The University of Melbourne
Department of Mathematics and Statistics
MAST10005
Calculus 1
Semester 1, 2014
Student name:
Email address:
Cover photos: Sir Isaac Newton (left) and Gottfried Leibniz, who each independently
discovered calculus and share the credit for its development. Newton (1642-1726) was an
English mathematician, physicist, astronomer, natural philosopher and alchemist, and is
regarded by many as the greatest scientist in the history of mankind. Leibniz (1646-1716)
was a German mathematician, philosopher, physicist, natural scientist and engineer, who
also wrote in politics, law, ethics, history and theology.
This compilation has been made in accordance with the provisions of Part VB of the
copyright act for the teaching purposes of the University.
For use of students of the University of Melbourne enrolled in the subject MAST10005
Calculus 1.
Contents
Subject Information . . . . . . . . . .
Useful Formulae . . . . . . . . . . . .
Problem Sheets . . . . . . . . . . . . .
Topic 1: Trigonometric Functions
Topic 2: Vectors . . . . . . . . . .
Topic 3: Complex Numbers . . .
Topic 4: Differential Calculus . .
Topic 5: Integral Calculus . . . .
Topic 6: Differential Equations .
Answers to Problem Sheets . . . . . .
Topic 1: Trigonometric Functions
Topic 2: Vectors . . . . . . . . . .
Topic 3: Complex Numbers . . .
Topic 4: Differential Calculus . .
Topic 5: Integral Calculus . . . .
Topic 6: Differential Equations .
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ii
vii
1
1
5
12
17
21
26
30
30
33
39
44
48
51
ii
MAST10005 Calculus 1
Semester 1, 2014
Subject Organisation
MAST10005 Calculus 1 is a mathematics subject designed for students who have not
completed a subject equivalent to VCE Specialist Mathematics 3/4. After successfully
completing MAST10005 Calculus 1, students may continue their study of mathematics by
proceeding to MAST10006 Calculus 2 and/or MAST10007 Linear Algebra. They may also
choose to study statistics by choosing MAST10010 Data Analysis 1.
Syllabus
Calculus 1 builds on knowledge you will have acquired in previous mathematics subjects,
such as VCE Mathematical Methods 3/4. It also introduces new topics.
One of the main topics is the study of trigonometric functions. It is therefore essential
that you brush up on your trigonometry skills! This subject will extend your knowledge
of trigonometric functions to reciprocal trigonometric functions and inverse trigonometric
functions. You will also learn how to manipulate simple trigonometric identities, as well as
compound and double angle formulae.
We will introduce you to the ideas of vectors. You will learn vector arithmetic and algebra,
study linear independence, scalar products and explore some geometric applications.
You will learn about a new number system: the complex numbers. As you will see, this set
is so large that it contains the real numbers as a proper subset. You will learn arithmetic of
complex numbers and how to represent them in cartesian and polar form. We can then use
them to find roots of polynomials that are not solvable over the reals (e.g., x2 + 1).
You will learn how to extend differentiation techniques to implicit differentiation, derivatives
of inverse trigonometric functions, and second and higher order derivatives. We will apply
these techniques to curve sketching and some simple modelling and optimisation problems.
You will also learn to integrate using algebraic and trigonometric substitutions. We will
apply integration techniques to calculate volumes of solids of revolution and to solve simple
ordinary differential equations.
iii
Classes
The subject MAST10005 Calculus 1 has
• three one-hour lectures per week;
• a one-hour tutorial each week.
Tutorials and lectures start in the first week of semester. Details of your lecture stream
and tutorial are given on your personal timetable. Please visit your student centre or the
Mathematics and Statistics Learning Centre if you are having difficulty accessing your
timetable.
Lecturers will be available for individual consultation, starting in week 1. Consultation
hours will be announced in lectures and put on a notice board in the Mathematics and
Statistics Learning Centre and on the LMS.
Lectures
There are three lecture streams:
• Stream 1: Monday, Wednesday and Friday at 9 am
Location: JH Michell Theatre, Richard Berry building
Lecturer: Dr Heather Lonsdale
Office: Room G43, Richard Berry building
Email: h.lonsdale@unimelb.edu.au
• Stream 2: Monday, Wednesday and Friday at 11 am
Location: JH Michell Theatre, Richard Berry building
Lecturer: Dr Alex Ghitza (Subject coordinator)
Office: Room 166, Richard Berry building
Email: aghitza@unimelb.edu.au
• Stream 3: Monday, Wednesday and Friday at 4:15 pm
Location, Monday and Friday: JH Michell Theatre, Richard Berry building
Wednesday: Theatre A, Elisabeth Murdoch building
Lecturer: Dr Nick Beaton
Office: MASCOS, 139 Barry St
Email: nrbeaton@unimelb.edu.au
Note that consultation hours may be held in a room other than the lecturer’s office; please
check the LMS for details.
iv
Reading and Resources
Textbook
The textbook for MAST10005 is:
• Hass, Weir and Thomas - University Calculus Early Transcendentals, 2nd edition,
packaged with a differential equations supplement from Hass, Weir and Thomas,
Calculus, Pearson 2012.
It is important that you obtain a copy (either print or electronic) of this book as it allows
you to access online resources that will be important for this course. This textbook can
also be used for MAST10006 Calculus 2.
Lecture Notes
Lecture notes will be available for sale in the bookshop. These contain an outline of the
theory of each topic, with space left for examples to be completed in lectures. These worked
examples are only available in the lectures.
Problem Sheets
There are six problem sheets in this booklet, corresponding to the six major topics covered
in lectures, accompanied by answers. You should be working through these problems in
your own time, to consolidate your understanding of the subject.
Website
Subject material such as the assignments, the consultation roster and other announcements
will be available on the subject webpage, from the address:
www.ms.unimelb.edu.au/~aghitza/teaching/2014calc
Note:
• The website can also be reached via the Learning Management System (LMS) at
www.lms.unimelb.edu.au
• If you have trouble viewing or printing any of the PDF documents in your browser,
you can get around this by ‘right clicking’ on the links and selecting ‘save link as’ or
‘download link as file’ (terminology depends on the web browser you use!). Then open
the downloaded PDF in Acrobat Reader or whatever program you prefer.
v
Assessment
The assessment is composed of two parts:
• a three-hour exam at the end of the semester;
• ten weekly assignments due at 2 pm on Mondays commencing 17 March 2014.
Each piece of assessment is compulsory.
Your final mark in MAST10005 is computed as follows:
Final Mark = 80% Exam + 20% Assignment
Assignments
• Assignments are due weekly at 2 pm on Mondays commencing 17 March 2014.
• The weekly assignment questions will be distributed during lectures, a week prior to
their due date, and posted on the subject webpage.
• Make sure that you write your name, student number, tutor’s name and tutorial time
on each assignment.
• Your assignment should be placed into the appropriate assignment box, ground floor,
Richard Berry building (just down the corridor from the vending machine). Your
tutor may be taking other subjects, so please ensure that you place the assignment in
your tutor’s box for MAST10005 Calculus 1.
• Late assignments will not be accepted.
• Assignment 1 must be submitted with a completed plagiarism cover sheet, which will
cover all assignments in MAST10005 during semester 1. The assignments will not be
marked until a completed plagiarism cover sheet has been submitted.
• Any medical certificates relating to the assignments should be given to Dr Alex Ghitza.
Please bring them to the General Office on the ground floor of the Berry building and
ask for them to be put in Dr Ghitza’s pigeonhole.
Calculators and Formula Sheets
Students are not permitted to use calculators and formula sheets in the end of semester
exam. Assessment in this subject concentrates on the testing of concepts and the ability to
conduct procedures in simple cases. There is no formal requirement to possess a calculator
for this subject. Nonetheless, there are some questions on the example sheets for which
calculator usage is appropriate. If you have a calculator, then you may find it useful
occasionally.
vi
How to Use This Book
Working through problems in this booklet is the best way to ensure you have understood the
lecture material. After each lecture topic, the relevant problems from this problem booklet
will be listed in the lecture notes. You should attempt these problems before attending
your next tutorial.
Getting help!
The first source of help is the person beside you in lectures and tutorials, who is doing the
same problems as you are and having similar but perhaps not exactly the same difficulties.
Remember though, that fellow students have no obligation to help you, nor you to help
them. Forming a small study group of two to four people is a preferred mode for “trading
secrets” for many students.
The next source is your tutor: students are expected to attend all tutorials, where you can
gain valuable practice and help with problems.
Lecturers and some tutors have rostered consultation hours to provide help on an individual
basis. Attendance is on a voluntary basis. Details will be announced in lectures and
provided on the subject webpage.
If something major goes wrong, and you need to be absent for more than a couple of days,
inform your lecturer or the subject coordinator.
Assumed Knowledge
The prerequisite for MAST10005 is VCE Mathematical Methods 3/4 or equivalent.
Why do people fail this subject? The answers are quite simple:
1. many people do not do enough work outside class;
2. many people do not attend all their lectures and tutorials; and
3. many people make algebraic mistakes.
The expectation in this subject is that each student spend at least 4 hours per week outside
class working on Calculus 1 material. This may consist of reading over lecture notes, doing
problems from the booklet or textbook, doing the weekly assignment or attempting problems
online.
vii
Useful Formulae
Pythagorean identity
cos2 x + sin2 x = 1
Compound angle formulae
sin(x + y) = sin x cos y + cos x sin y
sin(x − y) = sin x cos y − cos x sin y
cos(x + y) = cos x cos y − sin x sin y
cos(x − y) = cos x cos y + sin x sin y
tan(x + y) =
tan x + tan y
1 − tan x tan y
tan(x − y) =
tan x − tan y
1 + tan x tan y
Half angle formulae
1
sin2 x = (1 − cos 2x)
2
1
cos2 x = (1 + cos 2x)
2
Standard integrals
Z
1
x
dx = arcsin + C
2
a
−x
Z
−1
x
√
dx = arccos + C
2
2
a
Z a −x
1
x
1
dx
=
arctan
+C
a2 + x2
a
a
√
a2
where a > 0 is constant, and C is an arbitrary constant of integration.
Exponential formulae
eiθ = cos θ + i sin θ
1
cos θ = (eiθ + e−iθ
2
1
sin θ = (eiθ − e−iθ
2i
viii
1
Topic 1: Trigonometric Functions
Reciprocal Trigonometric Functions
1. Exact Values. If sin(x) =
(a) cos(x)
1
2
and
π
2
< x < π, find the exact values of:
(b) tan(x)
2. More Exact Values. If cos(x) =
(a) sin(x)
√1
2
(c) cot(x)
and
3π
2
< x < 2π, find the exact values of:
(b) tan(x)
(c) sec(x)
3. Even More Exact Values. If sin(x) =
of:
(a) cos(x)
(b) cot(x)
√1
2
π
2
and
< x < π, find exact the values
(c) sec(x)
(d) cosec(x)
4. Graph Sketching. Sketch the graphs of the following functions over the domain
[0, 2π].
