Mathematics 111 (Calculus II)
Laboratory Manual
Department of Mathematics & Statistics
University of Regina
2nd edition
prepared by Patrick Maidorn
University of Regina Department of Mathematics and Statistics
Contents
Module 1. Inverse Functions
1.1 Inverse Functions . . . . . . . . . . . . . . . . . . . .
1.2 Exponential and Logarithmic Functions . . . . . . .
1.3 Calculus of Exponential and Logarithmic Functions .
1.4 Inverse Trigonometric Functions . . . . . . . . . . .
1.5 L’Hopital’s Rule . . . . . . . . . . . . . . . . . . . .
Module 2. Techniques of Integration
2.1 Integration by Parts . . . . . . . . . . . . . . . . . .
2.2 Trigonometric Integrals . . . . . . . . . . . . . . . .
2.3 Trigonometric Substitution . . . . . . . . . . . . . .
2.4 Partial Fractions . . . . . . . . . . . . . . . . . . . .
2.5 Challenge Integration Practice . . . . . . . . . . . .
2.6 Improper Integrals . . . . . . . . . . . . . . . . . . .
Module 3. Integration Applications
3.1 Review - Areas Between Curves . . . . . . . . . . . .
3.2 Volumes by Cross Sections . . . . . . . . . . . . . . .
3.3 Volumes by Cylindrical Shells . . . . . . . . . . . . .
3.4 Arclength . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Applications in the Physical Sciences . . . . . . . . .
3.6 Taylor Series . . . . . . . . . . . . . . . . . . . . . .
Module 4. First Order Differential Equations
4.1 Introduction to Differential Equations . . . . . . . .
4.2 Separable Equations . . . . . . . . . . . . . . . . . .
4.3 Application: Growth and Decay Models . . . . . . .
4.4 Linear Equations . . . . . . . . . . . . . . . . . . . .
4.5 Partial Derivatives . . . . . . . . . . . . . . . . . . .
4.6 Exact Equations . . . . . . . . . . . . . . . . . . . .
4.7 Homogeneous Equations . . . . . . . . . . . . . . . .
4.8 Applications: Cooling and Mixing . . . . . . . . . .
Answers
References
i
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3
3
3
4
5
6
7
7
7
8
8
9
9
11
11
11
12
12
13
13
15
15
15
16
16
17
17
18
18
19
27
ii
Introduction
“One does not learn how to swim by reading a book about swimming,” as surely everyone agrees.
The same is true of mathematics. One does not learn mathematics by only reading a textbook and
listening to lectures. Rather, one learns mathematics by doing mathematics.
This Laboratory Manual is a small set of problems that are representative of the types of problems
that students of Mathematics 111 (Calculus II) at the University of Regina are expected to be able
to solve on quizzes, midterm exams, and final exams. In the weekly lab of your section of Math
111 you will work on selected problems from this manual under the guidance of the laboratory
instructor, thereby giving you the opportunity to do mathematics with a coach close at hand.
These problems are not homework and your work on these problems will not be graded. However,
by working on these problems during the lab periods, and outside the lab periods if you wish, you
will gain useful experience in working with the central ideas of elementary calculus.
The material in the Lab Manual does not replace the textbook. There are no explanations or short
reviews of the topics under study. Thus, you should refer to the relevant sections of your textbook
and your class notes when using the Lab Manual. These problems are not sufficient practice to
master calculus, and so you should solidify your understanding of the material by working through
problems given to you by your professor or that you yourself find in the textbook.
To succeed in calculus it is imperative that you attend the lectures and labs, read the relevant
sections of the textbook carefully, and work on the problems in the textbook and laboratory manual.
Through practice you will learn, and by learning you will succeed in achieving your academic goals.
We wish you good luck in your studies of calculus.
1
2
Module 1
Inverse Functions
1.1
Inverse Functions
1. Determine whether each of the following functions is invertible on its domain.
3π
(a) f (x) = 1 + x4
(d) f (x) = cos x on x ∈ 0,
2
(b) f (x) = sin x + cos x
x if x ≤ 0
x
(e) f (x) =
(c) f (x) =
2 if x > 0
x
x−1
Answers:
Page 19
2. In each case, find a formula for the inverse function f −1 (x).
(a) f (x) = x3 − 1
(b) f (x) = x2 − 4x on x ≥ 2
(c) f (x) =
1−x
2+x
3. In each
0 case, check whether f (x) is a one to one differentiable function, and if it is, find
−1
f
(a).
