AP Calc AB First Semester Review
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the limit.
1) lim (7 - 7x)
x→7
1)
A) -42
B) -56
C) 42
D) 56
Determine the limit by sketching an appropriate graph.
1 - x2
0≤x<1
lim f(x), where f(x) = 1
1≤x<3
2) x → 1 3
x=3
2)
y
x
A) 1
B) 0
C) Does not exist
D) 3
Find the limit, if it exists.
x4 - 1
3) lim
x→1 x - 1
A) 0
3)
B) 2
C) Does not exist
D) 4
x2 + 4x - 5
4) lim
x→1 x2 + 3x - 4
A) -
5)
lim
h→0
A) 0
6
5
4)
B)
6
5
C) Does not exist
D) -
4
5
(x + h)3 - x3
h
5)
B) 3x2
C) Does not exist
1
D) 3x2 + 3xh + h2
Find the limit.
6)
1
lim
x
+1
x → -1 A) -1
7)
7)
B) -∞
C) Does not exist
D) ∞
8)
B) 0
C) -∞
D) -1
9)
B) 1
C) -2
D) 2
-19x2 + 6x + 4
lim
x→-∞ -4x2 + 4x + 10
A)
11)
D) ∞
3
lim
-2
x→∞ x
A) -5
10)
C) -∞
5
lim
2
x → -1 - x - 1
A) ∞
9)
B) 0
1
lim
x→-2 x + 2
A) 1/2
8)
6)
2
5
10)
B) 1
C) ∞
D)
19
4
x2 + 6x + 7
lim
x→∞ x3 + 8x2 + 4
A) 0
11)
B)
7
4
C) 1
D) ∞
Divide numerator and denominator by the highest power of x in the denominator to find the limit.
36x2 + x - 3
12) lim
(x
- 13)(x + 1)
x→∞
A) 0
B) 6
C) 36
D) ∞
Find all points where the function is discontinuous.
13)
A) x = 1
12)
13)
B) None
C) x = -2, x = 1
2
D) x = -2
Find the intervals on which the function is continuous.
3
14) y =
2
x -9
14)
A) discontinuous only when x = -3
C) discontinuous only when x = -9 or x = 9
B) discontinuous only when x = 9
D) discontinuous only when x = -3 or x = 3
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
15) Use the Intermediate Value Theorem to prove that -3x4 + 7x3 + 5x - 7 = 0 has a solution
15)
between 2 and 3.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find numbers a and b, or k, so that f is continuous at every point.
16)
x2 , if x ≤ 6
16)
f(x) =
x + k, if x > 6
A) k = 30
B) k = 42
C) k = -6
D) Impossible
Calculate the derivative of the function. Then find the value of the derivative as specified.
8
17) f(x) = ; f ′(-1)
x
A) f ′(x) = C) f ′(x) =
8
; f ′ (-1) = -8
x2
B) f ′(x) = - 8x2 ; f ′(-1) = - 8
8
; f ′(-1) = 8
x2
D) f ′(x) = 8; f ′(-1) = 8
3
17)
The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable,
continuous but not differentiable, or neither continuous nor differentiable?
18) x = -1
18)
y
4
2
-4
-2
2
4
x
-2
-4
A) Differentiable
B) Continuous but not differentiable
C) Neither continuous nor differentiable
Compare the right -hand and left -hand derivatives to determine whether or not the function is differentiable at the
point whose coordinates are given.
19)
19)
y
x
(-1, -1)
y=
1
x
y = -1
A) Since limx→-1 + f ′(x) = -1 while limx→-1 - f ′ (x) = 0, f(x) is not differentiable at x = -1.
B) Since limx→-1 + f ′(x) = 0 while limx→-1 - f ′(x) = 1, f(x) is not differentiable at x = -1.
C) Since limx→-1 + f ′(x) = 0 while limx→-1 - f ′(x) = -1, f(x) is not differentiable at x = -1.
D) Since limx→-1 + f ′(x) = 0 while limx→-1 - f ′(x) = 0, f(x) is differentiable at x = -1.
4
Find y ′ .
20) y = (3x - 5)(4x3 - x2 + 1)
A) 48x3 - 23x2 + 69x + 3
20)
B) 36x3 + 69x2 - 23x + 3
D) 12x3 + 23x2 - 69x + 3
C) 48x3 - 69x2 + 10x + 3
Find an equation of the tangent line at x = a.
