ( ) ( ) ( )a t v t x t

( ) ( ) ,

( ) ( )

v t dt x t c

a t dt v t c

Total distance traveled from time 1t t to t t2 is given by

2

1

( )t

tTDT v t dt .

Speeding Up or Slowing Down

Sign Convention: When the object is moving in the right direction or moving upward then the velocity is positive (Graph of velocity vs. time is above the t axis). When the object is moving in the left direction or moving downward then the velocity is negative. (Graph of velocity vs. time is below the t axis

t

V

Time

0 2t ,v a Object is slowing down

2 4t ,v a Object is speeding up

4 6t ,v a Object is slowing down

6 8t ,v a Object is speeding up

What you need to know about motion along the x-axis: There are four closely related concepts that you need to keep straight: position, velocity, speed and acceleration. You need to understand how these relate to one another. If x(t) represents the position of a particle along the x-axis at any time t, then the following statements are true.

1. “Initially” means when time t = 0.

2. “At the origin” means x(t) = 0.

3. “At rest” means velocity v(t) = 0.

4. If the velocity of the particle is positive, then the particle is moving to the right.

5. If the velocity of the particle is negative, then the particle is moving to the left.

6. To find average velocity, divide the change in position by the change in time.

7. Instantaneous velocity is the velocity at a single moment (instant!) in time.

8. If the acceleration of the particle is positive, then the velocity is increasing.

9. If the acceleration of the particle is negative, then the velocity is decreasing.

10. Speed is the absolute value of velocity.

11. If the velocity and acceleration have the same sign (both positive or both negative),

then speed is increasing.

12. If the velocity and acceleration are opposite in sign (one is positive and the other is

negative), then speed is decreasing.

13. To determine total distance traveled over a time interval, you must find the sum of

the absolute values of the differences in position between all resting points.

14. There are 3 ways to use an anti-derivative that are easily confused. Watch out!

Position, Velocity, and Acceleration (Motion) Page 1 of 7

1. The velocity of a particle moving on a line at time t is 1 323 5v t t 2 meters per second.

How many meters did the particle travel from 0t to 4t ? (A) 32 (B) 40 (C) 64 (D) 80 (E) 184 2. If the position of a particle on the x-axis at time t is 25t , then the average velocity of the

particle for 0 3 is t (A) –45 (B –30 (C) –15 (D) –10 (E) –5 3. A particle moves along the x-axis so that at any time its position is given by 0t

3 23 9 1x t t t t . For what values of t is the particle at rest? (A) No values (B) 1 only (C) 3 only (D) 5 only (E) 1 and 3

4. The position of a particle moving along the x-axis is sin 2 cos 3x t t t for time . When 0t t , the acceleration of the particle is

(A) 9 (B) 19

(C) 0 (D) 19

(E) 9

5. A particle moves along a line so that at time t, where 0 t , its position is given by 2

4cos 102ts t t . What is the velocity of the particle when its acceleration is

zero? (A) –5.19 (B) 0.74 (C) 1.32 (D) 2.55 (E) 8.13


6. A particle moves along the x-axis so that its acceleration at any time t is . If

the initial velocity of the particle is 6, at what time t during the interval 0 is the particle farthest to the right?

2 7a t t4t

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4 7. A particle starts from rest at the point 2,0 and moves along the x-axis with a constant

positive acceleration for time . Which of the following could be the graph of the distance of the particle from the origin as a function of time?

0ts t

8. Two particles start at the origin and move along the x-axis. For 0 1t 0 , their

respective position functions are given by 1 sinx t and 22 1tx e . For how many

values of t do the particles have the same velocity? (A) None (B) One (C) Two (D) Three (E) Four


Questions 9 – 10 refer to the following situation.

t

v

1 2 3 4 5 6 7 8

-2

-1

1

2

3

4

0

A bug begins to crawl up a vertical wire at time 0t . The velocity v of the bug at time t,

, is given by the function whose graph is shown above. 0 t 8 9. At what value of t does the bug change direction? (A) 2 (B) 4 (C) 6 (D) 7 (E) 8 10. What is the total distance the bug traveled from 0t to 8t ? (A) 14 (B) 13 (C) 11 (D) 8 (E) 6

