Mathematics HL - OPTION - Calculus - Course Companion

2014·9.94 MB·english

by  Josip Harcet

O X F O R D I B D I P L O M A P R O G R A M M E
:
MATHEMATICS HIGHER LEVEL
C A LC U LU S
Josip Harcet
Lorraine Heinrichs
Palmira Mariz Seiler
Marlene Torres-Skoumal
3
Great Clarendon Street, Oxford OX2 6DP
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Course Companion denition
The IB Diploma Programme Course Companions are Diploma Programme core requirements, theory of
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curriculum in terms of a whole-course approach; the and authoritative without being prescriptive.
use of a wide range of resources, international
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It is of vital importance to acknowledge and What constitutes malpractice?
appropriately credit the owners of information
Malpractice is behaviour that results in, or may
when that information is used in your work. After
result in, you or any student gaining an unfair
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advantage in one or more assessment component.
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Malpractice includes plagiarism and collusion.
work, it must be based on your individual and
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acknowledged. Therefore, all assignments, written
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Collusion is dened as supporting malpractice by
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magazines, newspaper articles, Internet-based
Other forms of malpractice include any action
resources, CDs and works of art) and providing
that gives you an unfair advantage or aects the
full information as to how a reader or viewer
results of another student. Examples include,
of your work can nd the same information.
taking unauthorized material into an examination
A bibliography is compulsory in the extended
room, misconduct during an examination, and
essay.
falsifying a CAS record.
iv
About the book
The new syllabus for Mathematics Higher Level Questions are designed to increase in diculty,
Option: Calculus is thoroughly covered in this strengthen analytical skills and build condence
book. Each chapter is divided into lesson-size through understanding.
sections with the following features:
Where appropriate the solutions to examples are
given in the style of a graphics display calculator.
Did you know? History Mathematics education is a growing, ever
changing entity. The contextual, technology
integrated approach enables students to become
Extension Advice
adaptable, lifelong learners.
Note: US spelling has been used, with IB style for
The Course Companion will guide you through
mathematical terms.
the latest curriculum with full coverage of all
topics and the new internal assessment. The
emphasis is placed on the development and
improved understanding of mathematical
concepts and their real life application as well as
prociency in problem solving and critical
thinking. The Course Companion denotes
questions that would be suitable for examination
practice and those where a GDC may be used.
About the authors
Lorraine Heinrichs has been teaching Marlene Torres-Skoumal has taught IB
mathematics for 30 years and IB mathematics for mathematics for over 30 years. During this time,
the past 16 years at Bonn International School. she has enjoyed various roles with the IB,
She has been the IB DP coordinator since 2002. including deputy chief examiner for HL,
During this time she has also been senior senior moderator for Internal Assessment,
moderator for HL Internal Assessment and calculator forum moderator, workshop leader,
workshop leader of the IB; she was also a and a member of several curriculum review
member of the curriculum review team. teams.
Palmira Mariz Seiler has been teaching Josip Harcet has been involved with and teaching
mathematics for over 25 years. She joined the IB the IB programme since 1992. He has served as a
community in 2001 as a teacher at the Vienna curriculum review member, deputy chief
International School and since then has also examiner for Further Mathematics, assistant IA
worked as Internal Assessment moderator in examiner and senior examiner for Mathematics
curriculum review working groups and as a HL as well as a workshop leader since 1998.
workshop leader and deputy chief examiner for
HL mathematics. Currently she teaches at
Colegio Anglo Colombiano in Bogota, Colombia.
v
Contents
Chapter 1 Patterns to innity 2 Chapter 4 The nite in the innite 96
1.1 From limits of sequences to limits of 4.1 Series and convergence 98
functions 3 4.2 Introduction to convergence tests
1.2 Squeeze theorem and the algebra of for series 104
limits of convergent sequences 7 4.3 Improper Integrals 110
1.3 Divergent sequences: indeterminate 4.4 Integral test for convergence 112
forms and evaluation of limits 10 4.5 The p-series test 114
1.4 From limits of sequences to limits 4.6 Comparison test for convergence 115
of functions 13 4.7 Limit comparison test for convergence 118
4.8 Ratio test for convergence 119
Chapter 2 Smoothness in mathematics 22 4.9 Absolute convergence of series 120
2.1 Continuity and dierentiability 4.10 Conditional convergence of series 122
on an interval 24
