Mathematics 111 (Calculus II) Laboratory Manual · 2020-04-20 · This Laboratory Manual is a small set of problems that are representative of the types of problems that students

Mathematics 111 (Calculus II)Laboratory Manual

Department of Mathematics & StatisticsUniversity of Regina

2nd edition

prepared by Patrick Maidorn, Fotini Labropulu, and Robert Petry

University of Regina Department of Mathematics and Statistics

Contents

Module 1. Inverse Functions 31.1 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Calculus of Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . 41.4 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 L’Hopital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Module 2. Techniques of Integration 72.1 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Trigonometric Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Trigonometric Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Challenge Integration Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Module 3. Integration Applications 113.1 Review - Areas Between Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Volumes by Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Volumes by Cylindrical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 Arclength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Module 4. Sequences and Series 134.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 The Integral Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4 The Comparison Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.5 The Alternating Series Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.6 Absolute Convergence and the Ratio and Root Tests . . . . . . . . . . . . . . . . . . 164.7 Strategies for Testing Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.8 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.9 Representations of Functions as Power Series . . . . . . . . . . . . . . . . . . . . . . 174.10 Taylor and Maclaurin Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Module 5. Parameteric Equations and Polar Coordinates 195.1 Curves Defined by Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Calculus with Parametric Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.3 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.4 Areas and Lengths in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 20

Answers 22References 31

i

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 andlistening 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 problemsthat students of Mathematics 111 (Calculus II) at the University of Regina are expected to be ableto solve on quizzes, midterm exams, and final exams. In the weekly lab of your section of Math111 you will work on selected problems from this manual under the guidance of the laboratoryinstructor, 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, youwill 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 shortreviews of the topics under study. Thus, you should refer to the relevant sections of your textbookand your class notes when using the Lab Manual. These problems are not sufficient practice tomaster calculus, and so you should solidify your understanding of the material by working throughproblems 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 relevantsections 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

Answers:Page 22

1. Determine whether each of the following functions is invertible on its domain.

(a) f(x) = 1 + x4

(b) f(x) = sinx+ cosx

(c) f(x) =x

x− 1

(d) f(x) = cosx on x ∈[0,

3π

2

](e) f(x) =

{x if x ≤ 0x2 if x > 0

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− x2 + x

3. In each case, check whether f(x) is a one to one differentiable function, and if it is, find(f−1

)′(a).

(a) f(x) = x5 + 3x3 + 4, a = 8

(b) f(x) = 1− sinx, a =1

2

(c) f(x) = x3 − x, a = 2

1.2 Exponential and Logarithmic Functions

Answers:Page 22

1. Write each expression as a single exponential.

(a)(23+x

)· 3x

23 (b)e

5x3

3√ex

2. Write as a single logarithm.

(a) 4 ln(x)− 1

2ln(3x) + 1 (b) 4 log(x+ 1)− 2 log(x) + 2

3

4 MODULE 1. INVERSE FUNCTIONS

3. Solve each equation for x.

(a) 3x−1 = 81

(b)

(1

2

)x+1

=

(1

4

)x+2

(c) 4e3x = 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) = N0ekt.

(b) In 1998, Earth’s population was 6 billion. Use the model in (a) to predict the populationin 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 lose80% of its original C-14 content?

1.3 Calculus of Exponential and Logarithmic Functions

Answers:Page 22

1. Find the indicated derivatives.

(a) f(x) = ex2

+ 3e4x, f ′(x)

(b) f(x) = esinx, f ′′(x)

(c) f(x) =e√x

x, f ′(x)

(d) f(x) = ln(1 + x2 + x), f ′(0)

(e) f(x) = ln(cosx), f ′′(x)

(f) f(x) = ln

(x2

x2 + 1

), f ′(x)

2. Integrate the following:

(a)

∫e4x +

4

xdx

(b)

∫ 1

0

e2x − 1

exdx

(c)

∫dx

x− 1

(d)

∫x2ex

3+3 dx

(e)

∫3x

x2 + 1dx

(f)

∫ 4

0

e√x

√xdx

3. Use logarithmic differentiation to find f ′(x).

(a) f(x) = xx

(b) f(x) = xsinx(c) f(x) =

(2x+ 3)12 (x− 7)5

(x+ 1)16

4. Find the equation of the tangent line to f(x) = e−x2

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 5

1.4 Inverse Trigonometric Functions

Answers:Page 23

1. Find the exact value of each expression.

(a) sin−1

(√3

2

)