(c) f (x) = x3 − x, a = 2
(a) f (x) = x5 + 3x3 + 4, a = 8
1
(b) f (x) = 1 − sin x, a =
2
1.2
Exponential and Logarithmic Functions
1. Write each expression as a single exponential.
3x
(a) 23+x · 3
2
5x
e3
(b) √
3 x
e
2. Write as a single logarithm.
(a) 4 ln(x) −
1
ln(3x) + 1
2
(b) 4 log(x + 1) − 2 log(x) + 2
3
Answers:
Page 19
4
MODULE 1. INVERSE FUNCTIONS
3. Solve each equation for x.
(a) 3x−1 = 81
x+1 x+2
1
1
=
(b)
2
4
3x
(c) 4e = 16
(d) 2 = 104x−2
(e) ln(ln(x)) = 1
4. Assume the world’s population doubles every 53 years.
(a) Find its annual growth rate k in N (t) = N0 ekt .
(b) In 1998, Earth’s population was 6 billion. Use the model in (a) to predict the population
in 2020.
(c) In what year will Earth’s population reach 10 billion, according to this model?
5. Radioactive carbon-14 has a half-life of 5730 years. How long will it take for an object to lose
80% of its original C-14 content?
1.3
Calculus of Exponential and Logarithmic Functions
1. Find the indicated derivatives.
Answers:
Page 19
2
(a) f (x) = ex + 3e4x , f 0 (x)
(d) f (x) = ln(1 + x2 + x), f 0 (0)
(b) f (x) = esin x , f 00 (x)
(e) f (x) = ln(cos x), f 00 (x)
x2
, f 0 (x)
(f) f (x) = ln
x2 + 1
√
(c) f (x) =
e
x
x
, f 0 (x)
2. Integrate the following:
Z
4
(a)
e4x + dx
x
Z 1 2x
e −1
(b)
dx
ex
0
Z
dx
(c)
x−1
Z
(d)
x2 e x
3 +3
Z
3x
Z
x2 +
√
4
e x
(e)
(f)
0
1
dx
dx
√ dx
x
3. Use logarithmic differentiation to find f 0 (x).
(a) f (x) = xx
(b) f (x) = xsin x
1
(c) f (x) =
(2x + 3) 2 (x − 7)5
1
(x + 1) 6
2
4. Find the equation of the tangent line to f (x) = e−x at the point x = 1.
5. Find the area between y = ex and y = e−x between x = −1 and x = 1.
1.4. INVERSE TRIGONOMETRIC FUNCTIONS
1.4
5
Inverse Trigonometric Functions
1. Find the exact value of each expression.
√ !
3
(a) sin−1
2
√ !
3
(b) cos−1 −
2
−1 1
(c) sin sin
2
(d) cos−1 (cos 3π)
2. Differentiate the following:
p
1 − x4
(d) f (x) = cos sin−1 x
(a) f (x) = 3 cos−1 x + 5 tan−1 x
√
(b) f (x) = tan−1 3x
(c) f (x) = sin−1
π
3. Find the equation of the tangent line to the graph of f (x) = sec−1 2x at the point (x, y) = 1,
.
3
4.
150 ft
θ
x
As the sun descends, the shadow cast by a 150 ft tall wall lengthens.
(a) Express the angle θ as a function of the shadow’s length x.
dθ
(b) Find
when the shadow’s length is 200 ft.
dx
5. Integrate the following:
√
Z
(a)
√
3
3
Z
(b)
3
√
1
dx
1 + x2
x
dx
1 − x4
Z
(c)
tan−1 x
dx
x2 + 1
Answers:
Page 20
6
MODULE 1. INVERSE FUNCTIONS
1.5
L’Hopital’s Rule
1-10: In each case, find the indicated limit. Note: not all limits allow the use of L’Hopital’s Rule.
Answers:
Page 20
sin x2
x→0
x
1. lim
6. lim x csc x
x→0
x2 + x + 1
2. lim
x→∞ 4x2 + 3
7. lim x2x
ln(1 + x)
3. lim
x→0
x
x−1
4. lim 2
x→1 x + 1
8. lim (x + cos x) x
x5 + 4x
5. lim
x→∞
ex
x→0+
1
x→0
9. lim
x→0+
10. lim
x→∞
1
1
−
sin x x
1
1
−
x 1 − ex
Module 2
Techniques of Integration
2.1
Integration by Parts
1-10: Integrate the following:
Z
1.
x sin 2x dx
Z
2.
2xe
Z
−3x
dx
√
x x + 1 dx
Z
cos(2x)e3x dx
Z
ax2 ebx dx
Z
x5 sin x dx
8.
x2 ln(x) dx
4.
Z
7.
e
Z
(ln(x))2 dx
6.
3x2 cos x dx
3.
Z
9.
Answers:
Page 20
1
2π
Z
5.
(3x + 5) cos
x
0
2.2
4
dx
10.
Trigonometric Integrals
1-9: Integrate the following:
Z
1.
sin2 x cos5 x dx
Z
2.
6.
sin5 x cos2 x dx
Z √
Z
7.
