Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the derivative of the function.
1)
y =3sec5 x
1)
A)
dy
dx =15 tan x sec5 x
B)
dy
dx =15 tan2 x sec5 x
C)
dy
dx =15 tan2 x sec4 x
D)
dy
dx =15 sec4 x
Solve the problem.
2)
The motion of a spring–mass system is described by the equation y =14 sin t –
4, where y is the
distance in feet from the equilibrium position and t is time in seconds. If the weight is 19 feet from
the ceiling in a state of equilibrium, find the closest the weight will ever be to the ceiling.
2)
A)
33 ft
B)
5 ft
C)
14 ft
D)
19 ft
Evaluate the definite integral.
3)
0
e–4x sin x dx
3)
A)
Diverges
B)
4
17
C)
1
17
D)
1
Solve the problem.
4)
A plant population experiences seasonal growth. At time t, the population, f(t), is modeled by
f(t) =600e1.5sin(t) .
Find the maximum and minimum values of f(t) and the values of t where they occur.
4)
A)
Maximum: 2689 when t =
2+ 2n, where n is any integer
Minimum: 134 when t =3
2+ 2n
B)
Maximum: 1733 when t =
2+ 2n, where n is any integer
Minimum: 208 when t =3
2+ 2n
C)
Maximum: 600 when t =
2+ 2n, where n is any integer
Minimum: –600 when t =3
2+ 2n
D)
Maximum: 2689 when t = 2n, where n is any integer
Minimum: 134 when t =+ 2n
D)
Find the indicated trigonometric function for , given that is an angle in standard position with the terminal side
defined by the given point.
5)
(–10, 24); find sin
5)
A)
5
13
B)
–5
13
C)
–12
13
D)
12
13
D
D)
Give the exact value.
6)
sec 150°
6)
A)
2
B)
–2 3
3
C)
–2
D)
2 3
3
B
D)
Graph the function.
2
A
7)
y =1
2 sin(x +)
7)
A)
B)
C)
D)
Solve the problem.
8)
The formula Z = R sec gives the impedance Z (in ohms) in an ac circuit with resistance R (in
ohms) and phase angle (in radians). Find the average value of Z for an 76.1– resistor as ranges
from /8 radians to /3 radians.
8)
A)
132.6
B)
69.5
C)
139
D)
106.2
Give the amplitude or period as requested.
9)
Amplitude of f(x) = – 4 sin 3x
9)
A)
4
B)
4
3
C)
3
D)
4
Find the integral.
10)
cot x
3 dx
10)
A)
3 ln cos x
3+ C
B)
–3csc2x
3+ C
C)
–1
3 ln sin x
3+ C
D)
3 ln sin x
3+ C
Find the slope of the line tangent to the curve at the given point.
11)
y =9 tan x; x =
3
11)
A)
9
B)
18
C)
12
D)
36
4
Find the exact value of the following expression without using a calculator.
12)
sec
4
12)
A)
2
B)
2 3
3
C)
2
2
D)
3
Find the integral.
13)
sin x
1 – cos x dx
13)
A)
– cos x
x – sin x + C
B)
– ln 1 – cos x + C
C)
ln 1 – cos x + C
D)
cos x
x – sin x + C
Find the exact value of the following expression without using a calculator.
14)
tan 5
6
14)
A)
3
2
B)
–3
C)
–3
3
D)
2 3
3
Find the integral.
15)
–9x sin 4x dx
15)
A)
–9
16 sin 4x +9
4x cos 4x + C
B)
–9
4 sin 4x +9
4x cos 4x + C
C)
9
16 sin 4x +9
4 cos 4x + C
D)
–9
16 sin 4x +9
4 cos 4x + C
Solve the problem.
16)
The intensity of light I at time t hours after sunrise and at a depth of x feet below the ocean surface is
given by
I(x,t) =Ioe–kx sin3t
D ,
where D is the length of daylight in hours, Io is the intensity of light on the surface of the water at
midday, and k is a positive constant. Assume D = 14, Io= 50, and k =0.006. In a study of the effect
of light on ocean plants, a scientist needs to determine the minimum depth x at which a plant
would never at any time be exposed to an intensity of light greater than 25. What is this minimum
depth?
16)
A)
116 ft
B)
76 ft
C)
374 ft
D)
108 ft
17)
The total sales in dollars of some small businesses fluctuates according to the equation
S = A + B sin
6x , where x is the time in months, with x = 1 corresponding to January, A =7800,
and B =2500. Determine the month with the greatest total sales and give the sales in that month.
17)
A)
September; $5300
B)
June; $7800
C)
March; $10,300
D)
December; $10,300
Find the exact value of the following expression without using a calculator.
18)
cot –11
6
18)
A)
–3
3
B)
–3
C)
3
D)
3
3
Solve the problem.
19)
The beacon on a lighthouse 30 m from a straight shoreline rotates twice per minute. How fast is the
beam moving along the shoreline at the moment when the light beam and the shoreline are at right
angles?
[Hint: Find an equation relating and x where :
is the angle between the beam of light and the line from the lighthouse to the shoreline
x is the distance along the shoreline from the point on the shoreline closest to the lighthouse and
the point where the beam hits the shoreline.
