Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1)
One hundred dollars is deposited in a savings account at 6% interest compounded continuously.
The function defined by f(x) shown in the figure gives the balance in the account after x years. At
what rate (in dollars per year) is the balance growing after 4 years?
1)
A)
$28/year
B)
$7/year
C)
$14/year
D)
$15/year
2)
If f(t) =1
x2/3 , then lim
h
0
f(–8 + h) – f(–8)
h equals
2)
A)
f'(0)
B)
f'(8)
C)
1
48
D)
none of these
Solve the problem.
3)
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 relaxed length.
The magnitude of the force required to hold the rubber band at the position x = a is equal to the rate
of change of the potential energy with respect to x, evaluated at the point x = a. Estimate the force
required to hold the band at a stretched position x =7. (Hint: the force in this problem has units of
“energy units per inch”.)
3)
A)
–0.6 energy units per inch
B)
2.7 energy units per inch
C)
1.6 energy units per inch
D)
3.4 energy units per inch
4)
If (x,y) is a point on the parabola y = 3x2, then the tangent line to y = 3x2 passing through (x,y) has
slope 6x. Find the equation of the line tangent to y = 3x2 through the point (–2, 12).
4)
A)
y = – 2x + 12
B)
y = 12x + 6
C)
y = – 12x – 12
D)
y = 6x + 12
5)
Which of the following lines is/are parallel to the tangent line of the graph of y = – x3 at the point
where x = 1 and the slope of the tangent line is –3x2.
(I) y = – 3x + 1
(II) y = 3x + 1
(III) y = – 3x – 4
(IV) y = 3x + 1
5)
A)
(I) and (III)
B)
(III) and (IV)
C)
(II) and (III)
D)
(I) and (II)
E)
none of these
6)
Which of the following is the best description of f'(t)?
6)
A)
f'(a) measures the rate of change of f(t) per unit change in t at the point t = a.
B)
The derivative as a function is the best approximation of the tangent to line to f(x).
C)
f'(t) =f(t)
t
D)
It is approximately equal to f(t + h) – f(t)
h, as t gets very small.
E)
It is a function which gives the slope of the secant line through any two points.
7)
During the month of February, a flu epidemic hit the University. The number of people sick at time
t (measured in days) is given by the function P(t). The rate at which the epidemic is spreading on
February 3 is 110 people per day. How is the information best represented mathematically?
7)
A)
dP
dt t = 3
= P'(3)
B)
P'(3) = 110
C)
P(3) = 110
D)
dP
dt t = 110
E)
none of these
Solve the problem.
8)
Refer to the figure, where f(t) is the interest rate (as a percent) on a 6–month certificate of deposit t
years after January 1, 1985. The straight lines are tangent to the graph of y = f(t) at t = 2, t = 4, and
t = 10. How fast was the interest rate changing on January 1, 1987?
8)
A)
–2%/year
B)
0%/year
C)
1%/year
D)
2%/year
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
9)
For the following function f(x), determine whether or not lim
x
2f(x) exists. If so, give the
limit.
Enter either a real number or the words “does not exist”.
9)
10)
Find the equation of the tangent line to the curve y =x3+ 4x2+ 4 at (1, 9).
Enter your answer in standard slope–intercept form.
10)
11)
A point P is moving along the x–axis. At any time t, the location of P on the x–axis is
described by x =t3– 4t2+ 3t. Determine the point’s instantaneous velocity when t = 5.
Enter just an integer.
11)
12)
For the following function f(x), determine whether or not lim
x
3f(x) exists. If so, give the
limit.
Enter either a real number or enter the words “does not exist”.
12)
13)
Let f(x) =1
2x . Compute f(3) using limits.
Enter a reduced fraction or an integer.
13)
Differentiate.
14)
f(x) =x
Enter your answer as just: axb. No parentheses.
14)
15)
Find the second derivative of f(x) = 3x4– 4x3+ 5x + 1.