(a) y = cosec x −
π
4
(c) y = cot x −
(b) y = sec (3x + π)
π
3
Trigonometric Identities
5. Simplification. Simplify the following expressions:
(a) sin2 (x) cot2 (x)
(c)
(b)
cos2 (x)
+ sin(x)
sin(x)
tan2 (x)
1 + tan2 (x)
(d) cosec2 (x) − cot2 (x)
6. Trigonometric Values. If sec(x) = −2 and x ∈
values of:
(a) cos(x)
(b) sin(x)
(d) cosec(x)
(e) cot(x)
π
2
, π , use identities to find the exact
(c) tan(x)
2
7. Solve for x. Solve for x over the interval [0, 2π] in each of the following equations:
√
(a) sec(x) = − 2
(b) cosec(x) = 2
(c) cot2 (x) = 3
Compound and Double Angle Formulae
8. Expand. Expand the following expressions:
(b) cos(3x − 2y)
(a) tan(x + 2y)
(c) sin(y − 2x)
9. Simplify. Simplify each of the following expressions:
(a) cos(3x) cos(2x) + sin(3x) sin(2x)
(b) sin(3x) cos(2y) + cos(3x) sin(2y)
(c) tan(y) − tan(x + y)
(d) sin(2x + y) cos(2x − y) − cos(2x + y) sin(2x − y)
10. Expand and Simplify. Simplify the following expressions:
(a) sin
π
2
−x
(d) cosec
3π
2
−y
(b) tan
π
2
−y
(e) sec
3π
2
+x
(c) cot
3π
2
+x
11. Exact Values with Compound Angle Formulae. Use compound angle formulae
to help find the exact values of:
(a) tan
π
12
(b) sin
17π
12
12. More Exact Values. If sin(x) =
find the exact values of:
(a) tan(x)
(b) sec(x)
4
5
(c) cosec
5π
12
for x ∈ 0, π2 and tan(y) =
(c) sin(x + y)
5
12
for y ∈ π, 3π
,
2
(d) tan(x + 2y)
13. Even More Exact Values. If tan(x) = 2 and x ∈ 0, π2 , use trigonometric identities
to find exact values for:
(a) tan(2x)
(b) sin(2x)
(c) cos(2x)
3
Inverse Trigonometric Functions
14. Inverse Trigonometric. Evaluate each of the following in radians:
(a) arcsin
1
2
(b) arccos
√ 3
2
√ (c) arctan − 3
15. Simplification. Simplify the following:
(a) sin arcsin
1
2
(c) arccos sin
5π
4
(e) arctan 2 cos
√1
2
(b) tan arcsin
(d) cos arcsin
1
2
5π
6
(f) arccos cos
11π
6
16. Graphing Inverse Trigonometric Functions. On the same graph sketch
(a) arccos(x)
(b) arccos(x/3)
(c) arccos(x/3 − 1/3)
(d) 2 − arccos(x/3 − 1/3)
Domain and Range
17. Step by Step. For each of part (a) and (b)
(i) Find the implied domain and range of f .
(ii) Find the implied domain and range of g.
(iii) Find dom (f ) ∩ range (g).
(iv) Use your answers to (i), (ii) and (iii) to find the implied
domain and range of f ◦ g.
√
√
(a) f (x) = x; g(x) = 4 − x2 ; (f ◦ g)(x) = 4 − x2 .
(b) f (x) = cot(x); g(x) = arcsin x; (f ◦ g)(x) = cot(arcsin(x)).
18. Standard. State the (i) implied domain and (ii) range, of each of the following:
(a) y =
(c) y =
(e) y =
1
1−x
√
4 − x2
p
2 sin(x)
(b) y = log(1 + x)
1
(d) y = √
4 − x2
(f) y = 2 log (cos(x))
4
19. Inverse Trigonometric. State the (i) implied domain and (ii) range, of each of the
following:
(a) y = arcsin(1 − x)
(b) y = arccos(2x + 3)
(c) y = arctan(4 − x)
(d) y = arccos(sin(2x))
√
(f) y = arccos( 3 tan(x))
(e) y = tan(2 arcsin(x))
20. An incremental example. State the (i) implied domain and (ii) range, of each of
the following:
(a) y = |x|
(b) y = 2|x|
(c) y = 2|x| − 1
(d) y = arcsin(2|x| − 1)
21. Trigonometric. Express the following as algebraic functions of x:
(a) f (x) = cos(arccos(x))
(b) g(x) = sin(arccos(x))
State the implied domain and range of each function.
5
Topic 2: Vectors
Vector Algebra
1. Introduction. Let a = 2i + j and b = −i + j.
On the same set of x-y axes draw the vectors:
(i) a
(ii) b
(iii) −a
(iv) 2b
(v) a + b
(vi) b − 3a
2. Vector Arithmetic. Let u = 3i + j, v = 2i + k, w = −i − 2j + 3k be vectors in R3 .
Find
(a) 2u
(b) −3u
(c) 12 w
(d) u + v
(e) w − v
(f) u − 5v + 2w
−→
−→
3. The Hexagon. ABCDEF is a regular hexagon with centre O. If OA = a and OC = c,
−−→ −−→ −−→ −→
−→
express the vectors OB, F B, DA, AC and EA in terms of a and c.
Position and Length
4. Division by Ratio.
(a) Find the point which divides the line segment connecting the point A with position
vector (1, 2, 3) and the point B with position vector (−1, 4, 2) in the ratio 2 : 3.
−→
−−→
−→
(b) Let OA = (2, −1, 4) and OB = (−3, −2, 1). Find OC where C is the point that divides
the line segment AB in the ration 9 : 7.
(c) Let ABC be a triangle and let D divide the segment BC in the ratio 3 : 5. Express the
−−→
−→
−→
vector AD as a combination of AB and AC.
−→
−−→
(d) Let a = OA and b = OB. Let C be the point that divides the line segment AB into
the ratio s : t. Show that
−→ ta + sb
OC =
.
s+t
5. Position and Length. For each of the following pairs of points find:
(i) the position vector in the direction from the first point to the second point
(ii) the length of this vector.
(a) (0, 2) (4, −5)
(b) (2, 3) (5, 4)
(c) (4, −5) (0, 2)
(d) (5, 4) (2, 3)
(e) (3, 7) (5, 7)
(f) (7, −3) (3, −3)
6
6. Unit Length, Unit Vector. Calculate unit vectors in the direction of each of the
position vectors you have found in the previous question.
7. Let’s Go Backwards. For each pair of vectors above, find the position vector in the
direction from the second point to the first point.
8. More Vector Lengths. Let u = 3i − 4j + k and let v, the vector from the point
(2, 1, −1) to (3, 2, −1), be vectors in R3 . Find
(a) kuk
(b) kvk
(c) û
(d) v̂
(e) 2u − v
(f) k2u − vk
(g) A unit vector parallel and in the same direction as 2u − v
Dots and Angles
9. Dot product. Find u · v for each of the following pairs of vectors:
(a) u = 2i + 3j + k and v = i + 2j − k
(b) u = 3i − 2j − k and v = 2i + j + k
(c) u = i + j + k and v = 3i − j − k
(d) u = i + j and v = 2i − j − k
(e) u = j + k and v = i + j
(f) u = i + j and v = k
10. Angles. Find the angle between each pair of vectors u and v above.
11. Dot and Angle. If a = 3i + j − 2k and b = i + 3j + k, find:
(a) a · b
(b) the angle between a and b
12. Parallel and Perpendicular. If u = 3i + 4j + 5k which of the following are either
parallel or perpendicular to u?
(a) −3i − 4j − 5k
(b) 5i + 4j + 3k
(d) −5i + 3k
(e) −3j + 5k
(c) −3i
13. Perpendicular. Find a constant a in each case below, such that the pairs of vectors
u and v are perpendicular.
(a) u = 3i + aj and v = 2i + 6j
(b) u = ai − 3aj + 2k and v = 4i − 2k + 3j
7
14. Vector Angles. Let a = (2, −1, 6) and b = (1, −1, −1).
(a) Show that a + b is perpendicular to b.
(b) Find the angle between the vectors a + b and a − b.
Problem Solving with Vectors
−→
−−→
15. Triangle. In the triangle OAB, OA = a and OB = b. P is a point on AB such that
−→
−→
the length of AP is twice the length of BP . Q is a point such that OQ = 3OP .
A
Q
a
O
P
b
B
(a) Find each of the following in terms of a and b:
−→
(i) AB
−→
(ii) AP
−→
(iv) OQ
−→
(v) AQ
−→
(iii) OP
−→
−−→
(b) Hence explain why AQ is parallel to OB.
16. Trapezium. A trapezium is a quadrilateral with two parallel sides. Show that the
points A(4, 3, 0), B(5, 2, 3), C(4, −1, 3) and D(2, 1, −3) form a trapezium and state the
ratio of the parallel sides.
−→
−→
−→
17. Another Trapezium. OABC is a trapezium with OC = 2AB. Suppose OA =
−→
(2, −1, −3) and OC = (6, −3, 2).
A
B
O
−→
(a) Find AB.
−−→
−→ −→
−→
(b) Express BC in terms of OC, OA and AB.
−−→
(c) Hence find BC.
C
8
18. Parallelogram. ABCD is a parallelogram. A, B and C are represented by the
position vectors i + 2j − k, 2i + j − 2k and 4i − k respectively.
(a) Draw a sketch of the parallelogram with the vertices labelled with their coordinates.
−−→
(b) Find AD.
(c) Find the angle at vertex A of the parallelogram.
19. Diagonals. OABC is a parallelogram, where O is at the origin and A and C are
defined by a = (2, 2, −1) and c = (2, −6, −3).
(a) Draw a diagram, and explain why the diagonals of the parallelogram are given by the
vectors a + c and a − c.
(b) Find:
(i) ka + ck
(ii) ka − ck
(iii) (a + c) · (a − c)
(c) Hence find an angle between the diagonals of the parallelogram.
20. Right-Angled Triangle. ABC is a right-angled triangle with the right angle at B.
−→
−→
−→
If AC = 2i + 4j and AB is parallel to i + j, find AB.
−→
HINT: What is a simple expression for AB? Use this to first find an expression for
−−→
BC.
21. Triangle and Parallelogram. Let OAB be a triangle such that the midpoint of
−→
OA is P , the midpoint of OB is Q, and the midpoint of AB is R. Let a = OA and
−−→
b = OB.
