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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