Enter just a polynomial in standard form, unlabeled.
15)
16)
Find the slope of the tangent line to the curve y =3
(x + 2)2 at x = 3.
Enter a reduced fraction only.
16)
17)
Find the equation of the following line: (2, 3) and (4, 6) on line.
Enter your answer in slope–intercept form.
Include both the slope and the intercept in your equation.
17)
Calculate the following limit(s) if they exist.
18)
lim
x
0
1
x3– 1
+ 1
Enter either a fraction in lowest terms, an integer, or “does not exist”.
18)
19)
A point P is moving along the x–axis. At any time t, the location of P on the x–axis is
described by x =t3– 4t2+ 3t. Determine the instantaneous acceleration at time t = 5 of the
point P.
Enter just an integer.
19)
20)
Find the derivative of f(x) =1
x2.
Enter your answer exactly in the form: axb
20)
Solve the problem.
21)
A manufacturer’s profit from producing x units of a product is given by
P(x) = 0.002x3– 0.01x2+ 0.5x. What is the marginal profit when the production level is at
50 units? Enter your answer as a real number to two decimal places, no units.
21)
22)
Find all points on the graph of y =x3 where the curve has slope 12. The slope of the
tangent line to the graph is 3x2.
Enter your answer exactly in the form: (a, b), (c, d) where a > c.
22)
23)
Find the derivative of f(x) = 4x5/4.
Enter your answer in the form: axb, where a, b are either fractions in lowest terms or
integers. No parentheses.
23)
Solve the problem.
24)
A manufacturer’s profit from producing x units of a product is given by
P(x) = 0.002x3– 0.01x2+ 0.5x. At what production level(s) will the marginal profit be $9.30
per unit?
Enter just an integer, no units.
24)
25)
Find the equation of the tangent line to the curve y =1
x + 2 at 2, 1
4.
Enter your answer in standard slope–intercept form using reduced fractions .
No parentheses.
25)
26)
Find the equation of the following line: 1
2, 1 and (2, 0) on line.
Enter your answer in point–slope form using 1
2, 1 .
26)
Calculate the following limit(s) if they exist.
27)
lim
x
0
g(x)
f(x) where lim
x
0f(x) = – 1
3 and lim
x
0g(x) =2
3.
Enter your answer as an integer, fraction in lowest terms, or the words “does not exist”.
27)
28)
Compute d
dt
dv
dt , where v = – 5t3+2
1 – t at t = – 1.
Enter just a reduced fraction of form a
b.
28)
7
Solve the problem.
29)
A winter storm front moves through campus. At t hours after the onset of the storm, the
temperature is given by T(t) = 35 – 2t2+ t. What is the temperature 3 hours after the storm
begins?
Enter just an integer, no units.
29)
30)
A rock is thrown off a cliff. Its distance from the ground below at t seconds is
s(t ) = – 16t 2 + 16t + 96 feet. What is the velocity of the rock when it slams into the ground?
Enter just an integer.
30)
31)
Let f(x) =x3– 9x
2x + 6 . Is f(x) continuous at x = – 3?
Enter your answer as either “yes” or “no” or “does not exist”.
31)
32)
Find the equation of the following line: Slope is –5
7; –1
2, 1 on line.
Enter your answer in point–slope form.
32)
33)
Compute f ”(2) when f(t) =3
(3t – 1)2.
Enter just a reduced fraction of form a
b.
33)
34)
Find the equation of the tangent line to the curve y =x3+ 3x – 8 at x = 2.
Enter your answer in standard point–slope form.
34)
Solve the problem.
35)
A rock is thrown off a cliff. Its distance from the ground below at t seconds is
s(t ) = – 16t2+ 16t + 96 feet. When will it hit the ground?
Enter your answer exactly as: t = a. No units.
35)
36)
Find the first derivative of z = 4t + (3 –2t + 1)3 at t =3
2.
