More Review for Midterm Exam
AP Calculus - Meinke
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Determine the limit algebraically, if it exists.
6 sin x
1) lim
x 0 7x
A)
6
7
3) lim
x 4
B) Does not exist
1
9
D) Does not exist
3)
B) 0
C)
3
2
D) -
1
2
4)
B) 2
C) 16
D) Does not exist
x3 + 12x2 - 5x
5x
lim
x -7
5)
B) 5
C) 0
D) -1
x2 + 11x + 28
x+7
6)
B) 154
C) Does not exist
D) 11
x2 - 4
lim
x -2 x + 2
A) Does not exist
8)
C) 0
x2 + 2x - 63
x-7
A) -3
7)
1
9
x2 + 4x - 32
x2 - 16
A) Does not exist
6)
2)
B) -
A) 0
5) lim
x 0
D) 0
x
A) Does not exist
4) lim
x 7
C) 1
1
1
x+3 3
2) lim
x 0
A)
1)
7)
B) -2
C) -4
D) 1
10 - x
lim
x 10 10 - x
A) Does not exist
8)
B) -1
C) 0
1
D) 1
Find all points where the function is discontinuous.
0,
x< 0
9) f(x) = x2 - 4x,
0 x 4
4,
9)
x> 4
A) x = 0
B) x = 4
C) x = 0 and x = 4
D) Nowhere
Determine the limit algebraically, if it exists.
x+6
10) lim
x 6 (x - 6)2
A) Does not exist
11) lim
x 2
10)
B) 6
C) -6
D) 0
x-3
A) 1
11)
B) 0
C) Does not exist
D) -1
Solve the problem.
12) Find the points where the graph of the function has horizontal tangents.
f(x) = 4x2 + 4x - 4
A) (0, 4)
B) -
1
,- 5
2
C)
1
, - 18
2
12)
D) (-12, 716)
Find the limit, if it exists.
x2 - 4x + 17
13) lim
x3 + 9x2 + 8
x
A)
14)
17
8
lim
x -
B) 1
C) 0
D)
-12x2 + 8x + 9
-15x2 + 2x + 8
A)
15)
13)
14)
B)
9
8
C)
4
5
D) 1
6x + 1
lim
13x - 7
x
A) 0
15)
B)
C)
2
6
13
D) -
1
7
Find all points where the function is discontinuous.
16)
16)
A) x = -2
B) x = 1
C) None
D) x = -2, x = 1
Find the limit, if it exists.
4x3 + 3x2
17) lim
x - 6x2
x A) 4
17)
B) -
1
2
C) -
D)
Find dy/dx.
18) y = sin3 x - cos 10x
18)
A) 3 sin2 x cos x + 10 sin 10x
B) 30 sin2 x cos x sin 10x
C) 3 sin3 x cos x - 10 sin 10x
D) 3 sin2 x + sin 10x
19) y = 2 tan4x
A) 8 tan4x sec x
19)
B) 8 tan3x sec2 x
C) 8 tan3x
D) 8 tan5x
Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values of
x. Find the derivative with respect to x of the given combination at the given value of x.
x f(x) g(x) f (x) g (x)
20) 3 1
20)
4
8
7
4 -3
3
5 -4
f(g(x)) at x = 4
A) -32
B) 24
C) -20
D) 8
x f(x) g(x) f (x) g (x)
21) 3 1
9
6
5
4 -3
3
5 -4
1/g2 (x) at x = 4
2
A) 27
21)
B)
1
32
C) -
3
8
27
D)
8
27
x f(x) g(x) f (x) g (x)
22) 3 1
9
6
7
4 3
3
2 -5
g(x) at x = 3
1
A) 2 7
22)
B)
7
6
C)
1
D)
2 7
1
6
At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve,
as requested.
23) x2 + y2 - 2x + 4y = 8, tangent at (4, 0)
3
1
A) y = - (x - 4)
B) y = (x - 4)
2
2
23)
C) x = 4
D) y = 0
C) 4
1
D)
4
Find the value of df-1 /dx at x = f(a).
24) f(x) = 4x + 8, a = 1
A) 8
25) f(x) =
A)
1
B)
8
24)
1
x + 10, a = -2
4
1
10
25)
B)
1
4
C) 10
D) 4
Find the derivative of the given function.
26) y = tan-1 5x
1
A)
1 + 5x
27) y = 3.1 cos-1 2t
6.2
A)
1 - 4t2
28) y = sin-1
A)
5
B)
2(1 + 5x) 5x
B)
1
C)
10 5x(1 + 5x)
3.1
C) -
1 + 4t2
6.2
1 - 4t2
1
1 - 5x
D)
D) -
3.1
x 1 - x10
27)
1 - 4t2
1
x5
-5
26)
28)
B)
-5
C)
1 + x10
-5x5
1 - x10
D)
-5
x x10 - 1
Find dy/dx.
29) f(x) = -7e8x
A) -7e8x
29)
B) -56e8x
C) 8e8x
4
D) -56ex
30) y = 7xex - 7ex
30)
A) 7x
B) 7xex
C) 7ex
D) 7xex + 14ex
31) y = log (2x - 9)
2x - 9
A)
2 ln 10
2
B)
ln 10
1
C)
(2x - 9) ln 10
2
D)
(2x - 9) ln 10
32) y = ln (ln 5x)
1
A)
x
1
B)
5x
1
C)
ln 5x
1
D)
x ln 5x
31)
32)
Use logarithmic differentiation to find dy/dx.
33) y = (cos x)x
33)
A) ln x(cos x)x - 1
B) (cos x)x (ln cos x + x cot x)
C) (cos x)x (ln cos x - x tan x)
D) ln cos x - x tan x
34) y = 5 8x
34)
A) 40 (ln 8) 58x
B) 40 (ln 5) 58x
C) 8 (ln 5) 58x
D) 5 (ln 8) 58x
Find the points of inflection.
35) y = x 7 - x2
35)
A) (0, 7)
B) (7, 0)
C) (0, 0)
D) No inflection points.
36) y =
1 4
x - x3 + 15
4
36)
A) (0, 0) and (2, 11)
B) (0, 0)
C) (0, 0) and (2, -4)
D) (0, 15) and (2, 11)
Use the Concavity Test to find the intervals where the graph of the function is concave up.
37) y = 6x - 5e-x
A) ( - ,
37)
)
B) (- , 0 )
C) ( 0 ,
)
D) None
Solve the problem.
38) Given the distance function s(t) = t2 + 9t + 10, where s is in feet and t is in seconds, find the velocity
function, v(t), and the acceleration function, a(t).
A) v(t) = 2t + 9; a(t) = 0
B) v(t) = 2t + 9; a(t) = 2
C) v(t) = 2t + 9; a(t) = 2t
D) v(t) = 2t + 19; a(t) = 2
5
38)
39) The radius of a right circular cylinder is increasing at the rate of 5 in./s, while the height is
decreasing at the rate of 8 in./s. At what rate is the volume of the cylinder changing when the
radius is 15 in. and the height is 20 in.?
A) -300 in.3/s
B) -300 in.3/s
C) -80 in.3 /s
D) 1200 in.3 /s
40) A man flies a kite at a height of 50 m. The wind carries the kite horizontally away from him at a rate
of 10 m/sec. How fast is the distance between the man and the kite changing when the kite is 130 m
away from him?
A) 51 m/sec
B) 10.9 m/sec
C) 9.2 m/sec
6
39)
D) 10 m/sec
40)