(b) cos−1

(−√

3

2

)(c) sin

(sin−1

(1

2

))(d) cos−1(cos 3π)

2. Differentiate the following:

(a) f(x) = 3 cos−1 x+ 5 tan−1 x

(b) f(x) = tan−1√

3x

(c) f(x) = sin−1√

1− x4

(d) f(x) = cos(sin−1 x

)3. Find the equation of the tangent line to the graph of f(x) = sec−1 2x at the point (x, y) =

(1,π

3

).

4.

x

150 ft

θ

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.

(b) Finddθ

dxwhen the shadow’s length is 200 ft.


(a)

∫ √3√33

1

1 + x2dx

(b)

∫x√

1− x4dx

(c)

∫tan−1 x

x2 + 1dx

6 MODULE 1. INVERSE FUNCTIONS

1.5 L’Hopital’s Rule

Answers:Page 23

1-10: In each case, find the indicated limit. Note: not all limits allow the use of L’Hopital’s Rule.

1. limx→0

sinx2

x

2. limx→∞

x2 + x+ 1

4x2 + 3

3. limx→0

ln(1 + x)

x

4. limx→1

x− 1

x2 + 1

5. limx→∞

x5 + 4x

ex

6. limx→0

x cscx

7. limx→0+

x2x

8. limx→0

(x+ cosx)1x

9. limx→0+

(1

sinx− 1

x

)

10. limx→∞

(1

x− 1

1− ex

)

Module 2

Techniques of Integration

2.1 Integration by Parts

Answers:Page 23

1-10: Integrate the following:

1.

∫x sin 2x dx

2.

∫2xe−3x dx

3.

∫3x2 cosx dx

4.

∫ e

1x2 ln(x) dx

5.

∫ 2π

0(3x+ 5) cos

(x4

)dx

6.

∫(ln(x))2 dx

7.

∫x√x+ 1 dx

8.

∫cos(2x)e3x dx

9.

∫ax2ebx dx

10.

∫x5 sinx dx

2.2 Trigonometric Integrals

Answers:Page 24


1.

∫sin2 x cos5 x dx

2.

∫sin5 x cos2 x dx

3.

∫ π

0sin2 x cos2 x dx

4.

∫cos4 x dx

5.

∫tan3 x sec4 x dx

6.

∫ √tanx sec4 x dx

7.

∫tan(2t) sec3(2t) dt

8.

∫ π2

0cos5 x dx

9.

∫tan5 x dx

7

8 MODULE 2. TECHNIQUES OF INTEGRATION

2.3 Trigonometric Substitution

Answers:Page 24


1.

∫x3√

1− x2 dx

2.

∫x2√

4− x2 dx

3.

∫ 25

15

√25x2 − 1

xdx

4.

∫dx

(4 + x2)2

5.

∫x3√

2− x2

6.

∫ √2

8

0

dx

(16− x2)32

7.

∫dx

2x2 − 12x+ 26

8.

∫ 4

1

√x2 + 4x− 5

x+ 2dx

9.

∫ (x2 − 6x+ 13

)− 12 dx

2.4 Partial Fractions

Answers:Page 24

1. In each case, find the partial fraction decomposition.

(a)1

(x− 1)(x− 2)

(b)2x+ 3

x2 − x− 6

(c)5x2 − 3x+ 2

x3 − 2x2

(d)7x2 − 13x+ 13

(x− 2)(x2 − 2x+ 3)


(a)

∫dx

x2 + 3x+ 2

(b)

∫2x+ 1

(x+ 1) (x2 + 1)dx

(c)

∫3x2 + 7x− 2

x3 − x2 − 2xdx

(d)

∫2x+ 3

9x2 + 6x+ 5dx

(e)

∫dx

x3 + 1

(f)

∫x3 + 10x2 + 3x+ 36

(x− 1) (x2 + 4)2dx

(g)

∫5x3 − 4x2 + 2x+ 1

5x2 − 4x− 1dx

2.5. CHALLENGE INTEGRATION PRACTICE 9

2.5 Challenge Integration Practice

Answers:Page 25


1.