Use vector methods to show that AP QR is a parallelogram.
22. Flying Direction. If a plane encounters a wind, the velocity of the plane and the
velocity of the wind can be described by vectors p and w respectively as shown below. The
magnitude of the vector p + w gives the ground speed of the plane and the direction of
p + w gives the true direction of the plane.
(a) Assume that a plane is flying due east with a velocity of 700 km/hour and is hit by a
gust of wind blowing exactly north east with a velocity of 100 km/hour. Find the ground
speed and the true course of this plane.
N
w
p
E
√
(b) Assume that the wind is blowing exactly north east with a steady velocity of 30 2
km/hour. What direction and velocity should the pilot choose if he wishes to fly with a
true course of due east and a ground speed of 700 km/hour?
9
Scalar and Vector Projections
23. Simple Projections. Let a = (−3, 4), b = (0, 2) and c = (3, −1).
(a) Find the scalar projection of b on a.
(b) Find the scalar projection of c on a.
(c) Find proja b, the vector projection of b in the direction parallel to a.
(d) Find proja c, the vector projection of c in the direction parallel to a.
(e) On the same set of x-y axes draw the vectors a, b, c, proja b and proja c.
24. Scalar Projection. For each of the following pairs of vectors, find (i) the scalar
projection of u onto v, (ii) the scalar projection of v onto u
(a) u = 3i + 2j + k and v = 4i + j
(b) u = i − 2j + 3k and v = 3i − 2j + k
(c) u = 2i − 3k and v = 2i + j − k
(d) u = 5i + j − 3k and v = 2i + 2k
(e) u = j − 2k and v = i − k
(f) u = i + j + k and v = j
25. Vector Projection. For each of the pairs of vectors above, find (i) the vector
projection of v in the direction parallel to u, (ii) the vector projection of v in the direction
perpendicular to u.
26. Scalar and Vector Projections.
Calculate
Let u = 3i + 4j + 2k and v = i + 2j − k.
(a) the scalar projection of u on v
(b) the scalar projection of v on u
(c) the vector projection of v in the direction parallel to u
(d) the vector projection of u in the direction perpendicular to v.
You should give a sketch describing your answers.
10
27. Gravity. [Optional question.] Assume that a box with mass m = 50 kgs is on a ramp
as shown below. The magnitude of the weight force, w, on the box is mg Newtons/sec in
a direction vertically downwards, where g = 9.8 m/sec2 . In order to stop the box sliding
a force of r is applied in the direction shown where krk = kwx k and wx = the vector
projection of w in the direction parallel to the ramp. Find r and krk.
r
k
wx
wy
w
i
30º
Graphs of Ellipses and Hyperbolae
28. Ellipses. Sketch the graphs of the following functions:
(a)
(x − 3)2 (y − 2)2
+
=1
16
9
(b)
(x + 2)2 (y − 3)2
+
=1
9
25
(c)
(x + 1)2 (y + 2)2
+
=1
4
4
(d)
(x − 2)2 (y + 3)2
+
=1
8
12
29. Hyperbolae. Sketch the graphs of the following functions:
(a)
(x − 2)2 (y + 1)2
−
=1
9
16
(c) x2 −
(b)
(x + 1)2 (y − 1)2
−
=1
25
4
(y − 2)2
=1
4
30. Harder Questions. Sketch the graphs of the following functions:
(a) 9(x + 3)2 + 7y 2 = 63
(b) 4(x − 4)2 + 9(y − 3)2 = 36
(c) 25(x − 3)2 − 9(y + 2)2 = 225
(d) 9x2 − 16(y + 1)2 = 144
11
Parametric Curves
31. Equation of a Path. Let a particle’s position as a function of time be as given below.
Find the equation of the path for time t ≥ 0. Sketch the graph of the path.
(a) u = ti − 2tj
(b) u = t2 i + tj
(c) u = (t − 4)i + 3t2 j
(d) u = t3 i + tj
(e) u = (t + 1)i + (t2 − 2t)j
32. Periodic Motion. For each of the following functions describing a particle’s position
at time t, (i) find the equation of the path, (ii) sketch its graph, (iii) state the period of
motion of the path.
(a) u = cos(2t)i + sin(2t)j
(b) u = 2 cos(2t)i + 2 sin(2t)j
(c) u = (2 + cos(t))i + (−3 + sin(t))j
33. Another Path. Let r(t) = (et − e−t )i + (et + e−t )j = xi + yj where t ∈ R.
(i) find y 2 − x2 .
(ii) Hence find the equation of the path and sketch this.
12
Topic 3: Complex Numbers
Note: In this problem sheet, the principal argument should be used whenever the complex
number is written in polar or exponential form, that is −π < θ ≤ π.
1. Square root of negative numbers. Represent the following square roots as imaginary
numbers.
(a)
√
−4
(b)
√
√
(c) 2 −12
−6
2. Real and Imaginary. For the complex numbers below, find Re(z), Im(z), and Re(z)–
Im(z).
(a) z = 3 − 4i
(d) z =
5
7
(b) z = 6i
(c) z = 2 + 7i
(e) z = 12 i + 2
Complex Arithmetic
3. Complex Arithmetic. Let z1 = 2 + 6i, z2 =
z5 = 5. Simplify:
1
2
+ 2i, z3 = −6i, z4 = −3 + 2i and
(a) z1 + z4
(b) z1 + z3
(c) z1 − 2z2
(d) z4 + z5
(e) z2 + z4
(f) 3z2 − 2z4
(g) z5 + z3
(h) z3 + z2
(i) 2z1 + z2 + 3z3
4. The Argand Plane. Let z1 = 2 + 4i, z2 = −2 + 4i, z3 = 2 − 4i and z4 = −2 − 4i.
(a) Sketch z1 , z2 , z3 and z4 on the same set of axes.
(b) Sketch z1 , 2z1 and −3z1 on the same set of axes. What do you observe?
(c) Sketch z1 + z2 , z1 + z4 and z2 + z3 on the same set of axes.
(d) Is
(i) z1 = −z2 ?
(ii) z1 = −z3 ?
(iii) z1 = −z4 ?
(iv) z2 = −z3 ?
13
5. Powers of i. Express i2 , i3 and i4 in cartesian form. Use your answers to express the
following in cartesian form:
(a) i23 = (i4 )5 × i3
(b) i14
(c) i259
6. Cartesian form. Simplify each of the following products. You should express your
answer in the form a + ib, a, b ∈ R.
(a) i(2 + 3i)
(b) (3 − 2i)2
(c) (−2 − i)(2 + i)
(d) 3(2 + 3i)
(e) (2 + i)(2 − i)
(f) (1 + i)3
(g) (3 − 2i)(1 + i)
(h) (2 + i)(−2 + i)
(i) i2 (3 + 2i)
7. Rationalising the Denominator. Write each of the following expressions
form a + ib, a, b ∈ R. In each case you should first write down z2 .
(a)
1
3 − 2i
(b)
2
2+i
(c)
i
2 + 3i
(d)
5 − 2i
1−i
(e)
4 + 3i
−2 + i
(f)
1−i
−3 − 2i
z1
z2
in the
8. Real and Imaginary. Find the real and imaginary parts:
(a) Im
1 − 5i
4+i
(b) Re
2 + 7i
4 − 6i
9. Harder Examples. Express each of the following complex numbers in the form
a + ib, a, b ∈ R.
(a)
i(1 + 2i)
3(2 − i)
(b)
3(2 − i)
i(1 + 2i)
(c)
(2 + i)2
(1 + 3i)(2 + 3i)
(d)
(3 + 2i)(5 − i)
i2 (2 − i)
10. Conjugates to Cartesian. Express the following numbers in the cartesian form
a + ib with a, b real:
(a) 12 + 7i
(b) (1 + i)2 + (1 + i)
(c) (3 + 2i)(4 + 5i)
(d) (3 − i)(2 − i)(−1 − 3i)
(e)
i2 (1
−
i)3
(f)
2 − 3i
1 + 5i
14
11. Complex conjugates. Let z = x + iy with x and y real, and let z̄ be the complex
conjugate. Show the following:
(a) z z̄ = x2 + y 2
(b) z + z̄ = 2Re(z)
(c) z − z̄ = 2iIm(z)
Sketch the following points in the complex plane:
(f) z and −z
(e) z and −z
(d) z and z̄
12. Complex Trigonometric. Express the following in cartesian form a + ib with a, b
real.
(a) (cos θ + i sin θ)(cos θ − i sin θ)
(c)
(b)
cos θ + i sin θ
cos φ + i sin φ
1
1 + cos α + i sin α
Modulus and Argument
13. Modulus and Argument. For each of the following complex numbers write down
|z| (the modulus of z) and Arg(z) (remember this is an angle in the range (−π, π]). You
may find it useful to sketch the complex numbers on an argand diagram first.
(a) z = 6
(b) z = −3
√
(c) z = − 3 + i
(d) z = 4i
(e) z = −2i
(f) z = −1 −
(g) z = 7i
(h) z = 1 + i
(i) z = 4 − 4i
√
3i
14. Something more complicated.
(a) Find the modulus of each of the following complex numbers without multiplying them
into cartesian form:
(i)
5 + 2i
2 + 5i
(ii)
−27i(8 + 2i)(2 + i)
(4 + i)(4 − 3i)(4 − 8i)
(iii)
(2 − 3i)(−2 − 2i)(−5)
(1 + i)(3 + 2i)
(b) Find an argument θ, −π < θ ≤ π, for the following complex numbers:
(i)
−2
√
1+i 3
(ii)
i
−2 − 2i
(iii) (1 + i)4
15
Regions in the Complex Plane
15. Regions in the Complex Plane. Sketch the following regions in the complex
plane:
(a) Re(z) = 1
(b) Im(z) > 1
(c) |z − 2i| ≤ 1
(d) |z − 3| = |z + 2i|
(e) |z| ≤ 4, 0 ≤ arg(z) ≤ π/4
(f) Re(z 2 ) = 4
Complex Exponential
16. Exponential Polar Form. Express each of the following complex numbers in polar
form reiθ . In each case, chose an angle θ, −π < θ ≤ π:
(a) −1
(b) 5 + 5i
(c) (1 − i)2
(d) 2 + 5i
17. Powers. Simplify the following powers of complex numbers:
√
(b) (2 3 + 2i)5
(a) (1 + i)20
(c)
1+i
√
3+i
12
(d)
1−i
√
3−i
7
Finding Complex Solutions
18. Complete the square. Find all solutions for z in each of the following equations,
completing the square where necessary.