Z
3.
4.
sin2 x cos2 x dx
Z
8.
4
cos x dx
Z
5.
tan(2t) sec3 (2t) dt
π
0
Z
tan x sec4 x dx
π
2
cos5 x dx
0
Z
tan3 x sec4 x dx
9.
7
tan5 x dx
Answers:
Page 21
8
MODULE 2. TECHNIQUES OF INTEGRATION
2.3
Answers:
Page 21
Trigonometric Substitution
1-9: Integrate the following:
Z
p
1.
x3 1 − x2 dx
√
Z
6.
2
8
dx
3
(16 − x2 ) 2
0
Z
p
x2 4 − x2 dx
2.
Z
2
5
Z
3.
1
5
Z
4.
2.4
dx
7.
x3
p
2 − x2
2x2
Z
1
Z
9.
dx
− 12x + 26
p
x2 + 4x − 5
dx
x+2
4
8.
(4 + x2 )2
Z
5.
p
25x2 − 1
dx
x
x2 − 6x + 13
− 1
2
dx
Partial Fractions
1. In each case, find the partial fraction decomposition.
Answers:
Page 21
(a)
(b)
1
(x − 1)(x − 2)
2x + 3
−x−6
x2
2. Integrate the following:
Z
dx
(a)
2 + 3x + 2
x
Z
2x + 1
(b)
dx
(x + 1) (x2 + 1)
Z
3x2 + 7x − 2
(c)
dx
3
2
Z x − x − 2x
2x + 3
(d)
dx
9x2 + 6x + 5
5x2 − 3x + 2
x3 − 2x2
7x2 − 13x + 13
(d)
(x − 2)(x2 − 2x + 3)
(c)
Z
dx
x3 + 1
Z
x3 + 10x2 + 3x + 36
(e)
(f)
(x − 1) (x2 + 4)2
Z
(g)
dx
5x3 − 4x2 + 2x + 1
dx
5x2 − 4x − 1
2.5. CHALLENGE INTEGRATION PRACTICE
2.5
Challenge Integration Practice
1-5: Integrate the following:
Z
ex
p
1.
dx
1 − e2x
Z
p
2.
x5 x3 + 1 dx
Z
3.
2.6
9
Z
4.
1p
x2 + 1 dx
Answers:
Page 22
0
Z
5.
e4x
p
1 + e2x dx
sec3 x dx
Improper Integrals
1-10: In each case, determine whether this integral converges or diverges. If it converges, evaluate
the integral.
Z ∞
Z π
x
−3
1.
x dx
6.
tan
dx
3
1
0
Z ∞
Z ∞
2.
e−3x dx
2
7.
xe−x dx
0
Z ∞
−∞
dx
3.
Z ∞
x−1
2
dx
8.
Z 3
2+1
x
dx
−∞
p
4.
−3
9 − x2
Z ∞
dx
Z 2
9.
x
−x
−∞ e + e
5.
f (x) dx, where
0
( −1
Z 1
x2
0<x≤1
f (x) =
10.
ln |x| dx
x−1 1<x≤2
−1
Answers:
Page 22
10
Module 3
Integration Applications
3.1
Review - Areas Between Curves
1. Find the area of the region R bound by the line y = x and the parabola y = 6 − x2 .
2. Find the area of the region R enclosed by y = sin x and y = cos x from x = 0 to x = 2π.
Answers:
Page 22
3. Find the area of the region R enclosed by y = 2x − 1, y = x2 − 4, x = 1, and x = 2.
3.2
Volumes by Cross Sections
1. The region R is bounded by the curves y = x2 and y = 1. R is rotated about the line y = 2,
generating a ring shaped solid. Sketch the region R as well as a typical cross section of the
solid. Find the volume of the solid.
2. Find the volume of the solid S obtained by rotating the region bounded by y = x2 and y = x3
about the x-axis.
3. Find the volume of the right-circular cone with base radius r and height h. Note: the cone is
generated by rotating the triangle with vertices (0, 0), (0, h), and (r, h) about the y-axis.
√
4. Consider the region R, bound by y = x3 and y = x. Find the volume of the resulting solid
if
(a) R is revolved around the x-axis.
(b) R is revolved around the y-axis.
(c) R is revolved around the vertical line x = −1.
5. (a) Sketch the curve given by x = 2y − y 2 .
(b) Find the volume obtained by rotating the region enclosed by x = 0 and x = 2y − y 2
about the y-axis.
6. (a) R is bounded by y = sin x, y = 0, x = 0, and x = π. Rotate R about the x-axis. Find
the volume of the resulting solid.
π
(b) R is bounded by y = sin x, y = cos x, x = 0, and x = . Rotate R about the x-axis.
4
Find the volume of the resulting solid.