2.2 Theorems about continuous functions 28 Chapter 5 Everything polynomic 130
2.3 Dierentiable functions: Rolle’s 5.1 Representing Functions by
Theorem and Mean Value Theorem 33 Power Series 1 132
2.4 Limits at a point, indeterminate forms, 5.2 Representing Power Series as
and L’Hopital’s rule 42 Functions 135
2.5 What are smooth graphs of functions? 49 5.3 Representing Functions by
2.6 Limits of functions and limits of Power Series 2 138
sequences 50 5.4 Taylor Polynomials 143
5.5 Taylor and Maclaurin Series 146
Chapter 3 Modeling dynamic 5.6 Using Taylor Series to approximate
phenomena 54 functions 156
3.1 Classications of dierential 5.7 Useful applications of power series 161
equations and their solutions 56
3.2 Dierential Equations with separated Answers 168
variables 61
3.3 Separable variables, dierential Index 185
equations and graphs of their solutions 63
3.4 Modeling of growth and decay
phenomena 69
3.5 First order exact equations and
integrating factors 73
3.6 Homogeneous dierential
equations and substitution methods 80
3.7 Euler Method for rst order
dierential equations 85
vi
1 Patterns
to innity
CHAPTER OBJECTIVES:
9.1 Innite sequences of real numbers and their convergence or divergence.
Informal treatment of limit of sum, difference, product, quotient, squeeze theorem.
Before you start
1 Simplify algebraic fractions. 1 Simplify:
1 1 n+1−n 1 3 3 n 1 n
e.g. − = = a − b −
n n+1 n(n+1) n(n+1) 4n+1 4n−1 n+2 n+3
2 Solve inequalities that involve absolute value. 2 Solve:
e.g. Solve |x – 3| < 2 a |3x – 1| < 1 b |2x – 3| ≥ 4
x − 3 <2 ⇒−2 < x − 3<2⇒1< x < 5
3 Rationalise denominators of fractions. 3 Rationalise the denominator:
( ) 2n n 1
3 3 n + n+2 a b
e.g. =
( )( ) 1+ n n + n+1
n − n+2 n − n+2 n+ n+2
( ) ( )
3 n+ n+2 3 n+ n+2
= =
n−(n+2) 2
4 Recognise arithmetic and geometric 4 Find the nth term of
sequences and use knowledge about their 1 1 1 1
a ,− , ,− ,…
general terms to determine the expression 2 4 8 16
of other sequences. 1 1 1 1
b , − , , − ,…
e.g. Find the nth term of the sequence 2 3 4 5
1 2 4 8 16
,− , ,− , ,… 1 3 5 7
3 5 7 9 11 c , , , ,…
2 4 6 8
Each term can be seen as the quotient of
terms of a geometric sequence (numerator) 3 5 7 3
d , − , , − ,…
and the quotient of terms of an arithmetic 2 4 8 16
sequence (denominator). The general term
of this sequence is
( 2)n 1
u = , n∈Z+
n 2n+1
2 Patterns to innity
1.1 From limits of sequences to limits of functions
Innity is a concept that has challenged mathematicians
Intuitively we think of innity as
and scientists for centuries. Throughout this time the something larger than any number.
concept of innity was sometimes denied and sometimes This sounds simple, but in fact innity is
accepted by mathematicians, to the point that it became anything but simple. In the late 1800s,
one of the main issues in the history of Mathematics. In Georg Cantor discovered that there are
the last 150 years, great advances were made: rst with the many different sizes of innity, in fact
there are innitely many sizes! Cantor
axiomatization of set theory; and then with the work
showed that the smallest innity is the
of philosopher Bertrand Russell and his collection of
one you would get to if you could count
paradoxes. At the climax of all discussions was the work
forever (a ‘countable’ innity), and that
of Georg Cantor on the classication of innities.
innities which aren’t countable, such as
But do innities really exist? After all, how many types real numbers, are actually larger in size.
of innity are there? Does it make sense to compare
them and operate with them?
Is there a ‘medium’ size of innity,
In this chapter we will explore the concept of innity,
bigger than the natural numbers
starting with an intuitive approach and looking at
but smaller than the real numbers? The
familiar number patterns: sequences. We will then supposition that nothing is in between
formalise the idea of ‘the pattern that goes on the two innities is called the ‘continuum
forever’ and formally dene the limit of a sequence. hypothesis’. You may want to explore
This may help you to better understand the theorems this topic further to learn about the
about sequences, although formal treatment of limits powerful methods invented by Cantor
of sequences will not be examined. For this reason, and the resulting problems which still
puzzle mathematicians today.
all proofs of results have been omitted.
At the end of the chapter we will explore the connections between limits of
sequences and limits of functions introduced in the core course. We will also
establish criteria for the existence of the limit of a function at a point. Chapter 1 3
Consider the following numerical sequences:
a :, 2, 4, 8, 6, …, 2n, …
n
1 1 1 1
b :1, , , , ,
n 2 3 4 n
c :–, , –, , –, , …, (–)n, …
n
What is happening to the terms of these sequences as n increases?
Do they approach any real number as n → +∞?
1.1 Divergent innity 1.1 Convergent 1.1 Oscillating
y y y
70
1.2 1.5
60
1.0 1
50
0.8 0.5 u1(n) = (–1)n
40 u1(n) = 2n
30 0.6 u1(n) = 1 –0.50 2 4 6 8 10 12 x
20 0.4 n
–1
10 0.2
–1.5
0 1 2 3 4 5 6 7x 0 4 8 12 16 20 24 x –2
If we graph the sequences {a }, {b }, and {c }, we can observe their
n n n
behaviour as n increases, and notice that:
lima =+∞ which means that {a } diverges;