(a) z 2 + 1 = 0
(b) z 2 + 16 = 0
(d) z 2 + 4z + 5 = 0
(e) z 2 + 3z + 3 = 0
(c) z 2 − 4 = 0
19. Quadratic formula. Find all solutions for z in each of the following equations by
using the quadratic formula.
(a) z 2 + 3z + 2 = 0
(b) z 2 + 3z + 4 = 0
(c) 2z 2 − z + 3 = 0
(d) 3z 3 + 2z 2 + z = 0
16
20. Any way you like. Find all solutions for z in each of the following equations. These
can be done by factorising, or by rearranging and finding the nth root.
(a) z 4 − 81 = 0
(b) z 8 − 256 = 0.
21. Roots and Factors. For each of the following find the roots of the given equations
and sketch the roots in the complex plane:
(a) cube roots z 3 = 1
(b) square roots z 2 = i
(c) sixth roots z 6 = −64
(d) fifth roots z 5 = 32e5πi/3
22. Roots of Polynomials. Find the roots of the following polynomials, using the
complex exponential and roots of unity where necessary:
(a) z 4 + 4z 2 + 4 = 0
(b) z 4 + 4z 2 + 16 = 0
(c)(z + i)5 − (z − i)5 = 0
(d) z 4 + z 3 + z 2 + z + 1 = 0
Powers of Sin and Cos
23. Polynomials in Cos and Sin. Using the complex exponential, express the following
as the sum of powers of cos θ and sin θ as indicated:
(a) sin(3θ) in terms of sin θ
(b) cos(4θ) in terms of cos θ and sin θ
(c) sin(6θ) in terms of sin θ and cos θ
24. Sin and Cos of Multiple Angles. Using the complex exponential, express the
following as sums of sines and cosines of multiple angles (cos nθ and sin nθ):
(a) sin3 θ
(b) cos4 θ
(c) sin7 θ
17
Topic 4: Differential Calculus
Second and Higher Order Derivatives
1. First and Second Derivatives. Find the first and second derivatives of the given
functions:
√
(a) f (x) = x4 − 3x3 + 16x
(b) h(x) =
(d) y = (1 − x2 )3/4
(e) f (x) = e2x−sin(5x)
x2 + 1
(c) f (x) =
x
1−x
(f) g(x) = cosec(x)
2. nth Derivatives. For the following functions evaluate the first few derivatives f 0 (x),
f 00 (x), f (3) (x), and so on. Deduce the pattern and write down an expression for the nth
derivative f (n) (x):
(a) f (x) = xn
(b) f (x) =
1
3x3
3. Related Derivatives. Assuming g(x) is twice differentiable, that is, we can write down
g 0 (x) and g 00 (x), write down expressions for f 00 (x) if:
√
(b) f (x) = g( x)
(a) f (x) = xg(x2 )
Implicit Differentiation
4. Implicit Derivatives. Find
dy
dx
by implicit differentiation:
(a) x2 − xy + y 3 = 8
(c) x loge y +
√
y = loge x
(e) cos(x − y) = xex
(b) 4 cos x sin y = 1
(d) x2 y = exy
(f) loge
x
y
= cot(xy)
5. Second Derivative. If x4 + y 4 = 16 , use the following steps to find y 00 :
(a) Use implicit differentiation to find y 0 .
(b) Use the quotient rule to differentiate the expression for y 0 from part (a). Express your
answer in terms of x and y only.
(c) Use the fact that x and y must satisfy the original equation x4 + y 4 = 16 to simplify
x2
your answer to part (b) to y 00 = −48 7 .
y
18
6. More Implicit Derivatives Suppose y is given implicitly as a function of x by the
equation
x2 + axy 2 + by 3 = 1.
Given that this curve passes through the point (3, 2) and that the tangent line that touches
the curve at this point has a gradient of −1, find a and b.
7. Implicit Tangent. Show, by implicit differentiation, that the equation of the tangent
line to the ellipse
x2 y 2
+ 2 =1,
a2
b
at the point (x0 , y0 ) is
x0 x y 0 y
+ 2 =1.
a2
b
Inverse Trigonometric Functions
8. Derivatives. Find the derivative with respect to x of each of the following expressions:
(a) arcsin
x
5
(b) arccos
(d) arccos (4x)
x
4
(c) arctan
(e) arctan (7x)
√x
7
(f) arcsin (2x)
9. More derivatives. Find f 0 (x) if f (x) is equal to:
(a) arccos
7x
4
(d) arccos (4x − 3)
(b) arcsin
8x
5
(e) arcsin
3−4x
5
(c) arctan
7x
2
(f) arctan
5x−3
4
10. Even more derivatives. Find f 0 (x) if f (x) is equal to:
(a) arccos (x3 )
(b) arctan (ex + 5)
√
(c) arcsin ( 3 x)
(d) arccos(2 log(x))
(e) arctan(cos(3x) + 1)
(f) arcsin
2x2 + 1
3
19
Graph sketching
11. Graph sketching. For each of the following functions:
(a) f (x) =
1 + x2
1 − x2
(b) f (x) =
(d) f (x) = e1/x
(e) f (x) =
√
(c) f (x) = x 9 − x2
x3
x2 + 1
x2
1
−1
Determine where possible
• domain of f
• asymptotes of f
• x and y intercepts of the graph y = f (x)
• local extrema of f
• intervals over which f is increasing
• intervals over which f is decreasing
• intervals over which f is concave up
• intervals over which f is concave down
• points of inflection of f
Use these results to sketch the graph of y = f (x).
12. The Dying Wave. Consider the function
y = e−t cos t ,
t≥0.
(a) Find the zeros and stationary points of y
(b) Sketch the graph of y
(c) Show that y is a solution of the differential equation
d2 y
dy
+ 2 + 2y = 0 .
2
dt
dt
Applications of Differentiation
13. Flying High. The height h, in metres, of a kite above the ground t minutes after it is
let off is given by
h(t) = 12t − 2t2 .
(a) Find when the kite reaches its maximum height, and the maximum height reached.
(b) Find the time when the kite falls to the ground.
20
14. Flying Higher. In a second attempt, the kite now flies at height
9
1
h(t) = t3 − t2 + 9t
6
4
for 0 ≤ t ≤ 10.
(a) What is the maximum height reached by the kite in the first 5 minutes?
(b) What is the maximum height reached by the kite in the whole 10 minutes?
15. Cylinder. The radius, r cm, and height, h cm, of a solid circular cylinder vary in such
a way that the volume of the cylinder is always 250π cm3 .
(a) Draw the cylinder and write down expressions for its total surface area, A cm2 , and its
volume, V cm3 , in terms of r and h.
(b) Show that the total surface area can be rewritten as:
A = 2πr2 +
500π
.
r
(c) Find the minimum total surface area of the cylinder.
16. Rectangle. Find the maximum possible area of√a rectangle which has its base on the
x-axis and its upper vertices on the semi-circle y = 9 − x2 , −3 ≤ x ≤ 3.
17. Manufacturing. The sales revenue (in dollars) that a manufacturer receives for selling
x units of a certain product can be approximated by the function
x .
R(x) = 900 log 1 +
300
However, each unit costs the manufacturer one dollar to produce, and the initial cost of
adjusting the machinery for the production is $200, so the total cost of production (in
dollars) of x units is
C(x) = 200 + x .
(a) Write down the profit, P (x) dollars, obtained by the production and sales of x units.
(b) Hence find the number of units which should be produced and sold for maximum profit,
and calculate this maximum profit.
18. Rock n Roll. A travelling rock band is having a trailer made to carry their equipment.
The roadies say the trailer must have a square base, an open top, and a volume of 32m3 .
Find the dimensions (length of base side and the height) for the trailer to be made with the
least amount of sheet metal.
21
Topic 5: Integral Calculus
Integration from Standard Integrals
1. Simple Integrals. Find an antiderivative of each of the following:
(a) 7x3 + 6x2 − 4x + 3
(d)
3
2x2
(g) sec2
(e) cos
x
2
(c) e3x
(b) sin(πx)
2πx 3
(h) 2x −
(f)
1
2x
4
x
x
(i) e− 5
2. Equation of Curve. The slope of a certain curve at any point (x, y) on the curve is
given by (x − 1)3 − 2, and the curve passes through the point (3, 0). Find the equation of
the curve.
3. Inverse Trigonometric. Integrate the following:
(a) √
−2
16 − x2
(b)
4
25 + x2
(c) √
7
3 − x2
Integration by Substitution
4. Derivative Present. Find each of the following:
Z
(a)
Z
5
2x(x + 1) dx
Z
(e)
2x2 +3x
(8x + 6)e
Z
(h)
2
(b)
Z
3
3x cos(x + 5) dx
Z
dx
2
(f)
√
5x 9 +
(c)
(x2
Z
x2
dx
x
dx
+ 1)3
6 sin 3x
dx
cos2 3x
(g)
5
sin(loge (x)) dx
x
5. Complete the Square. Find antiderivatives of the following:
(a)
1
x2 + 2x + 2
(b) √
1
−x2 − 4x + 21
(c) √
−1
7 − x2 + 6x
22
6. Linear Substitutions. Evaluate the following:
√
x 1 − x dx
Z
(a)
Z
(b)
Z
(c)
Z
1
3
(x + 3)(x + 4) dx
(d)
2x − 1
dx
(x − 1)2
(2x + 1)(x + 3)20 dx
Integration using Trigonometric Identities
7. Trigonometric Powers. Find the following:
Z
(a)
Z
4
2 sin 2x cos 2x dx
Z
(d)
(b)
sin x dx
Z
2
cos 7x dx
Z
2
(e)
2
(c)
2
cos 5x sin 5x dx
Z
(f)
cos5 2x dx
sin4 x cos5 x dx
8. Even Powers of Tan and Sec. Evaluate the following integrals:
Z
(a)
Z
2
sec 2x dx
Z
2
(b)
tan x − sec x dx
(d)
tan 2x dx
Z
2
2
(e)
Z
(c)
tan5 x sec2 x dx
sec4 3x dx
Integration by Partial Fractions
9. Partial Fractions. Antidifferentiate the following:
(a)
(d)
3x
(x − 2)(x + 4)
(b)
x+3
− 3x + 2
(e)
x2
x2
3
− 5x + 4
2x + 1
x2 − 1
(c)
(f)
x2
2x + 1
+ 4x + 4
x2
4x + 17
+ 10x + 25
10. Long Division. Find antiderivatives of the following, after first applying polynomial
long division.