21) y = x3 - 4x - 5; a = 2
A) y = 8x - 5
21)
B) y = 8x - 21
C) y = -5
D) y = 3x - 21
Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the
value of the indicated derivative.
22) u(1) = 4, u ′ (1) = -7, v(1) = 7, v ′(1) = -4.
22)
d v
at x = 1
dx u
A) -
65
16
B)
33
16
C)
33
4
D) -
33
16
23) u(2) = 7, u ′ (2) = 3, v(2) = -1, v ′(2) = -5.
d
(uv) at x = 2
dx
A) 26
23)
B) -38
C) 38
D) -32
Solve the problem.
24) The power P (in W) generated by a particular windmill is given by P = 0.015V 3 where V is the
velocity of the wind (in mph). Find the instantaneous rate of change of power with respect to
velocity when the velocity is 7.3 mph.
A) 5.3 W/mph
B) 0.3 W/mph
C) 11.7 W/mph
D) 2.4 W/mph
Find the derivative of the function.
x2 + 8x + 3
25) y =
x
24)
25)
A) y ′ =
3x2 + 8x - 3
x
B) y ′ =
2x + 8
x
C) y ′ =
2x + 8
2x3/2
D) y ′ =
3x2 + 8x - 3
2x3/2
Provide an appropriate response.
26) Find an equation for the tangent to the curve y =
A) y = 5x
10x
x2 + 1
B) y = 0
at the point (1, 5).
C) y = 5
5
26)
D) y = x + 5
Find the second derivative of the function.
(x - 8)(x2 + 3x)
27) y =
x3
27)
A)
d2 y
10 144
=x
dx2
x2
B)
d2 y
5
48
=
+
dx2 x2 x3
C)
d2 y
10 144
=2
dx
x3
x4
D)
d2 y 10 144
=
+
dx2 x3
x4
Find the derivative.
10
28) y =
+ 5 sec x
x
28)
A) y ′ = -
10
+ 5 sec x tan x
x2
B) y ′ =
C) y ′ = -
10
+ 5 tan 2 x
x2
D) y ′ = -
29) y =
10
- 5 sec x tan x
x2
10
- 5 csc x
x2
2
1
+
sin x cot x
29)
A) y ′ = 2 csc x cot x - sec 2 x
C) y ′ = 2 cos x - csc2 x
B) y ′ = - 2 csc x cot x + sec 2 x
D) y ′ = 2 csc x cot x - csc2 x
Find the indicated derivative.
30) Find y ′′ if y = 6x sin x.
A) y ′′ = - 12 cos x + 6x sin x
C) y ′′ = 6 cos x - 12x sin x
30)
B) y ′′ = 12 cos x - 6x sin x
D) y ′′ = - 6x sin x
Solve the problem.
31) Find the tangent to y = cot x at x =
A) y = -2x +
π
2
π
.
4
31)
B) y = 2x + 1
C) y = 2x -
π
+1
2
D) y = -2x +
π
+1
2
Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x.
32) y = (-3x + 7)5
dy
A) y = u5 ; u = -3x + 7;
= -15(-3x + 7)4
dx
C) y = 5u + 7; u = x5 ;
dy
B) y = u5; u = -3x + 7;
= 5(-3x + 7)4
dx
dy
= -15x4
dx
D) y = u5; u = -3x + 7;
6
dy
= -3(-3x + 7)5
dx
32)
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)
33) 3 1
33)
4
8
7
4 -3
3
5
-6
g(x), x = 3
1
A)
4
B) -
1
C)
2 7
7
4
D)
1
2 7
x f(x) g(x) f ′(x) g ′(x)
34) 3 1 16
8
5
4 3
3
2
-4
f(g(x)), x = 4
A) -32
34)
B) 24
C) 8
D) -8
Find the derivative of the function.
cos x 5
35) h(x) =
1 + sin x
A)
35)
-5 cos4 x
(1 + sin x)5
C) -
B) -5
4 sin x
cos x 4
cos x 1 + sin x
Find dy/dt.
