Chapter 3 PRACTICE TEST
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the definition f'(a) = lim f(x) - f(a) to find the derivative of the given function at the given value of a.
x-a
x->a
1) f(x) =
A)
5
x + 2, a = 7
5
6
Solve the problem.
2) Find f'(x) if f(x) = 12x - 7.
A) f'(x) = 12
1)
1
B)
6
5
C)
B) f'(x) = 12x
C) f'(x) = -12
6
5
D)
6
2)
D) f'(x) = 5
3) Find the equation of the normal line to the curve y = 3 - 3x3 at the point (5, -372)
A) x + 225y + 83,705 = 0
B) x + 225y - 83,705 = 0
C) x - 225y + 83,705 = 0
D) x - 225y - 83,705 = 0
4) Find an equation of the tangent line to the graph of y = x2 - x, at the point (3, 6).
A) y = 5x + 12
B) y = 5x - 12
C) y = 5x - 9
1
3)
4)
D) y = 5x + 9
The graph of a function is given. Choose the answer that represents the graph of its derivative.
5)
15
5)
y
10
5
-15 -10
-5
5
15 x
10
-5
-10
-15
A)
B)
15
-15 -10
y
15
10
10
5
5
-5
5
10
15 x
-15 -10
-5
-5
-5
-10
-10
-15
-15
C)
y
5
10
15 x
5
10
15 x
D)
15
-15 -10
y
15
10
10
5
5
-5
5
10
15 x
-15 -10
-5
-5
-5
-10
-10
-15
-15
y
Solve the problem.
6) The graph shows the amount of potential energy V(x) (in arbitrary energy units) stored in a large rubber
band that is stretched a distance of x inches beyond its normal length.
V(x)
16
14
12
10
8
6
4
2
1
2
3
4
5
6
7
8
x
2
6)
The magnitude of the force required to hold the rubber band at the position x = a is the derivative of the
potential energy with respect to x, evaluated at the point x = a. Sketch a graph of the magnitude of the
force versus x.
A)
B)
F(x)
F(x)
2
4
1.5
3.5
1
3
0.5
2.5
2
1
-0.5
1.5
2
3
4
5
6
7
x
8
-1
1
-1.5
0.5
-2
1
2
3
4
5
6
7
8
x
-2.5
C)
D)
5
F(x)
F(x)
10
9
8
7
6
5
4
3
2
1
4
3
2
1
-1
1
2
3
4
5
6
7
8
x
-2
-1
-2
-3
-4
-3
-4
1
2
3
4
5
6
7
8
x
If the function is not differentiable at the given value of x, tell whether the problem is a corner, cusp, vertical tangent, or a
discontinuity.
7) y = -6 - 3 x, at x = 0
7)
A) discontinuity
C) vertical tangent
B) cusp
D) function is differentiable at x = 0
3
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?
8) x = 1
8)
y
4
2
-4
-2
2
4
x
-2
-4
A) Differentiable
B) Continuous but not differentiable
C) Neither continuous nor differentiable
Determine the values of x for which the function is differentiable.
1
9) y =
2
x + 64
A) All reals
C) All reals except -8 and 8
9)
B) All reals except 8
D) All reals except 64
Find the numerical derivative of the given function at the indicated point. Use h = 0.001. Is the function differentiable at the
indicated point?
10) f(x) = x3 - 3x, x = 2
10)
A) 2, yes
B) 9.000001, yes
C) 0, no
D) 9, yes
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.
′
′
11) u(1) = 2, u (1) = -6, v(1) = 7, v (1) = -4.
11)
d u at x = 1
dx v
A) - 34
7
Find dy/dx.
12) y = 11 - 5x2
A) -10x
13) y = 3x(8x4 - 8x)
A) 96x4 - 48x
B) - 17
8
C) - 50
49
D) - 34
49
B) -10
C) 11 - 5x
D) 11 - 10x
12)
13)
B) 120x4 - 48x
C) 120x4 - 24x
4
D) 96x4 - 24x
14) f(x) = (x + 3)(x + 2)
(x - 3)(x - 2)
14)
A) f'(x) =
12 - x2
(x - 3)2(x - 2)2
B) f'(x) =
10x - 60
(x - 3)2(x - 2)2
C) f'(x) =
10x2 - 60
(x - 3)2(x - 2)2
D) f'(x) =
60 - 10x2
(x - 3)2(x - 2)2
Find an equation for the line tangent to the curve at the given point.