∫ex√

1− e2xdx

2.

∫x5√x3 + 1 dx

3.

∫sec3 x dx

4.

∫ 1

0

√x2 + 1 dx

5.

∫e4x√

1 + e2x dx

2.6 Improper Integrals

Answers:Page 25

1-10: In each case, determine whether this integral converges or diverges. If it converges, evaluatethe integral.

1.

∫ ∞1

x−3 dx

2.

∫ ∞0

e−3x dx

3.

∫ ∞2

dx

x− 1

4.

∫ 3

−3

dx√9− x2

5.

∫ 2

0f(x) dx, where

f(x) =

{x−12 0 < x ≤ 1

x− 1 1 < x ≤ 2

6.

∫ π

0tan

(x3

)dx

7.

∫ ∞−∞

xe−x2dx

8.

∫ ∞−∞

dx

x2 + 1

9.

∫ ∞−∞

dx

ex + e−x

10.

∫ 1

−1ln |x| dx

10

Module 3

Integration Applications

3.1 Review - Areas Between Curves

Answers:Page 25

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 = sinx and y = cosx from x = 0 to x = 2π.

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

Answers:Page 25

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 thesolid. 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 isgenerated 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 − y2.(b) Find the volume obtained by rotating the region enclosed by x = 0 and x = 2y − y2

about the y-axis.

6. (a) R is bounded by y = sinx, y = 0, x = 0, and x = π. Rotate R about the x-axis. Findthe volume of the resulting solid.

(b) R is bounded by y = sinx, y = cosx, x = 0, and x =π

4. Rotate R about the x-axis.

Find the volume of the resulting solid.

11

12 MODULE 3. INTEGRATION APPLICATIONS

7. A circular man-made lake has a 200m diameter and a maximum depth of 10m. Its cross

section is the parabola y = 10

[( x

100

)2− 1

]. Find the capacity of the lake.

3.3 Volumes by Cylindrical Shells

Answers:Page 26

1. Consider the bowl obtained by revolving the region bounded by y = x2, y = 1, and x = 0about 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 =√x. Use cylindrical shells to find the volume

of the resulting solid if

(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.

5. Find the volume of the solid obtained by rotating the region bounded by y = (x − 1)12 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 centerof 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

Answers:Page 26

1. Find the length of the curve f(x) = x32 between x = 0 and x = 4.

2. Find the length of the curve f(x) = 2ex +1

8e−x between x = 0 and x = ln 2

3. Find the length of the curve y =

(3x

2

) 23

+ 1 between x = 0 and x =2(3)

32

3. Hint: consider

the curve as a function x(y) instead.

4. Find the length of the curve x =1

6y3 +

1

2ybetween y = 1 and y = 2.

Module 4

Sequences and Series

4.1 Sequences

Answers:Page 26

1-5: Determine whether the given sequence is convergent or divergent.

1. an =n4 + 5n+ 1

n4 + 2

2. an =n6 − 4n2 + 3

n5 − 4n+ 7

3. an = e−n lnn

4. an = ln

(2n

3n+ 5

)

5. an =(3n)!

(3n+ 2)!

6-9: Determine whether the given sequence is increasing or decreasing. Also, determine whetherit is bounded.

6. an =2n

5n+ 3

7. an = n2(4−n

)8. an = ln

2n

n+ 5

9. an =1

3 + lnn

13

14 MODULE 4. SEQUENCES AND SERIES

4.2 Series

Answers:Page 27

1-6: Determine whether the given series is convergent or divergent. If it is convergent, find itssum.

1.

∞∑n=1

2n+ 3

3n+ 4

2.

∞∑n=1

2 + 3n

4n

3.

∞∑n=1

(2

3n+

3

4 (3n+2)

)

4.

∞∑n=1

5

(n+ 2)(n+ 3)

5.

∞∑n=1

ln

(5n

7n+ 4

)

6.

∞∑n=1

[3

n(n+ 1)− 5

n

]

4.3 The Integral Test

Answers:Page 27

1-6: Determine whether the given series is convergent or divergent.