36) y = cos( 8t + 11)
1
A) sin( 8t + 11)
2 8t + 11
D) 5
cos x 4
1 + sin x
B)
4
sin( 8t + 11)
8t + 11
36)
C) -sin( 8t + 11)
Find
sin x 4
cos x
D) -sin
4
8t + 11
d2 y
for the given function.
dx 2
37) y = 5 sin(2x + 7)
A) -10 sin(2x + 7)
37)
B) -20 sin(2x + 7)
C) 10 cos(2x + 7)
D) -20 cos(2x + 7)
Solve the problem.
38) The position of a particle moving along a coordinate line is s = 5 + 4t, with s in meters and t in
seconds. Find the particle's velocity at t = 1 sec.
1
4
1
2
A) m/sec
B) m/sec
C) - m/sec
D) m/sec
6
3
3
3
7
38)
Use implicit differentiation to find dy/dx.
39) x3 + 3x 2 y + y 3 = 8
A) -
x2 + 3xy
x2 + y 2
40) cos xy + x6 = y6
6x5 - x sin xy
A)
6y5
B)
39)
x2 + 2xy
x2 + y2
C) -
x2 + 2xy
x2 + y 2
D)
x2 + 3xy
x2 + y2
40)
B)
6x5 + y sin xy
6y5 - x sin xy
C)
6x5 + x sin xy
6y5
D)
6x5 - y sin xy
6y5 + x sin xy
At the given point, find the slope of the curve or the line that is tangent to the curve, as requested.
41) y6 + x3 = y2 + 12x, tangent at (0, 1)
3
A) y = - x
2
B) y = 3x + 1
C) y = 2x + 1
Use implicit differentiation to find dy/dx and d2 y/dx 2.
42) xy - x + y = 2
dy
1 + y d2 y
y+1
A)
;
==
dx
x + 1 dx2 (x + 1)2
C)
41)
D) y = - 2x - 1
42)
dy y + 1 d2 y
2y + 2
B)
;
=
=
dx
x + 1 dx2 (x + 1)2
dy
1 + y d2 y
2y - 2
;
==
dx
x + 1 dx2 (x + 1)2
D)
dy 1 - y d2 y
2y - 2
;
=
=
dx 1 + x dx2 (x + 1)2
Solve the problem.
43) A piece of land is shaped like a right triangle. Two people start at the right angle of the triangle at
the same time, and walk at the same speed along different legs of the triangle. If the area formed
by the positions of the two people and their starting point (the right angle) is changing at 4 m2 /s,
43)
then how fast are the people moving when they are 5 m from the right angle? (Round your answer
to two decimal places.)
A) 0.80 m/s
B) 0.40 m/s
C) 1.60 m/s
D) 6.25 m/s
Solve the problem. Round your answer, if appropriate.
44) As the zoom lens in a camera moves in and out, the size of the rectangular image changes.
Assume that the current image is 8 cm × 5 cm. Find the rate at which the area of the image is
changing (dA/df) if the length of the image is changing at 0.6 cm/s and the width of the image is
changing at 0.2 cm/s.
A) 11.6 cm 2/sec
B) 9.2 cm 2 /sec
C) 5.8 cm 2 /sec
D) 4.6 cm 2 /sec
44)
45) Water is being drained from a container which has the shape of an inverted right circular cone.
The container has a radius of 5.00 inches at the top and a height of 6.00 inches. At the instant when
the water in the container is 4.00 inches deep, the surface level is falling at a rate of 0.7 in./sec. Find
the rate at which water is being drained from the container.
A) 27.7 in.3 /s
B) 24.4 in.3 /s
C) 22.0 in.3 s
D) 23.3 in.3 /s
45)
46) The radius of a right circular cylinder is increasing at the rate of 8 in./sec, while the height is
decreasing at the rate of 10 in./sec. At what rate is the volume of the cylinder changing when the
radius is 6 in. and the height is 18 in.?
A) 504 in.3 /sec
B) 1368π in.3 /sec
C) -228 in.3 /sec
D) 504π in.3 /sec
46)
8
Determine all critical points for the function.
47) f(x) = x3 - 3x2 + 6
47)
A) x = 0 and x = 1
C) x = 0 and x = 2
B) x = -1 and x = 1
D) x = 0
Find the absolute extreme values of the function on the interval.