11
Answers:
Page 22
12
MODULE 3. INTEGRATION APPLICATIONS
7. A circular man-made lake has a 200m diameter
and a maximum depth of 10m. Its cross
x 2
section is the parabola y = 10
− 1 . Find the capacity of the lake.
100
3.3
Answers:
Page 23
Volumes by Cylindrical Shells
1. Consider the bowl obtained by revolving the region bounded by y = x2 , y = 1, and x = 0
about the y-axis.
(a) Find its volume using cross sections.
(b) Find its volume using cylindrical shells.
(c) Compare your answers.
2. Consider the region bounded by y = x3 and y =
of the resulting solid if
√
x. Use cylindrical shells to find the volume
(a) R is revolved about the x-axis.
(b) R is revolved about the y-axis.
(c) R is revolved about the vertical line x = −1.
Note: Compare your answers with those of question 4 in Section 3.2.
3. Find the volume V of the solid generated by revolving the region enclosed by y = 3x2 − x3 ,
y = 0, x = 0, and x = 3 about the y-axis.
√
4. Determine the volume of the solid obtained by rotating the region bounded by y = 2 x and
y = x about the line x = 5.
1
5. Find the volume of the solid obtained by rotating the region bounded by y = (x − 1) 2 and
y = (x − 1)2 about the y-axis.
6. Consider the solid sphere of radius R. A cylinder of radius r < R is bored through the center
of the sphere. Find the volume of the remaining solid.
√
7. Consider f (x) = sin(x2 ) and g(x) = − sin(x2 ) from x = 0 to x = π. Find the volume of
revolution if the region enclosed by f (x) and g(x) is rotated about the y-axis.
8. Let R be the region in the first quadrant bounded by y = (x − 2)1/2 and let y = 2.
(a) Find the resulting volume if R is rotated about the x-axis.
(b) Find the resulting volume if R is rotated about the line y = −2.
3.4
Arclength
3
1. Find the length of the curve f (x) = x 2 between x = 0 and x = 4.
Answers:
Page 23
1
2. Find the length of the curve f (x) = 2ex + e−x between x = 0 and x = ln 2
8
2
3
3x 3
2(3) 2
3. Find the length of the curve y =
+ 1 between x = 0 and x =
. Hint: consider
2
3
the curve as a function x(y) instead.
1
1
4. Find the length of the curve x = y 3 +
between y = 1 and y = 2.
6
2y
3.5. APPLICATIONS IN THE PHYSICAL SCIENCES
3.5
13
Applications in the Physical Sciences
1. Find the centroid of the triangle with vertices (0, 0), (1, 0), and (0, 1).
2. Find the centroid of the region bounded by the curves y = sin(x) + 1 and y = 1 between
π
x = 0 and x = .
2
Answers:
Page 23
3. Find
h π ithe centroid of the region bounded by the curves y = 2 sin 2x and y = 0 on the interval
0, .
2
1
4. Find the centroid of the region bounded by the curves y = x3 and y = x 2 .
5. A vertical dam in the shape of a symmetric trapezoid has height of 30 m, a width at the base
of 20 m, and a width at the top of 40 m. What is the total force on the face of the dam when
the reservoir is full?
6. A circular plate of radius 2 m is held vertically and submerged 6 m into water (i.e. its center
point is 8 m below the water’s surface). Find the hydrostatic force on the plate.
3.6
Taylor Series
1. Find the Taylor series representation of
(a) f (x) = ln(1 + x) an a = 0.
π
(b) f (x) = cos x at a = .
4
(c) f (x) =
1
at a = 1.
x
2. Use the third degree Taylor polynomial of f (x) = ln x with center at a = 1 to estimate ln(1.1).
3. Use the Maclaurin series representation of f (x) = ex to find the Maclaurin series representation of the functions
(a) x4 ex
(b) e−2x
(c) ex
2
4. (a) Find the Maclaurin series for both y = sin x and y = cos x.
d
(b) Differentiate the above series term by term to verify that
(sin x) = cos x and
dx
d
(cos x) = − sin x.
dx
Answers:
Page 24
14
Module 4
First Order Differential Equations
4.1
Introduction to Differential Equations
x
1. Verify that y = e− 2 is a solution to the differential equation 2y 0 = −y.
√
dy
2. Verify that y = x + 4 x + 2 is a solution to the differential equation (y − x)
=y−x+8
dx
Answers:
Page 24
3. For what value(s) of k is y = ekx a solution to the differential equation y 0 + 3y = 0?
4. For what value(s) of k is y = ekx a solution to the differential equation y 00 − y 0 + 12y = 0?
5. Given that the general solution to the differential equation y 0 = y − y 2 is y =
1
the solution to the initial value problem y 0 = y − y 2 subject to y(0) = − .