11. A particle moves along the x-axis so that at any time its velocity is given by . The total distance traveled by the particle from t = 0 to t = 2 is

0t( ) ln( 1) 2 1v t t t

(A) 0.667 (B) 0.704 (C) 1.540 (D) 2.667 (E) 2.901

t (sec) 0 2 4 6 a(t) (ft/sec2) 5 2 8 3

12. The data for the acceleration of a car from 0 to 6 seconds are given in the table

above. If the velocity at t = 0 is 11 feet per second, the approximate value of the velocity at t = 6, computed using a left-hand Riemann sum with three subintervals of equal length, is

( )a t

(A) 26 ft/sec (B) 30 ft/sec (C) 37 ft/sec (D) 39 ft/sec (E) 41 ft/sec


Free Response Questions 1999 AB 1, Calculator allowed. A particle moves along the y-axis with velocity given by for 2( ) sin( )v t t t 0.t (a) In which direction (up or down) is the particle moving at time 1.5t ? Why? (b) Find the acceleration of the particle at time 1.5t . Is the velocity of the particle

increasing at ? Why or why not? 1.5t (c) Given that is the position of the particle at time t and that , find . ( )y t (0) 3y (2)y (d) Find the total distance traveled by the particle from 0t to 2t .


2003 AB 4 Form B A particle moves along the x-axis with velocity at time given by . 0t 1( ) 1 tv t e (a) Find the acceleration of the particle at time t = 3. (b) Is the speed of the particle increasing at time t = 3? Give a reason for your answer. (c) Find all values of t at which the particle changes direction. Justify your answer. (d) Find the total distance traveled by the particle over the time interval . 0 3t


2001 AB 3 and BC 3

A car is traveling on a straight road with velocity 55 ft/sec at time 0.t For 0 seconds, the car’s acceleration a(t), in , is the piecewise linear function defined by the graph above.

1t 8

8

18

8

2ft/sec (a) Is the velocity of the car increasing at t = 2 seconds? Why or why not? (b) At what time in the interval 0 , other than t = 0, is the velocity of the car 1t

55 ft/sec? Why?

(c) On the time interval 0 , what is the car’s absolute maximum velocity, in ft/sec, and at what time does it occur? Justify your answer.

t

(d) At what times in the interval 0 1t , if any, is the car’s velocity equal to zero? Justify

your answer.


2002 AB 3 Form B A particle moves along the x-axis so that its velocity v at any time t, for 0 t 16 , is given by

. At time , the particle is at the origin. v(t ) e2sin t 1 t 0 (a) Sketch the graph of for v(t ) 0 t 16 . (b) During what intervals of time is the particle moving to the left? Give a reason for your

answer. (c) Find the total distance traveled by the particle from t 0 to t 4. (d) Is there any time t, , at which the particle returns to the origin? Justify your

answer. 0 t 16

Advanced Integration Topics

Advanced Integration Topics

Integration by Parts

udv u v vdu

Integration by Partial Fractions with Linear Factors

1. Divide if Improper degree of the numerator greater than the

2. inator tor into m linear factors and create a sum of m

3. decomposition

If the integrand is a fraction with thedegree of the denominator, divide the numerator by the denominator and then apply steps 2 and 3. Factor the DenomCompletely factor the denominafractions, each having one of the factors as a denominator. Solve for the numerator values and integrate the

1 2 3 mA A A A dxax b cx d ex f rx t

Improper Integrals with Infinite Integration Limits

1. If f is continuous on the interval

a , then

b

a abf x dx lim f x dx, .

2. If f is continuous on the interval b b

aaf x dx lim f x dx,b , then .

3. If f is continuous on the interval ,

c

, then c

f x dx f x dx f x dx , where c is any real number.

the first two cases, the improper integral converges if the limit exists --- otherwise, the In

improper integral diverges. In the third case, the improper integral on the left diverges if either of the improper integrals on the right diverges.