1.∞∑n=1

2

n3

2.∞∑n=1

2n

4n2 + 3

3.∞∑n=1

n3e−2n4

4.∞∑n=1

3 lnn

n4

5.∞∑n=1

1√n (3√n+ 4)

6.∞∑n=1

4

4n2 + 3

4.4. THE COMPARISON TESTS 15

4.4 The Comparison Tests

Answers:Page 27


1.

∞∑n=1

1

n3 + 2n+ 5

2.∞∑n=1

14√

16n3 + 4n2 + 3

3.

∞∑n=1

2n3 − 4n2 + 5

5n6 + 3n4 + 2n+ 1

4.

∞∑n=1

1

n(5n)

5.

∞∑n=1

2 + 3n

5 + 6n

6.

∞∑n=1

(n+ 2)3

2n (1 + 3n+ 8n3)

4.5 The Alternating Series Test

Answers:Page 27


1.∞∑n=1

(−1)n1

n3 − 3n+ 4

2.∞∑n=1

(−1)n 2−n

3.

∞∑n=1

(−1)nlnn

n2

4.∞∑n=1

(−1)nn

4n

5.

∞∑n=1

(−1)ne2n

n3 + 1

16 MODULE 4. SEQUENCES AND SERIES

4.6 Absolute Convergence and the Ratio and Root Tests

Answers:Page 27


1.

∞∑n=1

4n

n(5n+1)

2.∞∑n=1

(−1)nne−n

3.∞∑n=1

nn

5n

4.

∞∑n=1

n!

(2n)!

5.

∞∑n=1

(3n)n

(2n+ 5)n

4.7 Strategies for Testing Series

Answers:Page 27


1.∞∑n=1

n3

3n

2.∞∑n=1

5

n(6n) + 3

3.∞∑n=1

ln(2n)

e2n

4.∞∑n=1

(n+ 1)!

n! 2n

5.∞∑n=1

(2n− 5)2

5n (6n2 + 3n+ 2)

4.8. POWER SERIES 17

4.8 Power Series

Answers:Page 28

1-5: Find the radius of convergence and the interval of convergence.

1.∞∑n=1

xn

n+ 3

2.∞∑n=1

4n

n3xn

3.∞∑n=1

(−1)n(x+ 2)n

n2

4.∞∑n=1

(3x− 5)n

52n

5.

∞∑n=1

1√2n+ 7

(4x− 5)n

4.9 Representations of Functions as Power Series

Answers:Page 28

1-5: Find a power series representation for f(x) and determine the interval of convergence.

1. f(x) = x2+x2

2. f(x) = 43x+7

3. f(x) = x(2+3x)2

4. f(x) = ln(x+ 3)

5. f(x) = 33+x2

4.10 Taylor and Maclaurin Series

Answers:Page 28

1-3: Find the Maclaurin series for f(x) and state the radius of convergence.

1. f(x) = xe−3x

2. f(x) = x2 sinx

3. f(x) = x cos 4x

4-5: Find the Taylor series for f(x) at the indicated number a and state the radius of convergence.

4. f(x) = sinx; a = π4

5. f(x) = e2x; a = 4

18

Module 5

Parameteric Equations and PolarCoordinates

5.1 Curves Defined by Parametric Equations

Answers:Page 29

1-4: In the following problems (a) Sketch the graph of the curve having the indicated parametricequations, and (b) Eliminate the parameter to find a Cartesian equation of the curve.

1. x = t2 + 2, y = t2 − 3; −2 ≤ t ≤ 1

2. x = 4t2 − 4, y = 2t+ 5; t ∈ R

3. x = 2 cos t, y = 2 sin t; 0 ≤ t ≤ 2π

4. x = 4 cos t, y = 3 sin t; 0 ≤ t ≤ 2π

5.2 Calculus with Parametric Curves

Answers:Page 29

1-3: Find the equation of the tangent line to the curve at the point corresponding to the givenvalue of the parameter.

1. x = 4t2 − 3, y = 3t+ 3; t = 1

2. x = et, y = e−3t; t = 0

3. x = 2 sin t, y = 4 cos t; t = π/4

4-5: Find the points on the curve at which the tangent line horizontal or vertical.