48) g(x) = -x2 + 11x - 30, 5 ≤ x ≤ 6
48)
1
11
A) absolute maximum is at x =
; absolute minimum is 0 at 6 and 0 at x = 5
4
2
B) absolute maximum is
241
11
at x =
; absolute minimum is 0 at 6 and 0 at x = 5
4
2
C) absolute maximum is
5
13
at x =
; absolute minimum is 0 at 6 and 0 at x = 5
4
2
D) absolute maximum is
1
13
at x =
; absolute minimum is 0 at 6 and 0 at x = 5
4
2
Find the extreme values of the function and where they occur.
49) f(x) = x3 - 12x + 2
49)
A) None
B) Local maximum at (0, 0).
C) Local maximum at (2, -14), local minimum at (-2, 18).
D) Local maximum at (-2, 18), local minimum at (2, -14).
50) f(x) =
1
2
x -1
50)
A) Local maximum at (1, 0), local minimum at ( -1, 0).
B) Local maximum at (-1, 0), local minimum at (1,0).
C) Local maximum at (0, -1).
D) None
9
Solve the problem.
51) Select an appropriate graph of a twice-differentiable function y = f(x) that passes through the
6 5
6 5
points (- 2,1) , , , (0,0),
, and ( 2,1), and whose first two derivatives have the
3 9
3 9
51)
following sign patterns.
+
y′ :
-
+
- 2
0
+
2
-
+
6
3
y′′ :
-
6
3
A)
B)
y
-3
-2
-1
y
16
16
12
12
8
8
4
4
1
-4
2
3
x
-3
-2
-1
-4
-8
-8
-12
-12
-16
-16
C)
1
2
3
x
1
2
3
x
D)
y
-3
-2
-1
y
4
2
3
1.5
2
1
1
0.5
-1
1
2
3
x
-3
-2
-1
-0.5
-2
-1
-3
-1.5
-4
-2
Find the largest open interval where the function is changing as requested.
52) Decreasing f(x) = x3 - 4x
A)
2 3
,∞
3
53) Increasing y = (x2 - 9)2
A) (3, ∞)
B) -
2 3 2 3
,
3
3
52)
C) -∞, ∞
D) -∞, -
C) (-∞, 0)
D) (-3, 0)
2 3
3
53)
B) (-3, 3)
10
Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up
and concave down.
54)
54)
10
y
5
-10
-5
5
10 x
-5
-10
A) Local minimum at
B) Local minimum at
(-∞, 0)
C) Local minimum at
D) Local minimum at
(-∞, 0)
x = 1; local maximum at x = -1; concave up on (-∞, ∞)
x = 1; local maximum at x = -1; concave down on (0, ∞); concave up on
x = 1; local maximum at x = -1; concave down on (-∞, ∞)
x = 1; local maximum at x = -1; concave up on (0, ∞); concave down on
11
Solve the problem.
55) Using the following properties of a twice-differentiable function y = f(x), select a possible graph of
f.
x
x<2
-2
-2 < x < 0
0
0<x<2
2
x>2
y
55)
Derivatives
y′ > 0,y′′ < 0
y′ = 0,y′′ < 0
y′ < 0,y′′ < 0
y′ < 0,y′′ = 0
y′ < 0,y′′ > 0
y′ = 0,y′′ > 0
y′ > 0,y′′ > 0
11
-5
-21
A)
B)
24
y
24
16
16
8
8
-4 -3 -2 -1
-8
1
2
3
4
x
y
-4 -3 -2 -1
-8
-16
-16
-24
-24
C)
1
2
3
4
x
1
2
3
4
x
D)
24
y
24
16
16
8
8
-4 -3 -2 -1
-8
1
2
3
4
x
-4 -3 -2 -1
-8
-16
-16
-24
-24
y
For the given expression y′, find y'' and sketch the general shape of the graph of y = f(x).
56) y′ = x2(4 - x)
y
x
12
56)
A)
B)
y
y
x
x
C)
D)
y
y
x
x
13
Solve the problem.
57) A private shipping company will accept a box for domestic shipment only if the sum of its length
and girth (distance around) does not exceed 120 in. What dimensions will give a box with a square
end the largest possible volume?
A) 20 in. × 20 in. × 40 in.