4
15) y = x - 5 , x = -1
x2
A) y = -12x + 16
B) y = -12x - 16
C) y = 8x - 14
15)
D) y = 8x + 14
Solve the problem.
16) Find the x- and y-intercepts of the line that is tangent to the curve y = x3 at the point (-2, -8).
A) x-intercept = - 4, y-intercept = -16
B) x-intercept = 8 , y-intercept = -32
3
C) x-intercept = - 4 , y-intercept = 16
3
16)
D) x-intercept = 0, y-intercept = 0
Find the fourth derivative of the function.
17) y = 3x3 + 3x2 - 2x
A) 18
B) 9
17)
C) 9x +18
D) 0
The figure shows the velocity v of a body moving along a coordinate line as a function of time t . Use the figure to answer the
question.
18)
v (ft/sec)
18)
6
5
4
3
2
1
-1
1
2
3
4
5
6
7
8
9
10
t (sec)
-2
-3
-4
-5
When is the body's acceleration equal to zero?
A) 2 < t < 3, 5 < t < 6
C) t = 2, t = 3, t = 5, t = 6
B) 0 < t < 2, 6 < t < 7
D) t = 0, t = 4, t = 7
Solve the problem.
19) The function V = s3 describes the volume of a cube, V, in cubic inches, whose length, width, and height
each measure s inches. Find the (instantaneous) rate of change of the volume with respect to s when s = 5
inches.
A) 75 in3/in.
B) 375 in3/in.
C) 76.51 in3/in.
D) 15 in3/in.
5
19)
20) Suppose that the dollar cost of producing x radios is c(x) = 400 + 20x - 0.2x2. Find the marginal cost
when 40 radios are produced.
A) -$880
B) $4
C) $36
D) $880
20)
21) At time t, the position of a body moving along the s-axis is s = t3 - 27t2 + 240t m. Find the body's
acceleration each time the velocity is zero.
A) a(10) = 6 m/sec2, a(8) = -6 m/sec2
B) a(10) = -6 m/sec2, a(8) = 6 m/sec2
2
2
C) a(20) = 120 m/sec , a(16) = 20 m/sec
D) a(10) = 0 m/sec2, a(8) = 0 m/sec2
21)
Find dy/dx.
22) y = x4 cos x - 5x sin x - 5 cos x
A) - x4 sin x + 4x3 cos x - 5x cos x - 10 sin x
22)
B) - x4 sin x + 4x3 cos x - 5x cos x
D) - 4x3 sin x - 5 cos x + 5 sin x
C) x4 sin x - 4x3 cos x + 5x cos x
Solve the problem.
23) Find the equations for the lines that are tangent and normal to y = cot x at x = π .
4
A) tangent: y = -2x + π + 1;
2
B) tangent: y = 2x - π + 1;
2
normal: y = 1 x - π + 1
2
8
normal: y = - 1 x - π + 1
2
8
C) tangent: y = -2x + π ;
2
D) tangent: y = 2x + 1;
normal: y = - 1 x + 1
2
normal: y = 1 x - π
2
8
Find the indicated′ ′derivative.
24) Find y if y = 4x sin x.
′ ′
A) y = - 8 cos x + 4x sin x
′ ′
C) y = 4 cos x - 8x sin x
B) y
D) y
′ ′
′ ′
24)
= 8 cos x - 4x sin x
= - 4x sin x
The equation gives the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds).
25) s = 4 sin t - cos t
Find the body's acceleration at time t = π/4 sec.
5 2 m/sec2
A) - 5 2 m/sec2
B)
2
2
C) - 3
2
23)
2 m/sec2
D)
6
3
2
2 m/sec2
25)