Advanced Integration Topics Page 1 of 8

x

What are advanced techniques of integration? There are two techniques that you are responsible for knowing on the AP exam. The first is called integration by parts and is based on your knowledge of the product rule.

udv uv vdu

1. sinx xd

2. ln xdx

3. Let’s review the tabular method: 3 2xx e dx


The second technique is called partial fractions.

4. 2

14dx

x

5. 2

12 3

dxx x

6. 2

52 1

x dxx x


Now, we should be able to use our knowledge of integration to deal with improper integrals. There are two types of improper integrals:

i) integrals that are improper because one or both of the limits of integration are infinite

7. dxx1

8. dxx2

11

ii) integrals that are improper because there is an infinite discontinuity either between or at the limits of integration

9. dxx

4

332

1

3


10. dxx

2

203

1

1

What might this look like on the AP exam? 1985 BC 5 Let f be the function defined by f (x ) ln x for 0 x 1 and let R be the region between the graph of f and the x-axis. (a) Determine whether region R has finite area. Justify your answer.


1980 BC 6 Let R be the region enclosed by the graphs of y e x , x k (k 0), and the coordinate axes. (a) Write an improper integral that represents the limit of the area of the region R as k

increases without bound and find the value of the integral if it exists. 1971 BC 5

Determine whether or not converges. If it converges, give the value. Show

your reasoning.

x e x dx0


1995 AB 2 A particle moves along the y-axis so that its velocity at any time is given by

. At time t = 0, the position of the particle is y = 3. 0t

( ) cosv t t t (c) Write an expression for the position of the particle. ( )y t 1996 BC 1 Consider the graph of the function h given by

2

( ) xh x e for 0 x . (a) Let R be the unbounded region in the first quadrant below the graph of h. Find the

volume of the solid generated when R is revolved about the y-axis.


2001 BC 5 Let f be the function satisfying ( ) 3 ( )f x x f x , for all real numbers x, with and .

(1) 4flim ( ) 0x

f x

(a) Evaluate

13 ( )x f x dx . Show the work that leads to your answer.

Multiple Choice: 11. 2secx x dx

(a) tanx x C (b) 2

tan2x x C (c) 2 2sec 2 sec tanx x x C

(d) tan ln cosx x x C (e) tan ln cosx x x C

12. 3

2

31 2

dxx x

(a) 3320

(b) 920

(c) 5ln2

(d) 8ln5

(e) 2ln5


13. 234

29

x dxx

(a) 237 (b)

233 7

2 (c)

2 239 73 (d)

2 23 33 9 7

2 (e) DNE

14. ( )x f x dx =

(a) ( )x f x x f x dx (b) 2 2

( )2 2x xf x f x dx

(c) 2

( )2xx f x f x C (d) ( )x f x f x dx

(e) 2

2x f x dx

15. 20

cosx x dx

(a) 2

(b) - 1 (c) 12

(d) 1 (e) 12

Area and

Volume

DEFINITION Area Between Curves If f and g are continuous with ( ) ( )f x g x throughout ,a b , then the area between the curves and from a to b is the integral of y f x( ) y g x( ) f g from a to b, ( ) ( )

b

aA f x g x dx .

DEFINITION Volume of a Solid The volume of a solid of known integrable cross section area ( )A x from x a to x b is the integral of A from a to b, . ( )

b

aV A x dx

Area and Volume P. 1 of 16

2006 AB 1 and BC 1 Let R be the shaded region bounded by the graph of lny x and the line 2y x , as shown above. (a) Find the area of R. (b) Find the volume of the solid generated when R is rotated about the horizontal line

. 3y (c) Write, but do not evaluate, an integral expression that can be used to find the

volume of the solid generated when R is rotated about the y-axis.


2006 AB 1 and BC 1 Form B

Let f be the function given by 3 2

( ) 3cos4 3 2x x xf x x . Let R be the shaded region in

the second quadrant bounded by the graph of f, and let S be the shaded region bounded by the graph of f and the line , the line tangent to the graph of f at x = 0, as shown above. (a) Find the area of R. (b) Find the volume of the solid generated when R is rotated about the horizontal line

. 2y (c) Write, but do not evaluate, an integral expression that can be used to find the area of S.