C) 20 in. × 20 in. × 100 in.
57)
B) 40 in. × 40 in. × 40 in.
D) 20 in. × 40 in. × 40 in.
58) A rectangular sheet of perimeter 33 cm and dimensions x cm by y cm is to be rolled into a cylinder
as shown in part (a) of the figure. What values of x and y give the largest volume?
A) x = 13 cm; y =
7
cm
2
B) x = 12 cm; y =
9
cm
2
C) x = 11 cm; y =
11
cm
2
D) x = 10 cm; y =
13
cm
2
14
58)
59) At noon, ship A was 12 nautical miles due north of ship B. Ship A was sailing south at 12 knots
(nautical miles per hour; a nautical mile is 2000 yards) and continued to do so all day. Ship B was
sailing east at 6 knots and continued to do so all day. The visibility was 5 nautical miles. Did the
ships ever sight each other?
A) Yes. They were within 3 nautical miles of each other.
B) No. The closest they ever got to each other was 6.4 nautical miles.
C) No. The closest they ever got to each other was 5.4 nautical miles.
D) Yes. They were within 4 nautical miles of each other.
59)
60) The 9 ft wall shown here stands 30 feet from the building. Find the length of the shortest straight
beam that will reach to the side of the building from the ground outside the wall.
60)
9' wall
30'
A) 53.3 ft
B) 51.3 ft
C) 39 ft
D) 52.3 ft
61) Find the optimum number of batches (to the nearest whole number) of an item that should be
produced annually (in order to minimize cost) if 300,000 units are to be made, it costs $2 to store a
unit for one year, and it costs $440 to set up the factory to produce each batch.
A) 26 batches
B) 20 batches
C) 28 batches
D) 18 batches
61)
62) Suppose c(x) = x3 - 22x2 + 30,000 x is the cost of manufacturing x items. Find a production level
that will minimize the average cost of making x items.
A) 12 items
B) 11 items
C) 10 items
D) 13 items
62)
Find the linearization L(x) of f(x) at x = a.
63) f(x) = 3x + 64, a = 0
3
3
A) L(x) =
x-8
B) L(x) = x - 8
16
8
63)
3
C) L(x) = x + 8
8
15
3
D) L(x) =
x+8
16
Express the relationship between a small change in x and the corresponding change in y in the form dy = f′(x) dx.
64) f(x) = 9x2 + 6x + 6
64)
A) dy = 18x + 12 dx
C) dy = 18x + 6 dx
B) dy = (18x + 6) dx
D) dy = 18x dx
Find the value or values of c that satisfy the equation
f(b) - f(a)
= f′(c) in the conclusion of the Mean Value Theorem for
b-a
the function and interval.
65) f(x) = x2 + 3x + 2, [1, 2]
A) 1, 2
66) f(x) = x +
65)
3 3
B) - ,
2 2
3
C)
2
3
D) 0,
2
32
, [2, 16]
x
A) 0, 4 2
66)
B) -4 2, 4 2
C) 2, 16
16
D) 4 2
Answer Key
Testname: AP CALC SEM 1 REVIEW
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
A
B
D
B
B
C
C
A
C
D
A
B
A
D
15) Let f(x) = -3x4 + 7x3 + 5x - 7 and let y0 = 0. f(2) = 11 and f(3) = -46. Since f is continuous on [2, 3] and since y0 = 0 is
between f(2) and f(3), by the Intermediate Value Theorem, there exists a c in the interval ( 2, 3) with the property that
f(c) = 0. Such a c is a solution to the equation -3x4 + 7x3 + 5x - 7 = 0.
16)
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
29)
30)
31)
32)
33)
34)
35)
36)
37)
38)
39)
40)
41)
42)
43)
44)
45)
46)
47)
A
A
B
C
C
B
B
B
D
D
C
C
A
B
B
D
A
C
A
A
B
B
D
C
D
B
D
A
D
B
B
C
17
Answer Key
Testname: AP CALC SEM 1 REVIEW
48)
49)
50)
51)
52)
53)
54)
55)
56)
57)
58)
59)
60)
61)
62)
63)
64)
65)
66)
A
D
C
D
B
A
D
C
C
A
C
C
D
A
B
D
B
C
D
18