Enter just a fraction of form a
b in lowest terms.
36)
Differentiate.
37)
y =2
2x + 1
Enter your answer exactly as: dy
dx = a(P(x))b where P(x) is a polynomial in standard form.
a, b reduced fractions or integers. No parentheses on coefficients or powers.
37)
38)
Find the derivative of f(x) =5
x2.
Enter your answer exactly in the form: axb, where a, b are integers. Do not use
parentheses.
38)
Calculate the following limit(s) if they exist.
39)
lim
x–2
x
(x3+ 8)–1
Enter either a fraction, integer, or the words “does not exist”.
39)
Differentiate.
40)
F(x) =3x + 1
Enter your answer as just: aP(x)b where P(x) is a polynomial in standard form.
40)
9
41)
Suppose that t hours after being placed in a freezer, the temperature of a piece of meat is
given by f(t ) = 70 – 12t +4
t + 1 . How fast is the temperature of the meat falling 3 hours
after being placed in the freezer?
Enter your answer as a reduced fraction of the form a
b, no units.
41)
42)
Find the slope of the line with equation 2x + 4 = 2(2y + 3).
Enter just a fraction of form a
b in lowest terms.
42)
43)
Determine: dy
dx if y =x – 2
5.
Enter your answer as just an integer or a reduced fraction.
43)
Solve the problem.
44)
An automobile‘s brakes are applied at time t = 0 when the vehicle is traveling at 48 ft/sec.
The brakes cause the automobile to decelerate so that after t sec the velocity is given by
v(t ) = 48 – 16t. How long will it take for the vehicle to come to a complete stop?
Enter just an integer, no units.
44)
45)
A ball is thrown straight up. Its height, in feet, at time t, in seconds, is represented by the
equation h(t) = 20t – 16t2+ 10. Determine the maximum height of the ball. (Hint: Consider
the velocity of the ball at the moment the ball reaches its maximum height.)
Enter just a reduced fraction of form a
b. No units.
45)
46)
Find the slope of the tangent line to the curve y = 3x4+ 2x3 at x = 1.
Enter just an integer.
46)
10
47)
Find the slope of the graph of y = (x2– 7)3 at x = 3.
Enter just an integer.
47)
48)
Determine: dy
dx if y = 4 – 6x.
Enter your answer as just an integer.
48)
49)
Find the equation of the following line: 3
2, –4 and –3
2, –4 on the line (in the xy –plane).
Enter your equation in the simplest possible form.
49)
Calculate the following limit(s) if they exist.
50)
lim
x
x3
x3– 1
Enter just an integer or a fraction.
50)
Differentiate.
51)
y =5
x3
Enter your answer in the form: axb. No parentheses.
51)
52)
Let f(x) =(2x + 1)2. Compute f(0) using limits.
Enter a reduced fraction or an integer.
52)
53)
Find the equation of the tangent line to the curve y = 1 + 3x –x2 at x = 5.
Enter your answer in standard slope–intercept form.
53)
54)
Let f(x) =x3– 9x
2x + 6 . Does lim
x–3 f(x) exist?
Enter your answer as either “yes” or “no” or “does not exist”.
54)
Calculate the following limit(s) if they exist.
55)
lim
x
1
x6– 1
x3– 1
Enter your answer as just an integer or a fraction in lowest terms, or the words “does not
exist”.
55)
56)
lim
x–1 (x3– 2x + 5).
Enter just an integer or “does not exist”.
56)
57)
Find the slope of the line with the equation y = 4 – 5x.
Enter just an integer.
57)
58)
A ball is thrown straight up. Its height, in feet, at time t, in seconds, is represented by the
equation h(t) = 30t – 16t2+ 6. Determine the instantaneous velocity of the ball at t = 2.
Enter just an integer (no units)
58)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the slope and the y–intercept of the line.