Use the chain rule to find dx/dt.
You need to express d/dt in radians per minute.]
19)
A)
30 m/min
B)
120 2 m/min
C)
303 m/min
D)
120 m/min
Find the exact value of the following expression without using a calculator.
20)
sin 7
6
20)
A)
1
2
B)
–3
2
C)
–1
2
D)
3
2
Graph the function.
21)
y = – cos(x)
21)
7
A)
B)
C)
D)
Find the integral.
22)
8 csc3 x cot x dx
22)
A)
2csc4 x cot x + C
B)
–8
3csc4 x + C
C)
–8
3csc3 x + C
D)
–8
3cot3 x + C
Find the derivative of the function.
23)
y = (csc x + cot x)(csc x – cot x)
23)
A)
dy
dx = – csc x cot x
B)
dy
dx = 1
C)
dy
dx = – csc2x
D)
dy
dx = 0
Use a calculator to find the function value to four decimal places.
24)
csc 26.7°
24)
A)
0.8934
B)
0.4493
C)
2.2256
D)
1.0000
Convert the radian measure to degrees.
25)
–
4
25)
A)
–32.5°
B)
–22.5°
C)
–45°
D)
–90°
Give the amplitude or period as requested.
26)
Amplitude of f(t) = – 4sin
11 t – 6
26)
A)
22
B)
–6
C)
4
D)
–4
Solve the problem.
27)
A population of animals varies periodically between a low of 700 on January 1 and a high of 900 on
July 1. Find an equation (using a trigonometric function) to describe the size of the population at
any time t, where t is measured in months from the beginning of the year, and use this equation to
find the rate of change of the population on September 1. Give your answer in exact form.
27)
A)
–253
6 animals per month
B)
–100
3 animals per month
C)
–50 3 animals per month
D)
–253
3 animals per month
Use a calculator to find the function value to four decimal places.
28)
sin 56.5°
28)
A)
–0.0486
B)
1.5108
C)
0.8339
D)
0.5519
Find the derivative of the function.
29)
y = x · csc 5x
29)
A)
dy
dx =csc 5x –5x · csc 5x cot 5x
B)
dy
dx =csc 5x –5x · csc 5x tan 5x
C)
dy
dx =csc 5x +5x · csc 5x cot 5x
D)
dy
dx =csc 5x – x · csc 5x cot 5x
30)
y =5tan3x
30)
A)
dy
dx =15 tan3x sec x
B)
dy
dx =15 tan4x
C)
dy
dx =15 tan2x
D)
dy
dx =15 tan2x sec2x
Solve the problem.
31)
A scientist studying ocean tides places an 8 ft high marker in the water at 6 am on a Monday
morning. At that time the water is about 5.5 ft high and receding. The scientist observes that the
water reaches its lowest level, 0.1 ft, at 9:18 am and then begins to rise. Assume that the water level,
in feet, is given by
h(t) = 4.9 sin 2
12.4 t+ 5,
where t represents the number of hours after midnight. (In other words, the marker was placed in
the water when t = 6.) Find the first time interval during which the marker is completely
underwater.
31)
A)
Approximately from 1:18 am to 4:54 am Tuesday
B)
Approximately from 2:06 pm to 4:24 pm Monday
C)
Approximately from 11:24 pm Monday to 2:00 am Tuesday
D)
Approximately from 1:42 pm to 5:18 pm Monday
Give the exact value.
32)
tan 300°
32)
A)
–3
B)
3
3
C)
3
D)
–3
3
Solve the problem.
33)
The motion of a spring–mass system is described by the equation y =12 sin t –
6, where y is the
distance in feet from the equilibrium position and t is time in seconds. If the weight is 22 feet from
the ceiling in a state of equilibrium, find the time at which the weight first passes the equilibrium
position.
33)
A)
1 sec
B)
4 sec
C)
1
4 sec
D)
1
8 sec
34)
From a boat on the lake, the angle of elevation to the top of a cliff is 25°15‘. If the base of the cliff is
936 feet from the boat, how high is the cliff (to the nearest foot)?
34)
A)
454 ft
B)
451 ft
C)
441 ft
D)
444 ft
Use a calculator to find the function value to four decimal places.
35)
sec 0.21
35)
A)
0.2085
B)
0.2131
C)
1.0225
D)
0.9780
Solve the problem.
36)
A contractor needs to know the height of a building to estimate the cost of a job. From a point 88
feet away from the base of the building, the angle of elevation to the top of the building is found to
be 44° 9 . Find the height of the building. Round your answer to the hundredths place.
36)
A)
83.90 ft
B)
85.43ft
C)
88.33 ft
D)
89.66 ft
Find the integral.
37)
sec2 x tan x dx
37)
A)
1
2tan x + C
B)
tan 2x + C
C)
1
2tan2 x + C
D)
sec 2x + C
Find the derivative of the function.
38)
y = cos x3
38)
A)
dy
dx = – 3x3 sin x3
B)
dy
dx =3 sin x3
C)
dy
dx = sin x3
D)
dy
dx = – 3x2 sin x3
Find the integral.
