ExamView - AP Calc AB Chapter 3 practice test

Name: ________________________ Class: ___________________ Date: __________

AP Calculus AB Chapter 3 Practice Test

Multiple Choice

Identify the choice that best completes the statement or answers the question.

____

1. Find any critical numbers of the function g (t ) = t

a.

b.

c.

d.

e.

____

7

3

14

3

7

?

3

14

?

3

0

2. Find all critical numbers of the function f (x ) = sin 2 6x + cos 6x , 0 < x <

a.

b.

c.

d.

e.

____

7 ? t , t < 7.

¦Ð

3

.

2¦Ð

36 6 9

¦Ð 5¦Ð 7¦Ð

,

,

6 24 24

¦Ð

18

¦Ð

24

,

,

,

¦Ð

¦Ð

¦Ð

12

¦Ð

,

,

¦Ð

9

,

¦Ð

4

¦Ð

2

5¦Ð

,

,

18 6 18

¦Ð

6

,

¦Ð

?

?

3. Locate the absolute extrema of the function f (x ) = x 3 ? 3x on the closed interval ??? 0, 4 ¨B¨B? .

a.

b.

c.

d.

e.

absolute max: ??? 4, 52 ??? ; no absolute min

absolute max: ??? 1, ? 2 ??? ; absolute min: ??? 4, 52 ???

no absolute max; absolute min: ??? 4, 52 ???

absolute max: ??? 4, 52 ??? ; absolute min: ??? 1, ? 2 ???

no absolute max or min

1

ID: A

Name: ________________________

____

4. Locate the absolute extrema of the function g (t ) =

a.

b.

c.

d.

e.

____

t2

on the closed interval

t2 + 3

??

?

?? ?2, 2 ¨B¨B? .

4

, and it occurs only at the right endpoint x = 2

7

The absolute minimum is 0 and it occurs at the critical number x = 0.

4

The absolute maximum is , and it occurs only at the left endpoint x = ?2 .

7

The absolute minimum is 0 and it occurs at the critical number x = 0.

4

The absolute maximum is , and it occurs at either endpoint x = ¡À2 .

7

The absolute minimum is 0, and it occurs at the critical number x = 0.

4

The absolute maximum is , and it occurs at the critical number x = 0.

7

4

The absolute minimum is , and it occurs at the left endpoint x = ?2 .

13

4

The absolute maximum is , and it occurs at the critical number x = 0.

7

4

The absolute minimum is , and it occurs at the right endpoint x = 2 .

13

The absolute maximum is

5. Determine whether Rolle's Theorem can be applied to the function f (x ) = x 2 ? 6x + 8 on the closed interval

[2,4]. If Rolle's Theorem can be applied, find all values of c in the open interval (2,4) such that f ¡ä (c) = 0.

a.

b.

c.

d.

e.

____

ID: A

Rolle's Theorem applies; c = 3

Rolle's Theorem applies; c = 3.5

Rolle's Theorem applies; c = 2.5

Rolle's Theorem applies; c = ¨C3

Rolle's Theorem does not apply

x 2 ? 15

?

?

on the closed interval ??? ?15,15 ¨B¨B? . If

x

Rolle's Theorem can be applied, find all values of c in the open interval ??? ?15,15 ??? such that f ¡ä (c) = 0.

6. Determine whether Rolle's Theorem can be applied to f (x ) =

a.

b.

c.

d.

e.

c = 7, c = 2

c = 2, c = 15

c = 15

c=7

Rolle's Theorem does not apply

2

Name: ________________________

____

ID: A

7. Determine whether the Mean Value Theorem can be applied to the function f (x ) = x 2 on the closed

interval [¨C2,8]. If the Mean Value Theorem can be applied, find all numbers c in the open interval

(¨C2,8) such that f ¡ä (c) =

a.

b.

c.

d.

e.

____

MVT applies; c = 3

MVT applies; c = 2

MVT applies; c = 4

MVT applies; c = 1

MVT applies; c = 5

8. The height of an object t seconds after it is dropped from a height of 650 meters is s (t ) = ?4.9t 2 + 650. Find

the time during the first 6 seconds of fall at which the instantaneous velocity equals the average velocity.

a.

b.

c.

d.

e.

____

f (8) ? f (?2)

.

8 ? (?2)

3 seconds

2.45 seconds

22.11 seconds

14.7 seconds

18 seconds

9. Identify the open intervals where the function f (x ) = x

a.

b.

c.

d.

e.

6 ? x 2 is increasing or decreasing.

?

? ?

?

?

?

increasing: ??? ? 6, 3 ??? ¡È ??? 3, 6 ??? ; decreasing: ??? ? 3, 3 ???

?

? ?

?

?

?

?

?

?

?

?

?

?

?

increasing: ?? ?¡Þ, 6 ?? ; decreasing: ?? 6,¡Þ ??

?

?

?

?

?

?

decreasing on ?? ?¡Þ,¡Þ ??

?

?

?

? ?

?

increasing: ??? ? 3, 3 ??? ; decreasing: ??? ? 6,? 3 ??? ¡È ??? 3, 6 ???

?

?

?