(a)
x3 + x2 − 3x + 3
x+2
x3 + 4x2 + 11x + 2
(d)
x2 + 4x + 10
(b)
4x3 + 8x2 + 5x + 13
4x2 + 5
4x3 + 42x + 1
(e)
x2 + 10
(c)
x3 + 3
x2 − x
23
11. Mixed Integrals. Find the following:
Z
2
(a)
(d)
3
cos x sin x dx
R
cos x dx
−2x − 3
dx
x2 − x
(g)
(b)
Z
4
Z
x3 + 3x − 2
dx
x2 − x
Z
2
(e)
Z
(c)
4
tan 5x sec 5x dx
(h)
Z
1
√
dx
5+x
Z
1
dx
8 + 2x2
(f)
x−2
√
dx
x+1
Z
cos 2x esin 2x dx
(i)
Definite Integrals
12. Simple Type. Evaluate the following definite integrals:
2
Z
Z
2
(3x + 2x + 4) dx
(a)
(cos θ + 2θ) dθ
0
0
1
Z
π/4
(b)
Z
−t
t
(e − e ) dt
(c)
2
(d)
−1
1
1
dt
3t
13. Substitution. Evaluate the following:
1
Z
(a)
√
x 1 − x dx
Z
sin3 t cos t dt
(b)
0
0
3
Z
π/4
(c)
x
√
Z
x2
+ 16 dx
(d)
0
e
e2
1
1
· dx
log x x
14. Mixed Type. Evaluate the following:
3
Z
(a)
2
2
Z
(d)
1
3
dx
x3
1
dx
x(x + 1)
2
(g)
sec x dx
π/3
(i)
−π/6
sin 2θ dθ
Z
(e)
2
Z
(h)
0
sin x
dx
10 + 6 cos x + cos2 x
√
0
3
Z
1/2
(c)
0
0
Z
2
(b)
π/4
Z
π/2
Z
1
2x + 6
dx
(x − 1)2
x3 + x2 + 4x + 1
dx
x2 + 1
Z
1
(f)
−1
3
dx
1 − x2
et
dt
et + 1
24
Areas
15. Area Under. Find the areas indicated below.
(a) Area under y = x +
1
x2
from x = 2 to x = 3.
(b) Area under y =
6
4 + x2
(c) Area under y =
2
log x from x = 1 to x = e.
x
from x = −2 to x = 2.
(d) Area enclosed by y = −1 +
2
x2 + 1
and the x-axis.
16. Vertical Strips. For each pair of curves below, sketch them and find their intersection
points. Hence find the area enclosed between the curves, using vertical strips.
(a) y = x2
(b) y = 4x2
and y = 2x.
and y = x2 + 3.
1
(c) y 2 = x and y = x.
3
17. Horizontal Strips. Sketch the curves below and find the areas enclosed by them
using horizontal strips.
(a) x =
√
y
(b) x = y 2
y
and x = .
2
and x = 3y.
(c) x = −y 2 + 2 and y = x.
18. Vertical and Horizontal. Consider the two curves
y =x−2
and
x = y2 .
(a) Draw the region enclosed by these curves, including all relevant intersection points.
(b) Write down (but do not solve) the integrals that give the area of this region:
(i) using horizontal strips,
(ii) using vertical strips.
HINT: One of these should be a sum of two integrals.
(c) Find the area of the region by evaluating one of the integrals in (b).
25
Volumes of Solids of Revolution
19. Rotation about x -axis. Find the volume of the solid of revolution formed when the
area under f (x) is rotated about the x-axis, between the given x values.
(a) f (x) = 2x + 1 from x = 1 to x = 2.
(b) f (x) = sin x from x = 0 to x =
(c) f (x) =
√
9 − x2
π
.
2
from x = −3 to x = 3.
20. Rotation about y -axis. Find the volumes of the solids formed when the following
regions are rotated about the y-axis.
(a) x2 = 4 + 4y 2
for 0 ≤ y ≤ 1.
(b) y = log x for 0 ≤ y ≤ 1.
21. Vase. A vase is defined by rotating the curve with equation
x
y = 10 loge
,
0 ≤ y ≤ 10 ,
5
about the y-axis, where x and y are measured in centimetres.
(a) Rewrite the vase equation to obtain x as a function of y.
(b) Find the maximum volume of water which the vase could hold.
22. Washers. Find the volume of the solid of revolution formed when the area enclosed
by the curves y = x2 and y = 2x is rotated:
(a) about the x-axis.
(b) about the y-axis.
26
Topic 6: Differential Equations
Verification of Solutions
1. Verification. For each of the following differential equations, verify that the given
function is a solution (i.e. substitute the function into both sides of the differential equation).
The given functions are the general solutions of the differential equations.
(a)
dy
=1 , y =x+C
dx
(c)
dy
x
= − , x2 + y 2 = C
dx
y
(d)
dy
= 2y − 4x , y = Ce2x + 2x + 1
dx
(b)
dy
= y , y = Cex
dx
(e)
dy
2y
C
=−
, y= 2
dx
x
x
(implicit)
2. Initial Value Problems. Find the particular solution for the following initial value
problems, by starting with the general solutions verified in the previous question (so you do
not need to solve them from scratch).
(a)
dy
= 1 , with y(0) = 3.
dx
(b)
dy
= y , with y(0) = 2.
dx
(c)
dy
x
= − , with y(3) = 4.
dx
y
(d)
dy
= 2y − 4x , with y(−1) = 0.
dx
(e)
2y
dy
=−
, with y(2) = 64.
dx
x
3. Second Order. Verify that the given function is a solution of the given second order
differential equation.
(a)
d2 y dy
−
−6y = 0 ,
dx2 dx
(b)
d2 x
= −n2 x ,
dt2
(c)
d2 y
2 dy 2
=
,
dx2
y dx
y = e−2x + e3x
x = C sin(nt)
y=
4
x+1
4. Cubic Solution. Find the constants a, b, c, d if y = ax3 + bx2 + cx + d is a solution of
the differential equation
d2 y
dy
+y = x3 .
+2
2
dx
dx
27
5. Exponential Solution. Find the possible values of k if y = ekx satisfies the differential
equation
y 00 + 3y 0 + 2y = 0 .
Solving by Direct Antidifferentiation
6. First Order. Find the general solution of the following differential equations using direct
antidifferentiation. For those with an initial condition, find the corresponding particular
solution.
(a)
dy
1
=√
dx
x
(c)
dy
= sin(3t + π) ,
dt
y(0) = 1
(b)
t2 + 3t − 1
dx
=
dt
t
(d)
dy
1
=
,
dx
2x − 1
y(1) = 3
7. Second Order. Find the general solution of the following by direct antidifferentiation.
(a)
x
d2 y
= e2
2
dx
(b)
d2 x √
= 1−t
dt2
(c)
d2 y
1
=
2
dx
(x + 1)2
Separable Differential Equations
8. y only. Solve the following separable differential equations whose right hand sides
depend only on y. For those with initial conditions, find the particular solution.
(a)
dy
1
= 2
dx
y
(c)
dy √
= y,
dx
y(3) = 1
(b)
dy
= 1 + y2
dx
(d)
dy
=y−4 ,
dx
y(0) = 5
9. Separable. Find the general solutions of the following separable differential equations.
In part (e) you do not need to express x as a function of t.
dy
= 5y 2 cos x
dx
p
1 − y2
dy
(c)
=
dx
x
(a)
(e)
dx
3t + e2t
= 2
dt
x + e−x
(b)
dy
= ex e−2y
dx
(d)
p
dy
= 3 9 − y 2 sin4 t cos t
dt
(f)
dy
(y 2 + 1) cos2 3t
=
dt
2y
28
10. Separable Initial Value. Find the particular solutions of the following initial value
problems.
(a)
dy
= 3xy ,
dx
y(0) = 3
(b)
dx
x
=
,
dt
t(t + 1)
x(1) = 1
Applications of Differential Equations
11. Projectile Motion. The height of a projectile fired vertically up from the ground
with an initial speed of 50m/s satisfies the differential equation
d2 h
= −10 .
dt2
(a) Express the two initial conditions of this problem mathematically.
(b) Find the height at any time t.
(c) What is the maximum height reached?
12. Balloon. The volume V cm3 of a balloon is increasing at a rate that is inversely
proportional to its current volume.
(a) Write down the differential equation governing the balloon’s volume.
(b) If the initial volume of the balloon is 10cm3 and after 5 seconds it is 40cm3 , find the
volume V at any time t.
(c) What is the volume after 8 seconds?
13. Radioactive Decay. The rate of decay of a radioactive substance is proportional to
the amount Q present at any time.
(a) Write down the differential equation governing the amount of radioactive substance.
(b) Find the amount Q present at any time, given that initially Q = 100 and when t = 10,
Q = 50.
29
Population Models
14. Something Fishy. The number of fish in a lake is growing according to
dF
= 0.1F
dt
where F is the number of fish t weeks after observation is started.
(a) Find an expression for F (t) if there are initially 10 fish in the lake.
(b) How many weeks after observation begins will the number of fish reach 1000?
15. Something Moosey. The population P of moose in a certain forest increases at a
rate proportional to the current population.
(a) Write down the differential equation governing the moose population.
(b) Find the general solution of this differential equation.
(c) If the initial population was 100 and after 2 years the population had risen to 110,
determine the population at any time.
(d) What is the population after 5 years?
Newton’s Law of Cooling
16. Metal rod. A metal rod that has been heated to 110◦ C is placed into a large tank of
water whose temperature is a constant 10◦ C. After 2 minutes the temperature of the rod is
70◦ C. Assume that Newton’s Law of Cooling applies, so that the temperature T of the rod
satisfies
dT
= −k(T − TS )
dt
where TS is the temperature of the surroundings of the rod, and k is a positive constant.
(a) Find an expression for the temperature of the rod at any time t > 0.
(b) Find the temperature of the rod after a further 2 minutes.