Area and Volume Free Response 2004 AB 1 Form B Let R be the region enclosed by the graph of 1y x , the vertical line x = 10, and the x-axis. (a) Find the area of R. (b) Find the volume of the solid generated when R is revolved about the horizontal line

y = 3. (c) Find the volume of the solid generated when R is revolved about the vertical line x = 10.


2003 AB 1 and BC 1 Form B Let f be the function given by 2 3( ) 4f x x x , and let be the line 18 3y x , where is tangent to the graph of f. Let R be the region bounded by the graph of f and the x-axis, and let S be the region bounded by the graph of f, the line , and the x-axis, as shown above. (a) Show that is tangent to the graph of ( )y f x at the point x = 3. (b) Find the area of S. (c) Find the volume of the solid generated when R is revolved about the x-axis.


2002 AB 1 and BC 1 Let f and g be the functions given by and f (x ) ex

g( x) ln x .

(a) Find the area of the region enclosed by the graphs of f and g between x 1

2 and

x 1. (b) Find the volume of the solid generated when the region enclosed by the graphs of f

and g between x 1

2 and x 1 is revolved about the line y 4.

(c) Let h be the function given by h(x ) f (x ) g( x). Find the absolute minimum

value of h(x ) on the closed interval 12

x 1, and find the absolute maximum

value of h(x ) on the closed interval 12

x 1. Show the analysis that leads to your

answers.


2000 AB 1 and BC 1

Let R be the shaded region in the first quadrant enclosed by the graphs of

2xy e , 1 cosy x , and the y-axis, as shown in the figure above.

(a) Find the area of the region R. (b) Find the volume of the solid generated when the region R is revolved about the

x-axis.

(c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Find the volume of this solid.


Area and Volume Multiple Choice Area 1. What is the area of the region between the graphs of 2y x and y x from x = 0 to x = 2?

(A) 23

(B) 83

(C) 4

(D) 143

(E) 163

2. The area of the region enclosed by the graph of 2 1y x and the line y = 5 is

(A) 143

(B) 163

(C) 283

(D) 323

(E) 8 3. The area of the region in the first quadrant between the graph of 24y x x and the x - axis is

(A) 2 23

(B) 83

(C) 2 2 (D) 2 3

(E) 163


4. Which of the following represents the area of the shaded region in the figure?

(A)

d

cf y dy (B) (C)

b

ad f x dx f b f a

(D) (E) b a f b f a d c f b f a

5. The area of the region enclosed by the curve 11

yx

, the x-axis, and the lines

3and 4x x is

(A) 5 36

(B) 213

n (C) 413

n (D) 312

n (E) 1 6n

6. The area of the region enclosed by the graphs of 2and 3 3y x y x x is

(A) 23

(B) 1 (C) 43

(D) 2 (E) 143

7. The area of the region in the first quadrant that is enclosed by the graphs of

3 8 and 8y x y x

(A) 14

(B) 12

(C) 34

(D) 1 (E) 654


8. The area of the region bounded by the lines 20, 2, and 0 and thecurvex

x x y y e

(A) 12

e (B) (C) 1e 2 1e (D) 2 1e (E) 2 e

9. What is the area of the region completely bounded by the curve 2 6y x x and the line 4y

(A) 32

(B) 73

(C) 92

(D) 316

(E) 332

10. The region bounded by the x-axis and the part of the graph of

cos between2

y x x x k is three times the area of the region for

, then 2

k x k

(A) arcsin 14

(B) arcsin 13

(C) 6

(D) 4

(E) 3

11. The area of the region bounded by the curve

2 , the -axis, the -axis, and the line 2xy e x y x is

(A) 4

2e e (B)

4

12e (C)

4 12 2e (D) 42e e (E) 42 2e


12. A region in the plane is bounded by the graph of 1yx

, the x-axis, the line x m ,

and the line 2 ,x m The area of this region 0.m(A) is independent of m (B) increases as m increases (C) decreases as m increases

(D) decreases as m increases when 1 ;2

m

increases as m increases when 12

m

(E) increases as m increases when 1 ;2

m

decreases as m increases when 12

m

13. The area in the first quadrant that is enclosed by the graphs of 3 and 4x y x y is (A) (B) 8 (C) 4 4 (D) 0


Area (calculator active)