4. x = 4t2, y = t3 − 27t

5. x = et, y = t+ e−2t

19

20 MODULE 5. PARAMETERIC EQUATIONS AND POLAR COORDINATES

5.3 Polar Coordinates

Answers:Page 29

1-4: Find a Cartesian equation for the given curve and identify it.

1. r = 4

2. r = 3 sin θ

3. r2(5 sin2 θ − 4 cos2 θ

)= 2θ

4. r2 cos 2θ = 4

5-7: Find a polar equation that has the same graph as the given Cartesian equation.

5. x2 + y2 = 25

6. x2 − y2 = 9

7. 4x2 + 25y2 = 49

8-10: Find the slope of the tangent line to the given curve at the indicated value of θ.

8. r = 4 cos θ; θ = π/6

9. r = 1 + 2 cos θ; θ = π/3

10. r2 = 2 cos 4θ; θ = π/4

5.4 Areas and Lengths in Polar Coordinates

Answers:Page 29

1-2: Sketch the graph of the equation and find the area of the region bounded by the graph.

1. r = 4 sin θ

2. r = 1 + 2 cos θ

3-4: Find the area of the region bounded by one loop of the graph of the give equation.

3. r = 6 cos 4θ

4. r2 = 2 cos 2θ

5-6: Find the area of the region that is outside the graph of the first equation and inside the graphof the second equation.

5. r = 9, r = 6 + 6 cos θ

6. r = 6 + 6 cos θ

5.4. AREAS AND LENGTHS IN POLAR COORDINATES 21

7-8: Find the length of the given polar curve.

7. r = e4θ, 0 ≤ θ ≤ 2

8. r = cos2(θ2

), 0 ≤ θ ≤ π

Answers

1.1 Exercises (page 3)

1. (a) No (b) No (c) Yes (d) No (e) Yes

2. (a) f−1(x) = (x+ 1)13 (b) f−1(x) = (x+ 4)

12 + 2 (c) f−1(x) =

1− 2x

x+ 1

3. (a)(f−1

)′(8) =

1

14(b)

(f−1

)′(1

2

)= − 2√

3(c) Not 1-1


1. (a) 6x (b) e4x3

2. (a) ln

(ex4

(3x)12

)(b) log

(100(x+ 1)4

x2

)

3. (a) x = 5 (b) x = −3 (c) x =ln(4)

3(d) x =

1

4(log(2) + 2) ≈ 0.575 (e) x = ee ≈ 15.154

4. (a) k =ln(2)

53(b) 8 Billion (c) In 2037

5. (a) 13300 years


1. (a) f ′(x) = 2xex2

+ 12e4x (b) f ′′(x) =(cos2 x− sinx

)esinx (c) f ′(x) =

12x

12 ex

12 − ex

12

x2

(d) f ′(0) = 1 (e) f ′′(x) = sec2 x (f) f ′(x) =2

x− 2x

x2 + 1

2. (a)1

4e4x + 4 ln |x| + c (b)

(e− 1)2

e(c) ln |x − 1| + c (d)

1

3ex

3+3 + c (e)3

2ln∣∣x2 + 1

∣∣ + c

(f) 2e2 − 2

3. (a) f ′(x) = xx(1 + ln(x)) (b) f ′(x) = xsinx(

cos(x) ln(x) +sinx

x

)(c) f ′(x) =

(x− 7)5(2x+ 3)12

(x+ 1)16

(5

x− 7+

1

2x+ 3− 1

6(x+ 1)

)

4. y = −2

ex+

3

e

22

ANSWERS 23

5. 2e+2

e− 4


1. (a)π

3(b)

5π

6(c)

1

2(d) π

2. (a) f ′(x) =−3√1− x2

+5

1 + x2(b) f ′(x) =

3

2√

3x(1 + 3x)(c) f ′(x) =

−2x√1− x4

(d) f ′(x) =−x√1− x2

3. y =x√3

+π

3− π√

3

9

4. (a) θ = tan−1(

150

x

)(b) −0.0024

Rad

ft

5. (a)π

6(b)

1

2sin−1 x2 + c (c)

