Math 166
Practice Questions
1. Find all the values of θ in radians, for 0 ≤ θ < 2π, in the following.
(a) sin θ = −1/2
Answers: (a) θ =
(b) tan θ = −2.5
7π 11π
6 , 6
(c) cos θ = 0.7
(b) θ ≈ 1.95, 5.09
(b) θ ≈ 0.795, 5.488
2. Consider the curve of equation y = −5 sin(2x + π4 ).
(a) Find the amplitude, period and phase shift.
(b) Sketch the graph of the curve over one period.
y
5
−18
Answers: (a) A = 5, T = π, s = − π8 . (b)
π
3
8
7
8
π
x
π
−5
3. Find an equation in the form y = a sin(bx + c) for the sine curve shown below.
y
3
1
−12
5
12
π
π
x
−3
Answer: y = 3 sin(4x + π3 )
4. Express the following as the logarithm of a single expression.
4 ln x − 5 ln(x + 1) + 12 ln(x + 2)
Answer: ln
√
x4 x+2
5
(x+1)
√
3
x
given that
5. Evaluate logb
y2z3
logb x = 12,
Answer: −12
6. Solve the following equations.
(a) 2x = 10
(b) ln(2x + 1) = 4
logb y = 2,
and
logb z = 4.
Math 166
Practice Questions
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(c) e2x = 4e−3x
(d) ln(ln x) = 0
(e) 5x+2 = 7x
(f) ln(9x) + ln x = 2
Answers: (a) x ≈ 3.322 (b) x ≈ 26.799 (c) x ≈ 0.277 (d) x = e (e) x ≈ 9.57 (f) x = e/3
7. The current i (in A) in a certain electrical circuit is given by
i = 10(1 − e−3t )
where t is the time in seconds.
(a) Find the current when t = 0.5 seconds.
(b) Find the time t when the current is 9.0 A.
Answers: (a) 7.77 A
(b) 0.77 seconds
8. An investment of $5000 earns interest at an annual rate of 4% compounded monthly. The value
V of the investment after t years is given by
0.04 12t
.
V = 5000 1 +
12
(a) What will the investment be worth in 5 years?
(b) How long will it take for the investment to be worth $12,000?
Answers: (a) V = $6104.98
(b) 21.9 years
9. Assume that a population of fruit flies after t days is given by
P = P0 ekt .
If 100 fruit flies are present initially, and 300 are present after 10 days. How many will be
present after 15 days?
Answer: About 520.
10. Perform the indicated operations, expressing all answers in the form a + bj.
√
√
(a) j 2 + −36 − 5j 3 −4
(b) (2 + j)(3 − j)2
8+j
(c)
2 + 3j
1
j
(d)
+
1 − 2j 1 + j
Answers: (a) −11 + 6j (b) 22 − 4j (c)
11. Evaluate z +
Answer:
6
5
−
1
z for
12
5 j
19
13
−
22
13 j
(d)
7
10
+
9
10 j
z = 1 + 2j. Express your answer in the form a + bj.
Math 166
Practice Questions
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12. Find the values of the real numbers x and y if
x − 2j 2 + 3j = 5 + yj + 2xj 3 .
Answers: x = 3, y = 9
13. (a) Convert to rectangular form: 2.6 6
262.5◦
(b) Convert to polar form: −12 + 5j.
(c) Convert to rectangular form: 3e2.5j .
30◦ .
(d) Convert to exponential form: 5 6
Answers: (a) −0.339 − 2.578 (b) 13 6
157.38◦
π
(c) −2.403 + 1.795j (d) 5e 6 j
14. Perform the following operations. Give your answers in polar form.
(a) (3j)(26 25◦ )4
(106 112◦ )(86 228◦ )
(b)
(46 48◦ )(56 72◦ )
(c) 36 115.2◦ + 56 195.6◦
Answers: (a) 486
190◦
(b) 46
220◦
(c) 6.26
167.3◦
15. Evaluate the following limits.
x2 − x − 6
x→3 x2 − 4x + 3
√
x−3
(b) lim
x→9 x − 9
2x2 + 3
(c) lim
x→∞ 5x2 − 2x + 1
Answers: (a) 5/2 (b) 1/6
(a) lim
(c) 2/5
16. Find the slope of the tangent line to the curve y = 3x2 −
√
x − 10x at point (4, 6).
Answer: 55/4
17. Find all the points at which a tangent line to the curve y = 2x3 + 1 is perpendicular to the line
of equation x + 6y = 12.
Answers: (1, 3) and (−1, −1)
18. Find all the points at which a tangent line to the curve y = x3 − 14x is parallel to the line of
equation 4x + 2y = 1.
Answers: (2, −20) and (−2, 20)
19. (a) What is the definition of the derivative of a function?