3
4.2
1
, find
1 + ce−x
Separable Equations
1-6: Find the general solution of each of the given differential equations:
1. x
Answers:
Page 24
dy
= 4y
dx
dy
= e3x+2y
dx
xy
3. y 0 = 2
x +1
2.
y ln x
x
p
dx
5.
= t 1 − x2
dt
4. y 0 =
6.
dy
4x − x3
=
dx
4 + y3
15
16
MODULE 4. FIRST ORDER DIFFERENTIAL EQUATIONS
7-9: Find the particular solution to each of the given initial value problems:
7. y 0 =
8.
2x
subject to y(1) = −2.
y
dy
xy
= 2
subject to y(0) = 1.
dx
x +1
9. yex + y 2 − 1
4.3
Answers:
Page 24
dy
= 0 subject to y(0) = 1.
dx
Application: Growth and Decay Models
1. The size of a bacteria culture with initial population N0 is given by N (t), where t is in hours.
The rate of growth is proportional to the number of bacteria present.
(a) State the differential equation that models the size of the bacteria culture.
(b) Find the solution to this differential equation.
(c) The size of the culture grows by 50% each hour. How long does it take for the size of
the culture to triple?
2. The population of a town grows at a rate proportional to the population present at time t.
The population doubles every 50 years. How long will it take for the population to triple?
dP
P
3. A population of bacteria grows according to the differential equation
= 0.03P 1 −
.
dt
2000
When t = 0, the population is 200 million
(a) Find the population P at time t.
(b) Find the population P at time t = 3.
4. A population of bacteria grows logistically. Suppose the initial population is 3 million bacteria
cells, the carrying capacity is 100 million, and the growth parameter is 0.2 per hour. At what
time will the population reach 90 million cells?
4.4
Linear Equations
1-4: Solve each differential equation:
Answers:
Page 25
dy
− 3y = 6
dx
dy
2. x2 − 9
+ xy = 0
dx
1.
5-7: Solve each initial value problem:
5.
dy
+ y = x subject to y(0) = 4.
dx
3. x2 y 0 + xy = 1
4. cos x
dy
+ (sin x)y = 1
dx
4.5. PARTIAL DERIVATIVES
17
6. xy 0 + y = ex subject to y(1) = 2.
7. xy 0 = x6 ex + 4y subject to y(1) = 1.
4.5
Partial Derivatives
1. For z = x4 + 3y 4 + xy 2 find
2. For f (z) = x2 exy
∂z
∂z
and
.
∂x
∂y
Answers:
Page 25
∂z
∂z
find
and
.
∂x
∂y
3. For f (x, y) = x2 − y 2 + 4, find fx (2, −4) and fy (2, −4).
4. For w = x sin(yz), find wx , wy , and wz .
2
5. For f (x, y) = sin(x)ex y , find fx (x, y) and fy (x, y).
6. For z = 3x4 y − 2xy + 5xy 3 find all four second partial derivatives.
7. Given f (x, y) = cos(xy) + x2 ey , verify that fx,y = fy,x .
8. If f (x, y, z) = e−xy cos z, find
∂2f
∂x∂z
∂3f
(b) 2
∂ x∂z
(c)
(a)
4.6
∂4f
∂z∂ 2 x∂z
Exact Equations
1-7: Determine if the following differential equations are exact. If they are, solve them. If not,
state so.
1. 2xy dx + x2 − 1 dy = 0
dy
2xe2y − x cos(xy) + 2y
=
dx
y cos(xy) − e2y
3. x2 − y 2 dx + x2 − 2xy dy = 0
2.
4. 2y + x2 + 1
5. y 0 =
6. x
7.
dy
= 9x2 − 2xy
dx
xy 2 − cos x sin x
y − yx2
dy
+ y = 2xex + 6x2
dx
dy
2xy
− 2x − 2 − ln x2 + 1
=0
2
x +1
dx
Answers:
Page 25
18
MODULE 4. FIRST ORDER DIFFERENTIAL EQUATIONS
4.7
Homogeneous Equations
1-5: Solve each differential equation:
Answers:
Page 25
1. y 0 =
x2 + y 2
xy − x2
2. (x − 2y)dx + xdy = 0
dy
+ 4x2 + y 2 = 0
dx
4. x4 + y 4 dx + 2x3 ydy = 0
3. xy
5. xy 0 =
4.8
Answers:
Page 25
x3
+x+y
y2
Applications: Cooling and Mixing
1. An object is heated to 212 ◦ C, then placed into a room at 25 ◦ C. After 10 minutes it has
cooled down to 152 ◦ C. Use Newton’s Law of Cooling to
(a) find the temperature of the object as a function of time.