1

2

(tan−1 x

)2+ c


1. 0

2.1

4

3. 1

4. 0

5. 0

6. 1

7. 1

8. e

9. 0

10. 0


1. −1

2x cos 2x+

1

4sin 2x+ c

2. −2

3xe−3x − 2

9e−3x + c

3. 3x2 sinx+ 6x cosx− 6 sinx+ c

4.1

3x3 ln(x)− 1

9x3]e1

=2

9e3 +

1

9

5. (12x+ 20) sin(x

4

)+ 48 cos

(x4

)]2π0

= 24π − 28

6. x (ln(x))2 − 2x ln(x) + 2x+ c

7.2

3x(x+ 1)

32 − 4

15(x+ 1)

52 + c

8.3

13cos(2x)e3x +

2

13sin(2x)e3x + c

9.a

bx2ebx − 2a

b2xebx +

2a

b3ebx + c

10. −x5 cosx+ 5x4 sinx+ 20x3 cosx− 60x2 sinx− 120x cosx+ 120 sinx+ c

24 ANSWERS


1.1

3sin3 x− 2

5sin5 x+

1

7x+ c

2. −1

3cos3 x+

2

5cos5 x− 1

7cos7 x+ c

3.1

8x− 1

32sin 4x

]π0

=π

8

4.3

8x+

sin 2x

4+

sin 4x

32+ c

5.1

6tan6 x+

1

4tan4 x+ c

6.2

3tan

32 x+

2

7tan

72 x+ c

7.1

6sec3 2t+ c

8. sinx− 2

3sin3 x+

1

5sin5 x

]π2

0

=8

15

9.1

4tan4 x− 1

2tan2 x+ ln | secx|+ c


1.1

5

(1− x2

) 52 − 1

3

(1− x2

) 32 + c

2. −4

3

(4− x2

) 32 +

4

5

(4− x2

) 52 + c

3. tan(θ)− θ]π

3

0=√

3− π

3

4.1

16tan−1

(x2

)+

1

8· x

4 + x2+ c

5. −2√

2− x2 +1

3

(2− x2

) 32 + c

6.1

16tan θ

]π4

0

=1

16

7.1

4tan−1

(x− 3

2

)+ c

8. 3(tan(θ)− θ)]π

3

0= 3√

3− π

9. ln

∣∣∣∣12 (x2 − 6x+ 13) 1

2 +1

2(x− 3)

∣∣∣∣+ c


1. (a)1

x− 2− 1

x− 1(b)

15

x+ 2+

95

x− 3(c)

1

x− 1

x2+

4

x− 2(d)

5

x− 2+

2x+ 1

x2 − 2x+ 3

2. (a) ln |x+ 1| − ln |x+ 2|+ c

(b) −1

2ln |x+ 1|+ 1

4ln∣∣x2 + 1

∣∣+3

2tan−1 x+ c

(c) ln |x| − 2 ln |x+ 1|+ 4 ln |x− 2|+ c

(d)1

9ln(9x2 + 6x+ 5

)+

7

18tan−1

(3x+ 1

2

)+ c

(e)1

3ln |x+ 1| − 1

6ln∣∣x2 − x+ 1

∣∣+1√3

tan−1(

2x− 1√3

)+ c

(f) 2 ln |x− 1| − ln∣∣x2 + 4

∣∣− 1

2tan−1

(x2

)− 1

2· 1

x2 + 4+ c

(g)1

2x2 − 1

15ln |5x+ 1|+ 2

3ln |x− 1|+ c

ANSWERS 25


1. sin−1 ex + c By substitution and trigonometric substitution

2.2

9x3(x3 + 1

) 32 − 4

45

(x3 + 1

) 52 + c By substitution and parts

3.1

2

(tan(x) sec(x)− ln

∣∣∣cos(x

2

)− sin

(x2

)∣∣∣+ ln∣∣∣cos

(x2

)+ sin

(x2

)∣∣∣)4.

√2 + ln

(√2 + 1

)2

By parts and trigonometric substitution

5.1

5

(1 + e2x

) 52 − 1

3

(1 + e2x

) 32 + c By substitution and trigonometric substitution


1. Convergent1

2

2. Convergent1

3

3. Divergent

4. Convergent π

5. Convergent5

2

6. Convergent 3 ln(2)

7. Convergent 0

8. Convergent π

9. Convergentπ

2

10. Convergent −2


1.125

62. 4√

2

3.11

3


1.56π

15

2.2π

35

3.1

3πr2h

4. (a)5π

14(b)

2π

5(c)

37π

30

26 ANSWERS

5. (a)y

x1

1

2

(b)16π

15

6. (a)1

2π2 (b)

1

2π

7. 50 000π m3


1. (a) V =π

2(b) V =

π

2(c) The same.