(b) Use the definition of the derivative to find f 0 (x) for
f (x) = 3x − 4x2 + 1.
Math 166
Practice Questions
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Answers:
(a)
f 0 (x) = lim
h→0
f (x + h) − f (x)
h
(b) After several algebraic steps, we should get f 0 (x) = 3 − 8x.
20. Find the derivative of the following functions. Simplify your answers.
1
1
(a) y = √
+ 2x5
+
3
x 3x + 1
√
(b) y = 4x2 + 9
(c) y = 2x4 (x3 + 1)5
3x
(d) y =
(2x + 1)2
Answers:
(c) y 0 =
3
−1
4
− (3x+1)
2 + 10x
3x4/3
√ 4x
4x2 +9
2x3 (x3 + 1)4 (19x3 + 4)
(d) y 0 =
3−6x
(2x+1)3
(a) y 0 =
(b) y 0 =
21. Find all the points on the curve
y=
3x
x2 + 4
where there is a horizontal tangent line?
Answers: (2, 3/4) and (−2, −3/4)
22. The resistance R (in Ω) of a certain wire as a function of the temperature T (in ◦ C) is given by
R = 12.0 + 0.35T + 0.0125T 2 .
Find the instantaneous rate of change of R with respect to T when T = 110◦ C.
Answer:
dR
dT
= 3.1 ◦ΩC
23. The position s of an object (in meters) after t seconds is given by
p
s = 2t2 + 5.
Find the instantaneous velocity of the object at t = 3 seconds.
Answer: 1.25 m/s
24. Find the second-derivative of the following function.
f (x) =
Answer: f 00 (x) =
110x2 −10
(x2 +1)7
1
(x2 + 1)5
Math 166
Practice Questions
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25. Find the second-derivative of
f (x) =
p
x3 + 1
evaluated at x = 2.
Answer: 2/3
26. Use implicit differentiation to find y 0 =
dy
given that
dx
x3 + y 3 = xy + 1.
y−3x2
3y 2 −x
Answer: y 0 =
27. Find the slope of the tangent line to the curve
(y 2 + 1)4 = x2 y + 5x + 2.
at point (2, 1).
Answer: 3/20
28. Find an equation for the normal line to the graph of y = 3x − 4x2 at the point corresponding
to x = 1.
Answer: y =
x
5
−
6
5
29. Find an equation for the tangent line at point (2, 4) on the curve x3 + y 3 = 9xy.
Answer: y = 45 x +
12
5
√
30. The impedence Z (in Ω) in a certain electric circuit is given by Z = 48 + R2 , where R is the
resistance. If R is increasing at the rate 0.45 Ω/min for R = 6.5 Ω, find the rate at which Z is
changing.
Answer: 0.308 Ω/min
31. Suppose the resistance of a certain resistor varies with temperature according to
R = 0.2 +
T2
50
where T is in ◦ C and R in Ω. If the temperature is increasing at the rate of 0.1◦ C/s, find the
rate of change of resistance when T = 100◦ C
Answer: 0.4 Ω/s
32. Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3 /s.
How fast is the radius of the balloon increasing when the diameter of the balloon is 20 cm?
Answer: 0.08 cm/s
33. A ladder 5 meters long is leaning against a vertical wall. If the bottom of the ladder is pulled
away at the constant rate of 2 m/s, how fast is the top of the ladder moving down the wall
when the bottom is 3 meters from the wall?
Answer: 1.5 m/s
Math 166
Practice Questions
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34. A water tank has the shape of an inverted circular cone with a top radius of 2 m and a height
of 6 m. If water is being pumped into the tank at a rate of 4 m3 /min, find the rate at which
the water level is rising when the water is 4 m deep. Hint: V = 31 πr2 h.
Answer: 0.716 m/min
35. Use Newton’s method to estimate the first three decimals of the root of f (x) = x4 − 2x2 − 1
located inside the interval 1 ≤ x ≤ 2.
Answer: 1.553
36. Let f (x) = x4 − 2x2 − 1.
(a) Find the intervals on which f is increasing and decreasing.
(b) Locate all relative maximum and relative minimum.
(c) Find the intervals where f is concave up and concave down and locate all inflection points.