(b) determine how long it will take to cool down to 40 ◦ C
2. An object is heated, then placed into a storage area with air temperature 5 ◦ C. After 1 minute
the object’s temperature is 55 ◦ C. After 5 minutes it is 30 ◦ C. To what temperature was the
object heated initially?
1
kg of
4
salt per litre is entering the tank at 3 litre/min, while the well mixed solution is leaving the
tank at the same rate.
3. At time t = 0, 10 kg of salt is dissolved in a 100 litre water tank. Water containing
(a) What is the salt concentration in the tank (in kg/litre) after 10 minutes?
(b) What limit does the salt concentration in this tank tend to?
4. A 1000 litre tank contains a 10% alcohol solution, which is constantly stirred. A 50% alcohol
solution is pumped into the tank at a rate of 10 litres/min, and a well-stirred mixture is
pumped out at the same rate. What is the alcohol percentage in the tank after one hour?
5. At time t = 0, 10 kg of salt is dissolved in a 100 litre water tank. A salt solution is pumped
into the tank at 3 litre/min, while the well mixed solution is leaving the tank at the same
rate. The concentration of the salt solution decreases over time, and is given by the function
t
1
Conc(t) = e− 100 (in kg/litre, after t minutes). Find the function A(t), the amount of salt
4
(in kg) in the tank at time t.
Answers
1.1 Exercises (page 3)
1. (a) No (b) No (c) Yes (d) No (e) Yes
1
1
2. (a) f −1 (x) = (x + 1) 3 (b) f −1 (x) = (x + 4) 2 + 2 (c) f −1 (x) =
0
0
1
3. (a) f −1 (8) =
(b) f −1
14
1 − 2x
x+1
1
2
= − √ (c) Not 1-1
2
3
1.2 Exercises (page 3)
1. (a) 6x (b) e
2. (a) ln
4x
3
!
ex4
1
(b) log
(3x) 2
100(x + 1)4
x2
3. (a) x = 5 (b) x = −3 (c) x =
4. (a) k =
ln(4)
1
(d) x = (log(2) + 2) ≈ 0.575 (e) x = ee ≈ 15.154
3
4
ln(2)
(b) 8 Billion (c) In 2037
53
5. (a) 13300 years
1.3 Exercises (page 4)
1
1. (a) f 0 (x) = 2xe
x2
+ 12e4x (b) f 00 (x) = cos2 x − sin x esin x (c) f 0 (x) =
2
2x
(d) f 0 (0) = 1 (e) f 00 (x) = sec2 x (f ) f 0 (x) = − 2
x x +1
1
1 12 x 2
− ex 2
2x e
x2
1 4x
(e − 1)2
1 3
3 e + 4 ln |x| + c (b)
(c) ln |x − 1| + c (d) ex +3 + c (e) ln x2 + 1 + c
4
e
3
2
(f ) 2e2 − 2
sin x
0
x
0
sin
x
3. (a) f (x) = x (1 + ln(x)) (b) f (x) = x
cos(x) ln(x) +
x
1 5
(x − 7) (2x + 3) 2
5
1
1
(c) f 0 (x) =
+
−
1
x − 7 2x + 3 6(x + 1)
(x + 1) 6
2. (a)
2
3
4. y = − x +
e
e
19
20
ANSWERS
5. 2e +
2
−4
e
1.4 Exercises (page 5)
1. (a)
π
5π
1
(b)
(c) (d) π
3
6
2
5
−2x
−x
−3
3
+
(c) f 0 (x) = p
(d) f 0 (x) = p
2. (a) f 0 (x) = p
(b) f 0 (x) = √
2
2
4
1
+
x
2 3x(1 + 3x)
1−x
1−x
1 − x2
√
x
π π 3
3. y = √ + −
9
3 3
150
Rad
−1
4. (a) θ = tan
(b) −0.0024
x
ft
5. (a)
2
1
1
π
(b) sin−1 x2 + c (c)
tan−1 x + c
6
2
2
1.5 Exercises (page 6)
1. 0
6. 1
1
4
7. 1
3. 1
8. e
4. 0
9. 0
5. 0
10. 0
2.
2.1 Exercises (page 7)
1
1
1. − x cos 2x + sin 2x + c
2
4
2
2
2. − xe−3x − e−3x + c
3
9
3. 3x2 sin x + 6x cos x − 6 sin x + c
1 3
1 3 e
2
1
4. x ln(x) − x
= e3 +
3
9
9
9
1
x
x i2π
5. (12x + 20) sin
+ 48 cos
4
4 0
= 24π − 28
6. x (ln(x))2 − 2x ln(x) + 2x + c
7.
3
5
2
4
x(x + 1) 2 − (x + 1) 2 + c
3
15
8.