30
Answers
Topic 1: Trigonometric Functions
1. Exact Values.
√
(a) −
3
2
√
(c) − 3
(b) − √13
2. More Exact Values.
(a) − √12
(b) −1
(c)
√
2
3. Even More Exact Values.
(a) − √12
√
(c) − 2
(b) −1
(d)
√
2
4. Graph Sketching.
(a)
(b)
(c)
y
y
y
1
1
-1
- 2
Π
4
3Π
4
x
7Π
4
5Π
4
Π
6
-1
Π
2
5Π
6
7Π
6
3Π
2
11 Π
6
1
x
-
1
3
Π
3
5. Simplification.
(b) sin2 (x)
(a) cos2 (x)
(c) cosec(x)
(d) 1
6. Trigonometric Values.
√
(a) − 12
(b)
3
2
√
(c) − 3
(d)
√2
3
(e) − √13
x=
5π
6
(c) x =
7. Solve for x.
(a) x =
3π
4 ,
x=
5π
4
(b) x =
π
6,
π
6,
x=
5π
6 ,
x=
7π
6 ,
x=
11π
6
5Π
6
4Π
3
11 Π
6
x
31
8. Expand.
(a)
tan(x)+tan(2y)
1−tan(x) tan(2y)
(b) cos(3x) cos(2y) + sin(3x) sin(2y)
(c) sin(y) cos(2x) − cos(y) sin(2x)
9. Simplify.
(a) cos(x)
(b) sin(3x + 2y)
(c)
− tan(x)(tan2 (y)+1)
1−tan(x) tan(y)
(d) sin(2y)
10. Expand and Simplify.
(a) cos(x)
(b) cot(y)
(c) − tan(x)
(d) − sec(y)
11. Exact Values with Compound Angle Formulae.
(a) 2 −
√
3
(b)
√ √
− 6− 2
4
(c)
√
6−
√
2
12. More Exact Values.
(a)
4
3
(b)
5
3
(c) − 63
65
(c) − 836
123
13. Even More Exact Values.
(a) − 43
(b)
4
5
(c) − 53
14. Inverse Trigonometric.
(a)
π
6
(b)
π
6
(c) − π3
15. Simplification.
(a)
1
2
(b) 1
(c)
3π
4
√
(d)
3
2
(e) − π3
(f)
π
6
(e) cosec(x)
32
16. Graphing Inverse Trigonometric Functions.
y
x-1
arccos
arccos x Π
3
x-1
2 - arccos
Π
2
3
-3
-2
3
2
x
arccos
1
-1
2
3
4
x
2-Π
17. Step by Step.
Let dom(f ) denote the domain of f and range(f ) denote the range of f .
(a)
(i) dom(f ) = [0, ∞); range(f ) = [0, ∞)
(ii) dom(g) = R; range(g) = (−∞, 4]
(iii) [0, ∞) ∩ (−∞, 4] = [0, 4]
(iv) dom(f ◦ g) = [−2, 2]; range(f ◦ g) = [0, 2]
(i)dom(f ) = R\{kπ : k ∈ Z} = {(kπ, π + kπ) : k ∈ Z}; range(f ) = R
(b)
(ii) dom(g) = [−1, 1]; range(g) = [−π/2, π/2]
(iii) {(kπ, π + kπ) : k ∈ Z} ∩ [−π/2, π/2] = [−π/2, 0) ∪ (0, π/2]
(iv) dom(f ◦ g) = [−1, 0) ∪ (0, 1]; range(f ◦ g) = R
Note that Z is the set of all integers.
18. Standard.
(a) (i) R\{1} (ii) R\{0}
(d) (i) (−2, 2) (ii) [ 12 , ∞)
(b) (i) (−1, ∞) (ii) R
(c) (i) [−2, 2] (ii) [0, 2]
√
(e) (i) [2kπ, 2kπ + π], k ∈ Z (ii) [0, 2]
(f) (i) (− π2 + 2kπ, π2 + 2kπ), k ∈ Z (ii) (−∞, 0]
19. Inverse Trigonometric.
(b) (i) [−2, −1] (ii) [0, π]
(a) (i) [0, 2] (ii) − π2 , π2
(d) (i) R (ii) [0, π]
(e) (i) [−1, 1]\{± √12 } (ii) R
(c) (i) R (ii) − π2 , π2
(f) (i) − π6 + kπ, π6 + kπ , k ∈ Z (ii) [0, π]
20. An incremental example.
(a) (i) R (ii) [0, ∞)
(b) (i) R (ii) [0, ∞)
(c) (i) R (ii) [−1, ∞)
(d) (i) [−1, 1] (ii) [−π/2, π/2]
21. Trigonometric.
(a) f (x) = x; domain is [−1, 1]; range is [−1, 1]
(b) g(x) =
√
1 − x2 ; domain is [−1, 1]; range is [0, 1]
33
Answers
Topic 2: Vectors
1. Introduction.
y
2b 2
a+b b
1
-7
b
-2
-3a
-a
b-3a
a
1
2
x
-2
2. Vector Arithmetic.
(a) 6i + 2j
(b) −9i − 3j
(c) − 21 i − j + 32 k
(d) 5i + j + k
(e) −3i − 2j + 2k
(f) −9i − 3j + k
3. The Hexagon.
−−→
−−→
−−→
−→
−→
OB = a + c, F B = a + 2c, DA = 2a, AC = c − a, EA = 2a + c
4. Division by Ratio.
(a) 51 (1, 14, 13)
(b)
1
16 (−13, −25, 37)
−−→
−→
(c) 18 (5b + 3c) where b = AB and c = AC
(d) Proof required.
5. Position and Length.
√
(a) (i) 4i − 7j
(ii) 65
√
(c) (i) −4i + 7j
(ii) 65
(b) (i) 3i + j
(e) (i) 2i
(f) (i) −4i
(d) (i) −3i − j
(ii) 2
(ii)
√
10
√
(ii) 10
(ii) 4
6. Unit Length, Unit Vector.
(a)
√1 (4i
65
(d)
√1 (−3i
10
− 7j)
− j)
(b)
√1 (3i
10
+ j)
(c)
√1 (−4i
65
(e) i
(f) −i
(a) −4i + 7j
(b) −3i − j
(c) 4i − 7j
(d) 3i + j
(e) −2i
(f) 4i
7. Let’s Go Backwards.
+ 7j)
34
8. More Vector Lengths.
√
(a) 26
(d)
√1 i
2
(g)
√ 1 (5i
110
+
(b)
√1 j
2
√
2
(c)
(e) 5i − 9j + 2k
(f)
√3 i
26
√
− 9j + 2k)
9. Dot product.
(a) 7
(b) 3
(c) 1
10. Angles.
(a) arccos √784
(d) arccos √112
(d) 1
(b) arccos
(e)
π
3
(e) 1
√3
84
(f) π2
(f) 0
(c) arccos
√ √1
33
11. Dot and Angle.
(a) 4
(b) arccos √144√11
12. Parallel and Perpendicular.
(a) is parallel and (d) is perpendicular
13. Perpendicular.
(a) a = −1
(b) a = − 35
14. Vector Angles.
(a) (a + b) · b = 0.
(b) arccos
√
√ 38
38 50
= arccos
19
5
15. Triangle.
(a) (i) b − a
(ii) 32 (b − a)
(iii) 13 (a + 2b)
(iv) a + 2b
−→
−−→
(b) Since AQ is a scalar multiple of OB, they are parallel.
16. Trapezium.
The side AB is parallel to DC, with the ratio being AB : DC = 1 : 2.
17. Another Trapezium.
(a) (3, − 23 , 1)
−−→ −→ −−→
(b) −AB − OA + OC
(c) (1, − 12 , 4)
(v) 2b
−
110
√4 j
26
+
√1 k
26
35
18. Parallelogram.
C(4,0,-1)
D(3,1,0)
B(2,1,-2)
(a) A(1,2,-1)
(b) 2i − j + k
(c) arccos
√ 2
3
19. Diagonals.
C
c
a
c
a+
a-c
a
O
(a)
√
(b) (i) 4 3
B
-c
A
√
(ii) 2 17 (iii) −40
(c) arccos
√ −5
√
3 17
20. Right-Angled Triangle.
3i + 3j
21. Triangle and Parallelogram.
Proof required.
22. Flying Direction.
p
√
(a)The ground
speed√is 100 50 + 7 2 ≈ 773.95 km/hour and the direction is
√
(700 + 50 2)i + 50 2j, approximately N84.76◦ E or 84.76 degrees east of north (.091 radians).
√
(b) p = 670i
10 4498 ≈ 670.67 km/hour and the required direction is
√ − 30j. The required velocity
(67i − 3j)/ 4498, approximately S87.43◦ E or 87.43 degrees east of south (−.045 radians).
23. Simple Projections.
(a)
8
5
(b) − 13
5
(c)
8
25 (−3, 4)
13
(d) − 25
(−3, 4)
y
4
a
w
2
projab b a
-3
x
c3
-1
projac
(e)
24. Scalar Projection.
(a) (i)
√14
17
(d) (i)
√4
8
(ii)
(ii)
√14
14
√4
35
(b) (i)
(e) (i)
√10
14
√2
2
(ii)
(ii)
√10
14
√2
5
(c) (i)
√7
6
(f) (i) 1 (ii)
(ii)
√1
3
√7
13
36
25. Vector Projection.
(d) (i)
(b) (i) 57 u
(ii) i − j − k
(a) (i) u
4
35 u
(ii)
1
35 (50i−4j+82k)
(ii) 17 (16i − 4j − 8k)
(e) (i) 25 u
(c) (i)
(ii) 15 (5i−2j−1k)
7
13 u
1
13 (12i + 13j + 8k)
(ii)
(f) (i) 13 u
(ii) 13 (−i+2j−k)
26. Scalar and Vector Projections.
(a)
√9
6
=
√
3 6
2
√9
29
(b)
(c)
27
29 i
+
36
29 j
+ 18
29 k
v-projuv
(d) 32 i + j + 72 k
v
u
projuv
27. Gravity. [Optional question.]
w = −490k;
r=
√
−245 3
i
2
+
245
2 k;
krk = 245
28. Ellipses.
(a)
(b)
(c)
(d)
y
y
2
8
3 -3
x
y
2
y
-3
1
-1
x
5
-3
3
-2
2
3
-1
-1
x
7
-5
1
-2
x
-4
-2
-3 - 2
3
29. Hyperbolae.
(a) Asymptotes: 4x − 3y = 11,
4x + 3y = 5
(b) Asymptotes: 2x − 5y = −7,
2x + 5y = 3
(c) Asymptotes: y − 2x = 2
2x + y = 2
y
y
y
3
1
H-1, -1L
2
x
-6
4
x
2
H5, -1L
-1
1
x
2+2
2
37
30. Harder Questions.
(a)
(b)
(c) Asymptotes:
5x + 3y = 9,
5x − 3y = 21
(d) Asymptotes:
3x − 4y = 4,
3x + 4y = −4
y
y
y
3
y
3
2
5
7
-3 -
7 -3
-3
3
-2
x
6
x
-4
4
-1
3
1
-3
1
4
x
7
31. Equation of a Path.
(a) y = −2x
(b) y =
√
(c) y = 3(x + 4)2
x
y
y
1
2
x
4
3
y
-2
2
2
1
1
1
-4
1
4
x
(e) y = (x − 3)(x − 1)
(d) y = x 3
y
3
2
y
1
2
1
1
1
8
x
-1
2
3
4
x
-4
-3
-2
-1
x
x
38
32. Periodic Motion.
(a) x2 + y 2 = 1, period π
(b) x2 + y 2 = 4, period π
(c) (x − 2)2 + (y + 3)2 = 1, period 2π
y
y
y
1
2
3
x
2
1
-1
1
-1
x
2
-2
-2
x
-3
-1
-2
-4
33. Another Path.
(i) y 2 − x2 = 4
(ii) This is a hyperbola and can be written y 2 /4 − x2 /4 = 1. Note that y ≥ 0 so it is only
part of the hyperbola. The asymptotes are y = ±x.
y
2
-2
2
x
39
Answers
Topic 3: Complex Numbers
1. Square root of negative numbers.
(a) 2i
(b)
√
6i
√
(c) 4 3i.