(d) Sketch the graph of y = f (x).
Answers:
(a)
f (x)
(−∞, −1)
&
(−1, 0)
%
(0, 1)
&
(1, +∞)
%
(b) The points (−1, −2) and (1, −2) are relative minimum and point (0, −1) is a relative
maximum.
(c)
(−∞, − √13 )
(− √13 , √13 )
( √13 , +∞)
∪
∩
∪
(− √13 , − 14
9 )
are inflection points.
f (x)
Both points
( √13 , − 14
9 )
and
(d) Since f (−x) = f (x), the graph is symmetric about the y-axis. From the previous question,
we know that the x-intercepts are (approximately) ±1.553.
y
6
4
2
−2
−1
0
−2
x
1
2
Math 166
Practice Questions
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37. Use the first and second derivative to sketch the graph of y = 3x5 − 5x3 . Identify all relative
extremum and inflection points.
3
2
1
Answer:
-2
-1
0
1
2
-1
-2
-3
√
2 −7 2
2 , 8 ),
√
Relative max at (−1, 2), relative min at (1, −2), inflection points at (0, 0), (
√
√
( −2 2 , 7 8 2 ).
and
38. An open-top box is to be made by cutting away four congruent squares of side length x from
the corners of a 50 cm × 40 cm piece of cardboard. Find the value of x that gives a box of
maximum volume.
Answer: x ≈ 7.36 cm
39. A peanuts manufacturer wishes to design a can to hold dry-roasted peanuts. The volume of the
cylindrical can is 1000 cm3 , and the circular top of the can is made from aluminum while the
sides and bottom are made from stainless steel. If aluminum is twice as expensive as stainless
steel, what are the most economical dimensions of the can?
q
1000
Answer: The radius is r = 3 1000
3π ≈ 4.73 cm, and the height is h = πr2 ≈ 14.2 cm.
40. Find the point (x, y) on the curve of equation y =
Answer: (x, y) = 21 , √12
√
x that is the closest to point (1, 0).
√
41. Use differentials to estimate 3 8.3.
√
Answer: 3 8.3 ≈ 2 + 0.3
12 = 2.025
42. The volume of a sphere of radius r is V = 43 πr3 . Use differentials to estimate both the absolute
error and the relative error of the volume if r = (1.5 ± 0.2) cm.
Answer: Absolute error: ∆V ≈ 5.7 cm3 . Relative error:
∆V
V
≈ 40%.
43. Find and simplify the derivative of the following functions.
(a) y = x3 cos(x2 + 1)
(b) y = e−2x sin(3x) + ln(cos(2x))
(c) y = sec2 (3x) + tan(4x)
(d) y = sin(πx) + cos3 (5x2 )
r
x
2
(e) y = ln 3 3
(f) y = 2xex − ln(1 + x2 )
x +1
Math 166
Practice Questions
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Answers:
(a) y 0 = 3x2 cos(x2 + 1) − 2x4 sin(x2 + 1)
(b) y 0 = −2e−2x sin(3x) + 3e−2x cos(3x) − 2 tan(2x)
(c) y 0 = 6 sec2 (3x) tan(3x) + 4 sec2 (4x)
(d) y 0 = π cos(πx) − 30x cos2 (5x2 ) sin(5x2 )
(e) y 0 =
1 − 2x3
3x(x3 + 1)
2
(f) (2 + 4x2 )ex −
2x
1 + x2
44. Consider the curve of equation:
sin(xy) + e−2x = 2 + y 2 − cos(2x).
Use implicit differentiation to find the derivative of y with respect to x.
Answer:
2e−2x + 2 sin(2x) − y cos(xy)
dy
=
dx
x cos(xy) − 2y
45. Find all values of x where the derivative of y = ln(4x2 + 1) − 2x2 is zero. Test if the points
correspond to relative maximum or minimum
Answer: x = 0 → relative min, x = ±1/2 → relative max.
46. Consider the following matrices.
1
2 1
A=
2 −3 2
Evaluate: (a) 2AC + 3D


1 2 4
B = 2 1 0
3 2 1
(b) BC


3 4
C = 1 0
2 2
(c) D−1 .
Answers:
20 15
(a)
23 36


13 12
(b)  7 8 
13 14
4/5 −1/5
(c)
−3/5 2/5
47. Solve the following systems of linear equations.
(a)


x + 2y − z = 2
3x + 7y − 5z = 5


−x − 2y = 1
2 1
D=
3 4
Math 166
Practice Questions
(b)


x + y + z = 2
x − 4y − z = 3


2x + 3y + 5z = 9
(c)
(
1.4x + 2.7y = 15.3
3.1x − 2.7y = 28.2
Answers:
(a) x = 13, y = −7, z = −3
(b) x = 1, y = −1, z = 2
(c) x = 9.67, y = 0.65
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