2
3
cos(2x)e3x +
sin(2x)e3x + c
13
13
9.
a 2 bx 2a bx 2a bx
x e − 2 xe + 3 e + c
b
b
b
10. −x5 cos x + 5x4 sin x + 20x3 cos x
− 60x2 sin x − 120x cos x + 120 sin x + c
ANSWERS
21
2.2 Exercises (page 7)
1
2
1
sin3 x − sin5 x + x + c
3
5
7
1
2
1
2. − cos3 x + cos5 x − cos7 x + c
3
5
7
π
1
1
π
3. x −
sin 4x =
8
32
8
0
1.
3
sin 2x sin 4x
4. x +
+
+c
8
4
32
1
1
5. tan6 x + tan4 x + c
6
4
6.
3
7
2
2
tan 2 x + tan 2 x + c
3
7
7.
1
sec3 2t + c
6
2
1
8. sin x − sin3 x + sin5 x
3
5
9.
π
2
=
0
8
15
1
1
tan4 x − tan2 x + ln | sec x| + c
4
2
2.3 Exercises (page 8)
1.
2.
3.
4.
5.
5 1
3
1
1 − x2 2 −
1 − x2 2 + c
5
3
3
5
4
4
− 4 − x2 2 +
4 − x2 2 + c
3
5
iπ
√
π
3
tan(θ) − θ = 3 −
3
0
1
x
1
x
tan−1
+ ·
+c
16
2
8 4 + x2
p
3
1
−2 2 − x2 +
2 − x2 2 + c
3
1
tan θ
6.
16
1
7. tan−1
4
π
4
=
0
1
16
x−3
2
+c
iπ
√
3
8. 3(tan(θ) − θ) = 3 3 − π
0
1 2
1 1
2
9. ln x − 6x + 13 + (x − 3) + c
2
2
2.4 Exercises (page 8)
1
9
1
1
1
4
5
2x + 1
1
5
−
(b)
+ 5 (c) − 2 +
(d)
+
1. (a)
x−2 x−1
x+2 x−3
x x
x−2
x − 2 x2 − 2x + 3
2. (a) ln |x + 1| − ln |x + 2| + c
3
1
1 (b) − ln |x + 1| + ln x2 + 1 + tan−1 x + c
2
4
2
(c) ln |x| − 2 ln |x + 1| + 4 ln |x − 2| + c
1
7
2
−1 3x + 1
tan
+c
(d) ln 9x + 6x + 5 +
9
18
2
1
1 2
1
−1 2x − 1
√
√
(e) ln |x + 1| − ln x − x + 1 +
tan
+c
3
6
3
3
x 1
1
1
(f ) 2 ln |x − 1| − ln x2 + 4 − tan−1
+c
− · 2
2
2
2 x +4
1
1
2
(g) x2 −
ln |5x + 1| + ln |x − 1| + c
2
15
3
22
ANSWERS
2.5 Exercises (page 9)
1. sin−1 ex + c By substitution and trigonometric substitution
3
5
2 3 3
4
x x +1 2 −
x3 + 1 2 + c By substitution and parts
9
45
x
x
x x 1
3.
tan(x) sec(x) − ln cos
− sin
+ sin
+ ln cos
2
2
2
2
2
√
√
2 + ln 2 + 1
4.
By parts and trigonometric substitution
2
5 1
3
1
1 + e2x 2 −
1 + e2x 2 + c By substitution and trigonometric substitution
5.
5
3
2.
2.6 Exercises (page 9)
1
2
1
2. Convergent
3
1. Convergent
7. Convergent 0
8. Convergent π
3. Divergent
4. Convergent π
5. Convergent
6. Convergent 3 ln(2)
5
2
9. Convergent
10. Convergent −2
3.1 Exercises (page 11)
1.
√
2. 4 2
125
6
3.
3.2 Exercises (page 11)
1.
56π
15
2.
2π
35
3.
1 2
πr h
3
4. (a)
5π
2π
37π
(b)
(c)
14
5
30
π
2
11
3
ANSWERS
23
5. (a)
y
2
1
1 x
(b)
6. (a)
16π
15
1 2
1
π (b) π
2
2
7. 50 000π m3
3.3 Exercises (page 12)
π
π
(b) V = (c) The same.
2
2
5π
2π
37π
2. (a)
(b) V =
(c) V =
14
5
30
243π
3.
10
272π
4.
15
1. (a) V =
5.
29π
30
6.
3
4π
R2 − r 2 2
3
7. 4π
8. (a) 16π (b)
128π
3
3.4 Exercises (page 12)
1. Length ≈ 9.1 units
3. Length =
14
units
3
33
units
16
4. Length =
17
units
12
2. Length =
3.5 Exercises (page 13)
1.
1 1
,
3 3
1+π
2. 1,
8
π π 3.
,
4 4
4.