2. (a)5π

14(b) V =

2π

5(c) V =

37π

30

3.243π

10

4.272π

15

5.29π

30

6.4π

3

(R2 − r2

) 32

7. 4π

8. (a) 16π (b)128π

3


1. Length ≈ 9.1 units

2. Length =33

16units

3. Length =14

3units

4. Length =17

12units


1. convergent

2. divergent

3. convergent

4. convergent

5. convergent

6. increasing, bounded

7. decreasing, bounded

8. increasing,bounded

9. decreasing,bounded

ANSWERS 27


1. divergent

2. convergent, sum=113

3. convergent, sum= 2524

4. convergent, sum=53

5. divergent

6. divergent


1. convergent

2. divergent

3. convergent

4. convergent

5. divergent

6. convergent


1. convergent

2. divergent

3. convergent

4. convergent

5. convergent

6. convergent


1. convergent

2. convergent

3. convergent

4. convergent

5. divergent


1. convergent

2. convergent

3. divergent

4. convergent

5. divergent


1. convergent

2. convergent

3. convergent

4. convergent

5. convergent

28 ANSWERS


1. R = 1, −1 ≤ x < 1

2. R = 14 , −1

4 ≤ x ≤14

3. R = 1, −3 ≤ x ≤ −1

4. R = 253 , −20

3 < x < 10

5. R = 14 , 1 ≤ x < 3

2


1.

∞∑n=0

(−1)nx2n+1

2n+1, |x| <

√2

2. 4

∞∑n=0

(−1)n3n

7n+1xn, |x| < 7

3

3.

∞∑n=0

(−1)n3n+1

2n+1nxn, |x| < 2

3

4.

∞∑n=0

(−1)n1

3n+1

xn+1

n+ 1+ ln 3, |x| < 3

5.

∞∑n=0

(−1)nx2n

3n, |x| <

√3


1.∞∑n=0

(−1)n3n

n!xn+1, R =∞

2.∞∑n=0

(−1)nx2n+3

(2n+ 1)!, R =∞

3.∞∑n=0

(−1)n42n

(2n)!x2n+1, R =∞

4.√22

[1 +

(x− π

4

)− 1

2!

(x− π

4

)2 − 13!

(x− π

4

)3+ 1

4!

(x− π

4

)4+ · · ·

], R =∞

5. e8[1 + 2 (x− 4) + 4

2! (x− 4)2 + 83! (x− 4)3 + · · ·

], R =∞

ANSWERS 29


1. line segment: y = x− 5, 3 ≤ x ≤ 6

2. parabola: x = (y − 5)2 − 4

3. circle: x2 + y2 = 4

4. ellipse: x2

16 + y2

9 = 1


1. y = 14x+ 23

4

2. y = −3x+ 4

3. y = −2x+ 4√

2

4. Horizontal if t = ±3, Vertical if t = 0

5. Horizontal if t = ln 22 , No Vertical


1. circle: x2 + y2 = 16

2. circle: x2 +(y − 3

2

)2= 9

4

3. hyperbola: y2

4 −x2

5 = 1

4. hyperbola: y2 − x2 = 4

5. r = 5

6. r2 cos(2θ) = 9

7. r2(4 + 21 sin2 θ

)= 49

8. dydx = −1

9. dydx = 1

3√3

10. dydx = −1


1. A = 8π

2. A = 3π

3. A = 9π8

4. A = 1

5. A = 81√3

2 − 9π

6. A = 81√3

2 + 18π

7. L =√174

(e8 − 1

)8. L = 2

30

Appendix A

References

The problems for this manual were collected from a variety of sources, including instructors’spersonal 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 CanadianEdition, Nelson.

Zill, Wright: Differential Equations with Boundary-Value Problems, 8th Edition, Brooks/Cole.

31

Documents

Mathematics 111 (Calculus II) Laboratory Manual · 2020-04-20 · This Laboratory Manual is a small set of problems that are representative of the types of problems that students