14. If 02

k and the area under the curve cosy x from x = k to 2

x is 0.1,

then k = (A) 1.471 (B) 1.414 (C) 1.277 (D) 1.120 (E) 0.436

15. What is the area of the region in the first quadrant enclosed by the graphs of y cos x , y = x, and the y-axis?

(A) 0.127 (B) 0.385 (C) 0.400 (D) 0.600 (E) 0.947


Volume 16. If the region enclosed by the y-axis, the line y = 2, and the curve y x is

revolved about the y-axis, the volume of the solid generated is

(A) 325

(B) 163

(C) 165

(D) 83

(E)

17. Let R be the region in the first quadrant enclosed by the graph of 131y x , the

line x = 7, the x-axis, and the y-axis. The volume of the solid generated when R is revolved about the y-axis is given by

(A) 273

01x dx (B)

173

02 1x x dx

(C) 223

01x dx (D)

123

02 1x x dx

(E) 7 23

01y dy

18. The region enclosed by the x-axis, the line x = 3, and the curve y x is rotated about the x-axis. What is the volume of the solid generated?

(A) 3 (B) 23

(C) 92

(D) 9 (E) 36 35


19. A region in the first quadrant is enclosed by the graphs of 2xy e , x = 1, and the coordinate axes. If the region is rotated about the y-axis, the volume of the solid that is generated is represented by which of the following integrals?

(A) 1 2

02 xxe dx (B)

1 (C) 2

02 xe dx

1 4

0

xe dx

(D) 0

1ey n y dy (E) 2

01

4e

n y dy

20. The volume of the solid obtained by revolving the region enclosed by the ellipse

about the x-axis is 2 29x y 9 (A) 2 (B) 4 (C) 6 (D) 9 (E) 12 21. The region enclosed by the graph of 2y x , the line x = 2, and the x-axis is revolved about the y-axis. The volume of the solid generated is

(A) 8 (B) 325

(C) 163

(D) 4 (E) 83

22. The region in the first quadrant bounded by the graph of y = sec x, 4

x , and the

axes is rotated about the x-axis. What is the volume of the solid generated?

(A) 2

4 (B) 1 (C) (D) 2 (E) 8

3

23. The first region in the first quadrant bounded by y = cos x, y = sin x, and the y-axis is rotated about the x-axis. The volume of the resulting solid is

(A) 2

(B) (C) 12

(D) 2 1 (E) 14 2


24. Which definite integral represents the volume of a sphere with radius 2 ?

(A) 2 2

24x dx (B)

2 2

24x dx

(C) 2 2

02 4 x dx (D)

2 2

22 4 x dx

(E) 2 2

04 x dx

25. The region in the first quadrant enclosed by the ellipse 2 24x y 36 and the coordinate axes is rotated about the y-axis. The volume of the resulting solid is (A) 9 (B) 12 (C) 18 (D) 36 (E) 72


Volume (calculator)

X

Y

8

4

0

26. The base of a solid is a region in the first quadrant bounded by the x-axis, the y-

axis, and the line , as shown in the figure above. If cross sections of the solid perpendicular to the x-axis are semicircles, what is the volume of the solid?

2 8x y

(A) 12.566 (B) 14.661 (C) 16.755 (D) 67.021 (E) 134.041 27. The base of a solid S is the region enclosed by the graph of lny x , the line x = e, and the x-axis. If the cross sections of S perpendicular to the perpendicular

to the x-axis are squares, then the volume of S is

(A) 12

(B) 23

(C) 1 (D) 2

(E) 31 13

e


The volume generated by revolving about the x axis the region enclosed by the graphs of 2y x and 22y x , for 0 x 1 is

(A) 1 2 2

0(2 2 )x x dx

(B) 1 2 4

0(4 4 )x x dx

(C) 1 2

02 (2 2 )x x x dx

(D) 2

2

0 22y y dy

(E) 222

0 2 2y y dy

Documents

Prep Session I...Calculus Test Format % of Grade Number of Questions Time allotted Calculator Use Section I 50% Part A 28 multiple choice 55 minutes no calculator Part B 17 multiple