2. Real and Imaginary.
(a) Re(z) = 3, Im(z) = −4, Re(z)−Im(z) = 7 (b) Re(z) = 0, Im(z) = 6, Re(z)−Im(z) = −6
(c) Re(z) = 2, Im(z) = 7, Re(z)−Im(z) = −5 (d) Re(z) = 75 , Im(z) = 0, Re(z)−Im(z) = 75
(e) Re(z) = 2, Im(z) = 21 , Re(z)−Im(z) = 32 .
3. Complex Arithmetic.
(a) −1 + 8i (b) 2 (c) 1 + 2i (d) 2 + 2i (e) − 25 + 4i
(g) 5 − 6i (h) 21 − 4i (i) 92 − 4i.
(f)
15
2
+ 2i
4. The Argand Plane.
(a)
æ
z2
(b)
Im z
4
z4
Im z
8
4
æ
z1
2
-2
æ
(c)
z1
4
Re z
æ
-3 z1
æ
-12
All points are on the same line.
(d) (i) No
(ii) No
(iii) Yes
(iv) Yes.
5. Powers of i.
i2 = −1, i3 = −i, i4 = 1
4
z1 + z4 = z2 + z3
z3 æ
-4
æ
2
-6
Re z
2 z1
Im z
8æ z1 + z2
æ
(a) −i
(b) −1
(c) −i
(d) 6 + 9i
(e) 5
6. Cartesian form.
(a) −3 + 2i (b) 5 − 12i (c) −3 − 4i
(g) 5 + i (h) −5 (i) −3 − 2i.
(f) −2 + 2i
2
4
Re z
40
7. Rationalising the Denominator.
(a)
3
13
(d)
7
2
+
2
13 i
+ 32 i
(b)
4
5
− 25 i
(c)
+
2
13 i
1
(f) − 13
+
−1 − 2i
(e)
3
13
5
13 i.
8. Real and Imaginary.
(a) -21/17
(b) -17/26
9. Harder Examples.
(a) − 13
(b) −3
(c)
3
26
−
11
26 i
(d) − 27
5 −
31
5 i.
10. Conjugates to Cartesian.
(a) 12 − 7i
(b) 1 + i
(c) 22 + 7i
(d) −10 − 20i
(f) − 21 + 21 i
(e) 2 − 2i
11. Complex conjugates.
(d) reflection in x axis
(e) reflection in origin
(f) reflection in y axis.
Im z
Im z
Im z
z
æ
æ
z
æ
-z
Re z
æ
-z
æ
-z
z
12. Complex Trigonometric.
(a) 1
(b) cos(θ − φ) + i sin(θ − φ)
sin α
(c) 12 (1 − i 1+cos
α)
13. Modulus and Argument.
(a) |z| = 6, Arg(z) = 0 (b) |z| = 3, Arg(z) = π (c) |z| = 2, Arg(z) = 5π
6
(d) |z| = 4, Arg(z) = π2 (e) |z| = 2, Arg(z) = − π2 (f) |z| = 2, Arg(z) = − 2π
3
√
√
(g) |z| = 7, Arg(z) = π2 (h) |z| = 2, Arg(z) = π4 (i) |z| = 4 2, Arg(z) = − π4 .
14. Something more complicated.
(a) (i) 1
(ii) 27/10
(iii) 10
(b) (i) 2π/3
(ii) −3π/4
z
Re z
Re z
æ
æ
(iii) π
41
15. Regions in the Complex Plane.
Graphs required. They should have the following:
(a) Line x = 1
(b) half plane y > 1
(c) boundary and interior of the circle
x2 + (y − 2)2 = 1
Im z
Im z
3
1
2
1
Im z
Re z
1
1
-1
Re z
1
-1
Re z
1
-1
(e) Sector of circle, centred at the ori-(f) Hyperbola x2 − y 2 = 4.
gin, radius 4, angle π/4 anticlockwise
from the real axis.
(d) line 6x + 4y = 5
Im z
Im z
Im z
4
2
1
2
1
2
3
2
Re z
2
-2
Re z
-1
-2
-2
Π/4
4
Re z
16. Exponential Polar Form.
(a) eiπ
√
(b) 5 2eiπ/4
(c) 2e−iπ/2
(d)
√
29eiα , where α = arctan(5/2)
17. Powers.
(a) −1024
√
(b) −512 3 + 512i
(c) −1/64
(d)
1
e−7πi/12
27/2
18. Complete the square.
(a) z1 = i, z2 √
= −i (b) z1 = 4i,
z2 = −4i
√
(e) z1 = − 23 + 23 i, z2 = − 32 − 23 i.
(c) z1 = 2,
z2 = −2 (d) z1 = −2 + i,
z2 = −2 − i
42
19. Quadratic formula.
√
(a) z1 = −2, z2 = −1 (b) z1 = − 32 + 27 i,
√
√
(d) z1 = − 13 + 32 i, z2 = − 13 − 32 i, z3 = 0
z2 = − 23 −
√
7
2 i
(c)
z1 =
1
4
+
√
23
4 i,
z2 =
1
4
√
−
23
4 i
20. Any way you like.
(a) z1 = 3, z2 = −3,
(b) z1 √
= 2, √z2 = −2,
z8 = − 2 + 2i.
z3 = 3i,
z3 = 2i,
z4 = −3i
z4 = −2i,
√
√
z5 = − 2 − 2i,
√
z6 =
2+
√
2i,
z7 =
√
2−
√
2i,
21. Roots and Factors.
(a) 1, e2πi/3 , e−2πi/3
æ
(b) eπi/4 , e−3πi/4
Im z
Im z
1
1
æ
2Π/3
Π/4
Re z
æ
-1
1
æ
-1
-3Π/4
Re z
1
æ
-1
-1
(c) 2eπi/6 , 2eπi/2 , 2e5πi/6 , 2e−5πi/6 , 2e−πi/2 , 2e−πi/6 (d) 2eπi/3 , 2e11πi/15 , 2e−13πi/15 , 2e−7πi/15 , 2e−πi/15
Im z
Im z
2æ
2
æ
æ
æ
æ
Π/3
2
-2
æ
Re z
-2
2
æ
Re z
æ
æ
-2 æ
-2æ
22. Roots of Polynomials.
√
(a) ±i 2 repeated
√
(b) ±1 ± i 3
q
q
√
√
(c) z = ± (5 + 2 5)/5, ± (5 − 2 5)/5
(d) e2πi/5 , e4πi/5 , e−2πi/5 , e−4πi/5
23. Polynomials in Cos and Sin.
(a) 3 sin θ−4 sin3 θ
(b) cos4 θ−6 cos2 θ sin2 θ+sin4 θ
(c) 6 cos5 θ sin θ−20 cos3 θ sin3 θ+6 cos θ sin5 θ
43
24. Sin and Cos of Multiple Angles.
a)
3
4
sin θ− 14 sin(3θ)
(b)
1
8
(cos(4θ) + 4 cos(2θ) + 3)
(c)
1
64
(− sin(7θ) + 7 sin(5θ) − 21 sin(3θ) + 35 sin θ)
44
Answers
Topic 4: Differential Calculus
1. First and Second Derivatives.
√
(b) h0 (x) = x/( x2 + 1), h00 (x) = 1/(x2 + 1)3/2
(a) f 0 (x) = 4x3 − 9x2 + 16, f 00 (x) = 12x2 − 18x
(c) y 0 = 1/(1 − x)2 , y 00 = 2/(1 − x)3
(d) y 0 = − 23 x(1 − x2 )−1/4 , y 00 = 34 (1 − x2 )−5/4 (x2 − 2)
(e) f 0 (x) = (2 − 5 cos(5x))e2x−sin(5x) , f 00 (x) = (25 sin(5x) + (2 − 5 cos(5x))2 )e2x−sin(5x)
(f) g 0 (x) = − cosec(x) cot(x), g 00 (x) = cosec3 (x) + cosec(x) cot2 (x)
2. nth Derivatives.
(a) n! = n × (n − 1) × (n − 2) × (n − 3)...
(b) (−1)n (n + 2)!/(6xn+3 )
3. Related Derivatives.
(a) f 00 (x) = 6xg 0 (x2 ) + 4x3 g 00 (x2 )
(b) f 00 (x) =
√
√
√
xg 00 ( x)−g 0 ( x)
√
4x x
4. Implicit Derivatives.
(a)
y−2x
3y 2 −x
(b) tan x tan y
(c)
2y−2xy loge y
√
2x2 +x y
(d)
yexy −2xy
x2 −xexy
x
e (1+x)
(e) 1+ sin(x−y)
5. Second Derivative.
(a) y 0 =
−x3
y3
(b) y 00 =
−3x2
y3
−
3x6
y7
6. More Implicit Derivatives
a = −9/5 and b = 17/10.
7. Implicit Tangent.
Proof required.
8. Derivatives.
(a)
√ 1
25−x2
4
(d) − √1−16x
2
1
(b) − √16−x
2
(e)
7
1+49x2
(c)
√
7
7+x2
(f)
√ 2
1−4x2
(f)
y+y 2 x cosec2 xy
x−x2 y cosec2 xy
45
9. More derivatives.
7
(a) − √16−49x
2
(d) − √
(b)
4
1−(4x−3)2
√
8
25−64x2
(e) − √
4
25−(3−4x)2
(c)
14
4+49x2
(f)
20
16+(5x−3)2
10. Even more derivatives.
2
(a) − √3x
1−x6
(d) − √
x
(b)
ex
1+(ex +5)2
(c)
3 sin(3x)
(e) − 1+(cos(3x)+1)
2
2
1−4(log(x))2
√1
3x2/3
(f) √
1−x2/3
4x
9−(2x2 +1)2
11. Graph sketching.
(a) Domain R except ±1. y intercept at y = 1, no
x intercept. Local minimum at (0, 1). Increasing for
0 < x < 1 and x > 1, decreasing for x < −1 and
−1 < x < 0. Concave down for x < −1 and x > 1,
concave up for −1 < x < 1. No points of inflection.