39)
x
6 tan x
6
2 dx
39)
A)
–3 ln cos x
6
2
+ C
B)
–1
12 x2 ln cos x
6
2
+ C
C)
3 ln sin x
6
2
+ C
D)
3sec2x
6
2
+ C
Find the derivative of the function.
40)
y =5 cos x
3– sin x
40)
A)
dy
dx =
–15 sin x +5sin2x –5cos2x
(3 – sin x)2
B)
dy
dx =
–15 sin x +5 sin2x –5 sin x cos x
(3 – sin x)2
C)
dy
dx =15 sin x –5
(3 – sin x)2
D)
dy
dx =
–15 sin x +5
(3 – sin x)2
41)
y = cos (2x2+ 4)
41)
A)
dy
dx =4x sin (2x2+ 4)
B)
dy
dx = – 4 sin 2x2
C)
dy
dx = – 4x sin (2x2+ 4)
D)
dy
dx = sin (2x2+ 4)
Find the slope of the line tangent to the curve at the given point.
42)
y =19 cos x; x =
4
42)
A)
19 2
2
B)
–19
2
C)
–19 2
2
D)
19 3
2
If is an angle in the indicated quadrant, determine whether the given function is positive or negative.
43)
II, sec
43)
A)
Negative
B)
Positive
44)
IV, sin
44)
A)
Positive
B)
Negative
45)
III, cot
45)
A)
Negative
B)
Positive
Solve the problem.
46)
A car moves along a straight road. The distance from the starting point is given by s(t) = 2 sin t +
cos t. Find the velocity at t = 0.
46)
A)
–1
B)
0
C)
1
D)
2
14
Find the derivative of the function.
47)
y =ecos x
47)
A)
dy
dx = sin x ecos x
B)
dy
dx = – sin x ecos x
C)
dy
dx = – esin x
D)
dy
dx = – cos x esin x
Solve the problem.
48)
A weight attached to a spring is pulled down 8 inches below the equilibrium position. Assuming
that the frequency of the system is 5
cycles per second, determine a trigonometric model that gives
the position of the weight at time t seconds.
48)
A)
y = – 8 cos 10t
B)
y =8 cos 5t
C)
y = – 8 cos 5t
D)
y =8 cos 10t
Graph the function.
49)
y =4 sin(x –) + 4
49)
A)
B)
15
C)
D)
Find the integral.
50)
sin x cos3 x dx
50)
A)
–3cos3 x + C
B)
1
4sin4 x + C
C)
3sin3 x + C
D)
–1
4cos4 x + C
Graph the function.
51)
y = – 3 cos x +
2
51)
16
A)
B)
C)
D)
Find the slope of the line tangent to the curve at the given point.
52)
y =11 sin x; x =
2
52)
A)
–11
B)
11
C)
0
D)
11
2
Solve the problem.
53)
Tides go up and down in a 14–hour period. The average depth of a certain river is 14 m and ranges
from 11 to 17 m. The depth of the river can be approximated by a sine curve. Write an equation that
gives the depth x hours after midnight given that high tide occurs at 7:00 am.
53)
A)
d = 7 sin x
7–
4
B)
d = 3 sin x
7+ 14
C)
d = 3 sin x
7–
2+ 14
D)
d = 3 sin x
14 –
2+ 1
Find the derivative of the function.
54)
y =sec x + csc x
csc x
54)
A)
dy
dx = – csc x cot x
B)
dy
dx = sec x tan x
C)
dy
dx =sec2x + 1
D)
dy
dx =sec2x
D)
Evaluate the definite integral.
55)
/6
7 cos x dx
55)
A)
14
B)
3.5
C)
–3.5
D)
7
D)
Solve the problem.
56)
The revenue received from the sale of electric heaters is seasonal, with maximum revenue in the
winter. Let the revenue received from the sale of heaters be approximated by R(x) =240 cos 2x +
600, where x is time in years, measured from January 1. Find R'(x) for March 1st.
56)
A)
–2402
B)
–240
C)
0
D)
–2403
D)
D)
Find the integral.
57)
9 sin 5x dx
57)
A)
–9
5 cos 5x + C
B)
–45 cos 5x + C
C)
9
5 cos 5x + C
D)
9 cos 5x + C
Solve the problem.
58)
The electric charge q, in Coulombs, passing a given point in a circuit is given by q = t csc (0.4t2+ 2),
where t is the time in seconds. Find the current i, to the nearest tenth of an amp, for
t =0.75 s. (i = dq/dt)
58)
A)
1.7 A
B)
2.0 A
C)
0.97 A
D)
0.43 A
Convert the radian measure to degrees.
59)
3
59)
A)
30°
B)
60°
C)
120°
D)
40°
Solve the problem.
60)
The chairlift at a ski resort has a vertical rise of 2700 feet. If the length of the ride is 1.8 miles, what
is the average angle of inclination of the lift (to the nearest tenth of a degree)?
60)
A)
10.5°
B)
13.5°
C)
16.5°
D)
19.5°
Graph the function.
19
61)
y = cos 2
3x
61)
A)
B)
C)
D)