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AP Calculus AB
Name: ________________
Chapter Three test
1)
Determine the function whose graph has vertical asymptotes at x = +2
and a horizontal asymptote at y = 0.
x
a)
f(x) =
(x 2)2
x
b)
f(x) = 2
x +4
c)
f(x) =
d)
f(x) =
e)
x3
x2 4
3x
x2 4
none of these
2)
Find the values of x that give relative extrema for the function
f(x) = 3x5 – 5x3.
a)
relative maximum: x = 0; relative minimum x = 5 3
b)
relative maximum: x = -1; relative minimum x = 1
c)
relative maximum: x = +1; relative minimum x = 0
d)
relative maximum: x = 0; relative minimum x = +1
e)
none of these
3)
Decide whether Rolle’s Theorem can be applied to f(x) = x2 – 2x on the
interval [0, 2]. If Rolle’s Theorem can be applied, find all values of c
in the interval such that f’(c) = 0. If Rolle’s Theorem cannot be
applied, state why.
a)
Rolle’s can be applied; c = 1
b)
Rolle’s cannot be applied: f(x) differentiable on (0, 2)
c)
Rolle’s cannot be applied: f(0) = f(2)
d)
Rolle’s cannot be applied; f(x) is not continuous on [0, 2]
e)
none of these
4)
Find all critical numbers for the function f(x) =
a)
b)
c)
d)
e)
-1
–1, -3
–3
f(x) has no critical numbers
none of these
x +1
x +3
5)
The product of two positive numbers is 972. Minimize the sum of the
first and three times the second.
a) 30 and 1625
b) both numbers are 18 3
c) 18 and 54
d) impossible, one number will be negative
e) none of these
6)
Find all the intervals on which the graph of the function is concave
upward: f(x) = 8x3 – 2x4
a)
(- , 3)
b)
(- , 0) (2, )
c)
(3,
)
d)
(0, 2)
e)
none of these
7)
Find all horizontal asymptotes for f(x) =
a)
b)
c)
d)
e)
y=+1
y=4
y=+4
y=0
none of these
4x
x2 + 9
8)
Find all points of inflection of the graph of f(x) = x4 – 6x3
a)
(0, 0)
b)
(0, 0) and ( 9 2 , - 218716 )
c)
(3, -81)
d)
(0, 0) and (3, -81)
e)
none of these
9)
Find all of the open intervals on which f(x) is increasing or decreasing:
f(x) = 1 2
x
a)
increasing (
); decreasing (
)
b)
decreasing (
); increasing (
)
c)
strictly increasing
d)
strictly decreasing
e)
none of these
10)
Determine whether the Mean Value Theorem applies to f(x) = 3x – x2
on the interval [2, 3]. If the “MVT” does apply, find all values of c in
f(b) f( a)
[2, 3] such that f’(c) =
. If it does not apply, state why.
b a
a)
MVT applies; c = 2 3
c)
MVT applies; c = 5 2
MVT does not apply; f(2) = f(3)
d)
MVT does not apply; f(x) is not differentiable on [2, 3]
e)
none of these
b)
11)
Determine from the graph whether f possesses extrema on the
interval (a, b)
a)
maximum at x = c, minimum at x = b
b)
maximum at x = c, no minimum
c)
no maximum, minimum at x = b
d)
no extrema
12)
13)
14)
Find the limit: lim 8 +
a)
81
b)
c)
d)
d)
-8
0
x
sin x
3x
3
none of these
A farmer has 160 feet of fencing to enclose 2 adjacent rectangular
pig pens. What dimensions should be used so that the enclosed area
will be a maximum?
80
a)
40 ft. by
ft.
3
8
b)
4 15 ft. by
15 ft.
5
c)
40 ft. by 40 ft.
d)
20 ft. by
e)
none of these
80
ft.
3
The side of a cube is measured to be 3.0 inches. If the measurement
is correct to within 0.1 inch, use differentials to estimate the
propagated error in the volume of the cube.
a)
+0.000001 in3.
b)
+0.06 in3.
c)
+2.7 in3.
d)
+0.27 in3.
e)
none of these
15)
Which of the following is the correct sketch of the graph of the
function y = -x3 – 12x + 20?
16)
Find the differential of y = tan3x
a)
sec23xdx
b)
3sec3xtan3xdx
c)
3sec23xdx
d)
3sec3xtan3xdx
e)
none of these