Vertical asymptotes at x = ±1, horizontal asymptote
at y = −1.
(b) Domain R. Oblique asymptote y = x. x and
y-intercept at the origin (0, 0). Stationary point√at
x = 0. Increasing
√ for all x. Concave up for
√ x<− 3
and 0 < √
x < 3. Concave down for − 3√< x <
√0
and x > 3. Points of inflection at x = − 3, 0, 3.
y
y
1
x
1
-1
-1
0.5
-1 -0.5
(c) Domain −3 ≤ x ≤ 3. Intercepts
√ are (-3,0),
(0,0), (3,0). Local minimum
at (−3 2/2, −9/2).
√
Local maximum
at (3 2/2,√
9/2). Decreasing for
√
x <
−3
2/2
and
x
>
3
2/2. Increasing on
√
√
−3 2/2 < x < 3 2/2. Concave up for x < 0,
concave down for x > 0. Point of inflection at x = 0.
y
x
1
(d) Domain R except 0. Vertical asymptote at x = 0,
horizontal asymptote at y = 1. No x or y intercepts.
No stationary points. Decreasing for all x ∈ R\{0}.
Concave up for − 21 < x < 0 and x > 0, concave down
for x < − 21 . Point of inflection at x = − 21 . As x → 0
from above, f (x) → ∞, while as x → 0 from below,
f (x) → 0.
y
9
2
ã
-3 -
3
2
3
2
- 92
3
x
1
1
x
46
y
(e) Domain R except {−1, 1}. Vertical asymptotes
at x = −1, 1, horizontal asymptote at y = 0. No
x intercept; y intercept (0, −1). Local maximum at
(0, −1). Increasing on (−∞, −1) ∪ (−1, 0). Decreasing
on (0, 1) ∪ (1, ∞). No points of inflection. Concave up
on (−∞, −1) ∪ (1, ∞), concave down on (−1, 1).
-1
-1
x
1
12. The Dying Wave.
(a) zeros: (2n+1)π/2, stationary points: 3π/4+kπ. To classify, use second derivative test, where y 00 = 2e−t sin t.
If k is even then sin(3π/4+kπ) > 0, so the turning point is a local minimum; if k is odd then sin(3π/4+kπ) < 0,
so the turning point is a local maximum.
y
1
et
3А4
-1
5А4
7А4
9А4
-et
13. Flying High.
(a) h = 18 at t = 3
(b) t = 6
14. Flying Higher.
(a) h = 11 41 at t = 3
(b) h = 31 32 at t = 10
15. Cylinder.
(a) A = 2πr2 + 2πrh, V = πr2 h
r
2πr
(b) Proof required.
(c) 150π cm2
h
16. Rectangle.
Maximum area is 9, when one vertex is at ( √32 , 0).
t
47
17. Manufacturing.
(a) P (x) = 900 log(1 +
x
300 )
− (200 + x)
18. Rock n Roll.
Base: 4m by 4m, Height: 2m.
(b) x = 600 gives maximum profit of $188.75.
48
Answers
Topic 5: Integral Calculus
1. Simple Integrals.
(b) − π1 cos(πx) + C
(a) 47 x4 + 2x3 − 2x2 + 3x + C
3
(d) − 2x
+C
(e)
(g) 2 tan( x2 ) + C
3
2π
sin( 2πx
3 )+C
(h) x2 −
1
2
(c) 13 e3x + C
(f) 4 log |x| + C
x
(i) −5e− 5 + C
log |x| + C
2. Equation of Curve.
y = 41 (x − 1)4 − 2x + 2
3. Inverse Trigonometric.
(a) 2 arccos x4 + C
(b)
4
5
arctan x5 + C
(c) 7 arcsin √x3 + C
4. Derivative Present.
(a) 16 (x2 + 1)6 + C
(b) sin(x3 + 5) + C
(g) 53 (9 + x2 )3/2 + C
2
cos 3x
(h)
(c)
−1
4(x2 +1)2
+C
2
(d) 2e2x
+3x
+C
(i) −5 cos(loge (x)) + C
+C
5. Complete the Square.
(b) arcsin x+2
5 +C
(a) arctan(x + 1) + C
(c) arccos x−3
4 +C
6. Linear Substitutions.
(a) − 23 (1 − x)3/2 + 25 (1 − x)5/2 + C
(b) 2 log |x − 1| −
(c) 37 (x + 4)7/3 − 34 (x + 4)4/3 + C
(d)
1
11 (x
+ 3)22 −
1
x−1
5
21 (x
+C
+ 3)21 + C
7. Trigonometric Powers.
(a)
1
5
sin5 2x + C
(d) 12 x +
1
28
(b) 12 x −
1
4
sin 2x + C
(e) 18 x −
sin 14x + C
1
160
(c)
1
2
sin 20x + C
sin 2x −
(f)
1
3
1
5
8. Even Powers of Tan and Sec.
(a)
1
2
tan 2x + C
(d) −x + C
(b)
(e)
1
2
tan 3x
3
tan 2x − x + C
+
tan3 3x
9
+C
(c)
1
6
tan6 x + C
sin3 2x +
sin5 x −
2
7
1
10
sin5 2x + C
sin7 x +
1
9
sin9 x + C
49
9. Partial Fractions.
(a) log |x − 2| + 2 log |x + 4| + C
(b) log |x − 4| − log |x − 1| + C
(d) 5 log |x − 2| − 4 log |x − 1| + C
(e)
1
2
(c) 2 log |x + 2| +
log |x + 1| + 32 log |x − 1| + C
3
x+2
+C
3
(f) 4 log |x + 5| + x+5
+C
10. Long Division.
(a) 13 x3 − 12 x2 − x + 5 log |x + 2| + C
(b)
(c) 21 x2 + x − 3 log |x| + 4 log |x − 1| + C
(e) 2x2 + log(x2 + 10) +
√
arctan(x/ 10)
√
10
x2
2
+ 2x +
(d)
x2
2
+
1
2
√
3 arctan(2x/ 5)
√
2 5
+C
log(10 + 4x + x2 ) + C
+C
11. Mixed Integrals.
(a)− 13 cos3 x +
(d)
3x
8
+
1
4
1
5
(b) 12 x2 + x + 2 log |x2 − x| + C
cos5 x + C
sin(2x) +
1
32
sin(4x) + C
1
25
(e)
tan5 5x +
1
15
√
(f) 2 5 + x + C
tan3 5x + C
√
(h) 23 (−8 + x) 1 + x + C
(g) 3 log |x| − 5 log |x − 1| + C
(c) 21 esin 2x + C
(i)
1
4
arctan x2 + C
12. Simple Type.
(a) 20
(b)
√1
2
+
π2
16
(c) 0
(d)
1
3
log 2
13. Substitution.
(a)
4
15
(b)
1
16
(c)
61
3
(d) log 2
14. Mixed Type.
(a)
5
24
(f) 1
(b)
(g) 1
π
4
(c)
π
2
(d) log
(h) 23 (1 + log 2)
4
3
(e) log 4 + 4
√
(i) − arctan(7/2) + arctan(3 +
3
2 )
15. Area Under.
(a)
8
3
units2
(b)
3π
2
units2
(c) 1 unit2
(d) π − 2 units2
16. Vertical Strips.
(a)
4
3
(b) 4
(c)
9
2
y
y
y
4
3
4
3
2
1
2
1
x
-1
1
x
1
9
x
50
17. Horizontal Strips.
(a)
4
3
(b)
9
2
(c)
9
2
y
y
1
y
4
-2
3
2
1
-1
x
2
-1
1
1
x
2
1
x
9
-2
18. Vertical and Horizontal.
(b) (i) Area =
y
2
(ii) Area =
1
1
(a)
2
3
x
4
-1
19. Rotation about x -axis.
(a)
49π
3
(b)
π2
4
(c) 36π
20. Rotation about y -axis.
(a)
16π
3
(b)
π 2
2 (e
− 1)
21. Vase.
(a) x = 5ey/10
(b) 125π(e2 − 1) cm3 ≈ 2509 cm3
22. Washers.
(a)
64π
15
(b)
8π
3
R2
−1
(y + 2) − y 2 dy
R1 √
R4√
2 x dx + 1 x − x + 2 dx
0
(c)
9
2
51
Answers
Topic 6: Differential Equations
1. Verification.
Verification required.
2. Initial Value Problems.
(a) y = x + 3
(b) y = 2ex
(c) y =
√
25 − x2
(d) y = e2x+2 + 2x + 1
(e) y =
256
x2
3. Second Order.
Verification required.
4. Cubic Solution.
a = 1, b = −6, c = 18, d = −24
5. Exponential Solution.
k = −1, −2
6. First Order.
√
(a) y = 2 x + C
1) + 3
(b) x = 12 t2 + 3t − log |t| + C
(c) y = − 31 cos(3t + π) +
2
3
(d) y =
7. Second Order.
x
(a) y = 4e 2 + Cx + D
8. y only.
p
(a) y = 3 3(x − C)
(b) x =
4
15 (1
5
− t) 2 + Ct + D
(b) y = tan(x − C)
(c) y = − log |x + 1| + Cx + D
(c) y = 41 (x − 1)2
(d) y = ex + 4
9. Separable.
(a) y =
−1
5 sin x+C
(b) y =
1
2
(d) y = 3 sin( 53 sin5 t + C)
log(2ex + D)
(c) y = sin(log |x| + C)
(e) 31 x3 − e−x = 23 t2 + 12 e2t + C
p
p
(f) y = ± exp(t/2 + sin 6t/12 + C) − 1 = ± A exp(t/2 + sin 6t/12) − 1 where A = eC .
10. Separable Initial Value.
3
(a) y = 3e 2 x
2
(b) x =
2t
t+1
1
2
log(2x −
52
11. Projectile Motion.
(a) h(0) = 0 (fired from ground level),
(b) h = −5t2 + 50t
dh
dt
= 50 (initial speed 50m/s)
(c) 125m
12. Balloon.
(a)
dV
dt
=
k
V
√
(b) V = 10 3t + 1
, where k is a constant.
(c) 50cm3
13. Radioactive Decay.
(a)
dQ
dt
= −kQ, where k is a positive constant.
(b) Q = 100( 12 )t/10
14. Something Fishy.
(a) F = 10e0.1t
(b) Approximately 46 weeks.
15. Something Moosey.
(a)
dP
dt
= kP , where k is a constant.
t
1
(c) P = 100e 2 log(1.1)t = 100(1.1) 2
(b) P = Aekt (A a constant)
(d) P (5) = 127
16. Metal rod.
1
3
(a) T = 100e 2 log( 5 )t + 10 = 100
3
5
2t
+ 10
(b) 46◦ C