n→∞ n n
limb =0 which means that {b } converges to 0;

n→∞ n n
limc does not exist (it oscillates from  to −) which means that {c } diverges.

n→∞ n n
The following investigation will help you to better understand what ‘convergent’ means.
Investigation 1
n+1
1 Use technology to graph the sequence dened by u =
n 2n+1
1
2 Hence explain why limu =
n→∞ n 2
1
3 Find the minimum value of m such that n ≥ m ⇒ u − <0 1 (i.e. nd the smallest
n 2
1
integer n for which the dierence between the value u and is less than 0.1).
n 2
4 Consider the positive small quantities ε = 0.01, 0.001, and 0.0001. In each case nd
1
the minimum value of m such that n ≥ m ⇒ u − <ε
n 2
5 Decide whether or not it is possible to nd the order m such
that n ≥ m⇒ |u – 0.4| < 0.1. Give reasons for your answer.
n
⎛ 1⎞n
Consider now the sequence dened by v = −
⎜ ⎟
n ⎝ 3⎠
6 Explain why limv = 0
n
n→∞
7 Consider the positive small quantities ε = 0.01, 0.001, and 0.0001. In each case nd
the minimum value of m such that n ≥ m⇒ |v | < ε
n
8 Explain the meaning of limu = L in terms
n You may want to use sequences dened by
n→∞
of the value of |u L|.
n expressions involving arithmetic and geometric
9 Explore further cases of your choice. sequences studied as part of the core course.
4 Patterns to innity
Denition: Convergent sequences
{u } is a convergent sequence with limu = L if and only if for
n n The Greek letter
n→∞
any ε > 0 there exists a least order m∈Z+ such that, for all epsilon, ε, is used
n ≥ m⇒ |u – L| < ε among mathematicians
n
all over the world to
This denition gives an algebraic criterion to test whether or not a
represent a small
given number L is the limit of a sequence. However, to apply this
positive quantity.
test, you must rst decide about the value of L.
Example 
3n 1
Show that the sequence dened by u = is convergent.
n n+1
1.1 *Convergent
y
4 3•n–1
u1(n) =
n+1
3
Graph the sequence and observe its behavior as
2
n increases.
1
0 1 2 3 4 5 6 x
–1
3n 1 3n−1−3n−3 4 Find a simplied expression for |u – 3|
u 3 = −3 = =
n n+1 n+1 n+1 n
4 4 4 The value of m is the least positive integer
u −3 <ε⇒ <ε⇒n+1> ⇒n> −1
n n+1 ε ε 4
greater than −1
ε
So, it is possible to nd m such that
Use the denition to show that limu = 3
n ≥ m⇒ |u – L|< ε n→∞ n
n
∴ lim u = 3
n
n→+∞
Note that this denitions tells you that from the order m onwards,
all the terms of the sequence lie within the interval ]L − ε, L + ε[
This means that the sequence can have exactly one limit, L.
Useful theorems about subsequences
of convergent and divergent sequences: It is usual to use set notation
when describing sequences and
If {b } ⊆ {a } is a subsequence of a convergent

n n their subsequences. For example
sequence {a }, then {b } is also a convergent
n n {b } ⊆ {a } means that the
sequence and limb = lima n n
n n sequence {b } can be obtained
n→∞ n→∞ n
If {b } ⊆ {a } and {c } ⊆ {a } are subsequences of from the sequence {a } by
 n
n n n n
removing at least one of the
a sequence {a } and limb ≠ limc then {a } is
n n n n terms of {a }. In this way, {b } can
n→∞ n→∞
n n
not convergent (i.e. {a } is a divergent sequence).
be seen as a part of {a }.
n n
Chapter 1 5