12 3
,
25 7
5. 1.18 × 108 N
6. 313920π N
24
ANSWERS
3.6 Exercises (page 13)
1. (a)
∞
X
(−1)n+1 xn
n
n=1
valid for −1 < x ≤ 1
√
√ √ √ √ 2
2
2
2
2
π
π 2
π 3
π 4
(b)
−
x−
−
x−
+
x−
+
x−
− . . . valid for all x
2
2
4
2!2
4
3!2
4
4!2
4
∞
X
(c)
(−1)n (x − 1)n valid for 0 < x < 2
n=0
2. ln(1.1) ≈ 0.095333
3. (a)
(b)
(c)
∞
X
xn+4
n=0
∞
X
n=0
∞
X
n=0
n!
(−1)n (2x)n
n!
(−1)n x2n
n!
∞
∞
X
X
(−1)n 2n+1
(−1)n 2n
4. (a) sin x =
x
, cos x =
x for all x.
(2n + 1)!
(2n)!
n=0
n=0
4.1 Exercises (page 15)
3. k = −3
4. k = −3, k = 4
5. y =
1
1 − 4e−x
4.2 Exercises (page 15)
1. y = cx4
6. y 4 + 16y + x4 − 8x2 = c
2. −3e−2y = 2e3x + c
p
3. y = c x2 + 1
7. y 2 = 2x2 + 2
4. 2 ln |y| = (ln x)2 + c
8. y =
1
5. arcsin(x) = t2 + c
2
9. y 2 − ln y 2 = −2ex + 3
p
x2 + 1
4.3 Exercises (page 16)
dN
ln 3
= kN (b) N (t) = N0 ekt (c) t =
≈ 2.7 hours
dt
0.4055
2. 79 years
1. (a)
3. (a) P (t) =
400000
(b) P (3) ≈ 216.8 million
200 + 1800e−0.03t
4. t = 10.45 hours
ANSWERS
25
4.4 Exercises (page 16)
1. y = ce3x − 2
4. y = sin x + c cos x
c
2. y = p
x2 − 9
5. y = x + 5e−x − 1
6. y = x−1 (ex + 2 − e)
3. y = x−1 ln x + cx−1
7. y = x5 ex − x4 ex + x4
4.5 Exercises (page 17)
1.
∂z
∂z
= 4x3 + y 2 ,
= 12y 3 + 2xy
∂x
∂y
2.
∂z
∂z
= xexy (xy + 2),
= x3 exy
∂x
∂y
3. fx (2, −4) = 4, fy (2, −4) = 8
4. wx = sin(yz), wy = xz cos(yz), wz = xy cos(yz)
5. fx (x, y) = cos xex
2y
2
+ 2xy sin xex y , fy (x, y) = x2 sin xex
2y
6. zxx = 36x2 y, zyy = 30xy, zxy = 12x3 − 2 + 15y 2 , zyx = 12x3 − 2 + 15y 2
8. (a)
∂2f
∂3f
∂4f
= ye−xy sin z (b) 2
= −y 2 e−xy sin z (c)
= −y 2 e−xy cos z
∂x∂z
∂ x∂z
∂z∂ 2 x∂z
4.6 Exercises (page 17)
1. x2 y − y = c
5. y 2 1 − x2 − cos2 x = c
2. Not exact
6. xy − 2xex + 2ex − 2x3 = c
3. Not exact
4. y 2 + x2 + 1 y − 3x3 + c = 0
7. y ln x2 + 1 − x2 − 2y = c
4.7 Exercises (page 18)
y
1. (x + y)2 = cxe x
4. ln |x| −
2. x − y = c
3. y 2 =
c − 4x2
2x2
5.
y
y
− tan−1 = ln |x| + c
x
x
4.8 Exercises (page 18)
1. (a) T (t) = 25 + 187e−0.03869t (b) 65.21 minutes
2. 64.46 ◦ C
x2
=c
x2 + y 2
26
ANSWERS
3. (a) 0.1389 kg/litre (b) 0.25 kg/litre
4. 28.5%
5. A(t) =
75 − t
55 −2t
e 100 − e 100
2
2
Appendix A
References
The problems for this manual were collected from a variety of sources, including instructors’s
personal class notes and exams, as well as the following resources:
Adams, Essex: Calculus: A Complete Course, 8th Edition, Pearson.
Briggs, Chocran: Calculus: Early Transcendentals, Addison Wesley.
Dawkins: Paul’s Online Math Notes, http://tutorial.math.lamar.edu/
Edwards, Penny: Calculus, Early Transcendentals, 7th Edition, Prentice Hall.
Tan, Menz, Ashlock: Applied Calculus for the Managerial, Life, and Social Sciences, 1st Canadian
Edition, Nelson.
Zill, Wright: Differential Equations with Boundary-Value Problems, 8th Edition, Brooks/Cole.
27