? ?

?

??

??

??

??

decreasing: ?? ?¡Þ, 3 ?? ; increasing: ?? 3,¡Þ ??

?

?

?

?

____ 10. Identify the open intervals on which the function y = 3x ? 6cos x, 0 < x < 2¦Ð is increasing or decreasing.

a.

b.

c.

d.

e.

?? 7¦Ð 5¦Ð ??

?

?

?

?

?? and ??? 5¦Ð ,2¦Ð ??? ; decreasing on ??? 5¦Ð , 11¦Ð ???

,

increasing on ????

?

?

?

?

?

? 6

?

? 6

6 ???

? 6 6 ?

?

?

?

?? 7¦Ð ??

?

?

?

?

?? and ??? 11¦Ð ,2¦Ð ??? ; decreasing on ??? 7¦Ð , 11¦Ð ???

increasing on ???? 0,

?

?

?

?

?

? 6

?

? 6

6 ???

? 6 ?

?

?

?

?? 11¦Ð ???

??

??

??

??

?? and ??? 7¦Ð ,2¦Ð ??? ; decreasing on ??? 0, 7¦Ð ???

increasing on ???? 0,

?

?

?

?

?

6 ?

?

? 6

?

? 6 ?

?? 7¦Ð ???

??

??

??

??

?? and ??? 5¦Ð ,2¦Ð ??? ; decreasing on ??? 7¦Ð , 5¦Ð ???

increasing on ???? 0,

?

?

?

?

?

? 6 ?

? 6

?

? 6 6 ?

?? 7¦Ð 11¦Ð ???

??

??

??

??

?? ; decreasing on ??? 0, 7¦Ð ??? and ??? 11¦Ð ,2¦Ð ???

,

increasing on ????

?

?

?

?

?

6 ?

? 6

? 6 ?

? 6

?

3

Name: ________________________

ID: A

4

____ 11. For the function f (x ) = (x ? 1 ) 7 :

(a) Find the critical numbers of f (if any);

(b) Find the open intervals where the function is increasing or decreasing; and

(c) Apply the First Derivative Test to identify all relative extrema.

Use a graphing utility to confirm your results.

a.

(a) x = 1

(b) increasing: ??? ?¡Þ,1 ??? ; decreasing: ??? 1,¡Þ ???

(c) relative max: f (1) = 0

b.

(a) x = 0,1

(b) decreasing: ??? ?¡Þ,0 ??? ¡È ??? 1,¡Þ ??? ; increasing: ??? 0,1 ???

(c) relative min: f (0) = 1 ; relative max: f (1) = 0

c.

(a) x = 0

(b) increasing: ??? ?¡Þ,0 ??? ; decreasing: ??? 0,¡Þ ???

(c) relative max: f (0) = 1

d.

(a) x = 1

(b) decreasing: ??? ?¡Þ,1 ??? ; increasing: ??? 1,¡Þ ???

(c) relative min: f (1) = 0

e.

(a) x = 0

(b) decreasing: ??? ?¡Þ,0 ??? ; increasing: ??? 0,¡Þ ???

(c) relative min: f (0) = 1

____ 12. Find all points of inflection on the graph of the function f (x ) =

a.

b.

c.

d.

e.

?? 0,0 ??

? ?

?? 0,0 ?? ?? ?2,?8 ??

? ??

?

?? ?2,?8 ??

?

?

?? ?2,?8 ??

?

?

?? 0,0 ?? , ?? ?4,0 ??

? ? ?

?

4

1 4

x + 2x 3 .

2

Name: ________________________

ID: A

____ 13. Find the point of inflection of the graph of the function f (x ) = 4sin

a.

b.

c.

d.

e.

x

on the interval

8

??

?

?? 0,16¦Ð ¨B¨B? .

?? 9¦Ð ,0 ??

?

?

?? 8¦Ð ,4 ??

?

?

?? 0,0 ??

? ?

?? 4¦Ð ,0 ??

?

?

?? 8¦Ð ,0 ??

?

?

____ 14. Find the points of inflection and discuss the concavity of the function f (x ) = x + 7cos x on the interval

??

?

?? 0,2¦Ð ¨B¨B? .

a.

concave down on ??? 0,2¦Ð ??? ; no points of inflection

b.

concave upward on

?? ¦Ð 3¦Ð ??

?? ,

?

?? 2 2 ??? ; concave downward on

?

?

3¦Ð

¦Ð

inflection points at x = and x =

2

2

?? ¦Ð

?? 0,

?? 2

?

c.

?? ¦Ð 3¦Ð ??

??? ; concave upward on

concave downward on ???? ,

?

?? ¦Ð ?? ?? 3¦Ð

??

?? 0, ?? , ??

?

?? 2 ?? ?? 2 ,2¦Ð ??? ;

?

? ?

?

?2

inflection points at x =

¦Ð

2

2 ?

and x =

3¦Ð

2

d.

concave up on ??? 0,2¦Ð ??? ; no points of inflection

e.

none of the above

5

?? ?? 3¦Ð

??

?? , ??

?

?? ?? 2 ,2¦Ð ??? ;

? ?

?

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