59)
y =2x –6
59)
A)
m =2, b =6
B)
m =2, b = – 6
C)
m = – 6, b =2
D)
m =6, b =2
Solve the problem.
60)
The graph shows the average cost of a barrel of crude oil for the years 1981 to 1990 in constant 1996
dollars. Find the approximate average change in price from 1981 to 1987.
1996 $/Barrel
Year
60)
A)
About –$6/year
B)
About –$7/year
C)
About –$33/year
D)
About –$12/year
61)
Find the first derivative of y =T7– 9T5+ 2T4+ 59
61)
A)
7T6– 45T4+ 8T3+ 59
B)
7T6– 45T4+ 8T3
C)
T6– 9T4+ 2T3+ 59
D)
T6– 9T4+ 2T3
Find f'(x) using limits.
62)
f(x) =1
5x2
62)
A)
f'(x) = – 2
5x3
B)
f'(x) = – 2
5x
C)
f'(x) =2
5x3
D)
f'(x) = – 1
5x3
63)
Which of the following is/are true of the lines 2x – 5y = – 15 and 5x + 2y = 6?
(I) They are parallel.
(II) They are perpendicular.
(III) They cross the x–axis at the same point.
(IV) They cross the y–axis at the same point.
63)
A)
II
B)
II and III
C)
II and IV
D)
I
E)
none of these
Find the derivative of f(x) at the given value of x.
64)
f(x) =x3+4, at x =4
64)
A)
49
B)
–48
C)
52
D)
48
D
65)
For the graphed function below, state the x–values for which the derivative does not exist.
65)
A)
x = 0, 9, –9
B)
x =3, –3
C)
x =9, –9
D)
x = 0, 3, –3
B
C
Solve the problem.
66)
On a hot day, a child can sell 28 cups of lemonade if she charges $2.50 per cup. If she raises the
price to $3.00 she will sell 24 cups. Let f(x) denote the number of cups of lemonade sold per day
when the price is x dollars. Assume that f(x) is a linear function of x. How many cups will the child
sell if she sets the price to $3.25 per cup? (If necessary, round to the nearest whole cup.)
66)
A)
22 cups
B)
19 cups
C)
25 cups
D)
20 cups
67)
The tangent line to the curve y =1
6x3–1
2x2– x + 4 is perpendicular to the line 18x + 9y = – 37 at
two points on the curve. Find the two points.
67)
A)
(–1, 0), (3, 0)
B)
–1, 1
2, 3, 1
2
C)
–1, 13
3, (3, 1)
D)
(–3, –2), 1, –10
3
C
Use the properties of limits to help decide whether the limit exists. If the limit exists, find its value.
68)
lim
x–2
x2–4
x +2
68)
A)
0
B)
–4
C)
Does not exist
D)
–2
B
69)
Find the equation of the following line: Perpendicular to –5x + y –8= 0; (3, 3) on line.
69)
A)
y = – 1
5x –18
5
B)
y = – 5x – 18
C)
y =1
5x –18
5
D)
y = – 1
5x +18
5
D
15
A
70)
Find the second derivative of f(x) =2x
3x2+ 4x for x
0.
70)
A)
2(3x + 4)–1
B)
12(3x + 4)–3
C)
36(3x + 4)–3
D)
(–2)3–2
E)
–6(3x + 4)–2
E)
71)
Find the second derivative of f(x) =2x4– 5x2+ 8.
71)
A)
8x2– 10
B)
8x2– 10x
C)
24x2– 10x
D)
24x2– 10
B)
Write an equation for the line described.
72)
Through (–5, 1) and (3, –2)
72)
A)
y =3
8x –7
8
B)
y = – 6
5x +8
5
C)
y = – 3
8x –7
8
D)
y =6
5x +8
5
B)
73)
Find the average rate of change for f(x) = – 3x2– 2x + 5 on [–2, 1]
73)
A)
0
B)
– 7
C)
3
D)
–3
E)
1
B)
E)
16
B)