MATH1131 1141 Calculus Notes 2020T1 PDF

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MATH1131 Mathematics 1AandMATH1141 Higher Mathematics 1ACALCULUS NOTESCRICOS Provider No: 00098G ©c2020 School of Mathematics and Statistics, UNSW SydneyivvContentsPreface iii Calculus Syllabus.................................. ....... ix1 Sets, inequalities and functions 1 1 Sets of numbers...........

Description

MATH1131 Mathematics 1A and

MATH1141 Higher Mathematics 1A

CALCULUS NOTES CRICOS Provider No: 00098G

c 2020 School of Mathematics and Statistics, UNSW Sydney

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Preface Please read carefully. These Notes form the basis for the calculus strand of MATH1131 and MATH1141. However, not all of the material in these Notes is included in the MATH1131 or MATH1141 calculus syllabuses. A detailed syllabus will be uploaded to Moodle. In using these Notes, you should remember the following points: 1. Most courses at university present new material at a faster pace than you will have been accustomed to in high school, so it is essential that you start working right from the beginning of the session and continue to work steadily throughout the session. Make every effort to keep up with the lectures and to do problems relevant to the current lectures. 2. These Notes are not intended to be a substitute for attending lectures or tutorials. The lectures will expand on the material in the notes and help you to understand it. 3. These Notes may seem to contain a lot of material but not all of this material is equally important. One aim of the lectures will be to give you a clearer idea of the relative importance of the topics covered in the Notes. 4. Use the tutorials for the purpose for which they are intended, that is, to ask questions about both the theory and the problems being covered in the current lectures. 5. Some of the material in these Notes is more difficult than the rest. This extra material is marked with the symbol [H]. Material marked with an [X] is intended for students in MATH1141. 6. Some of the problems are marked [V]. These have a video solution available from Moodle. 7. It is essential for you to do problems which are given at the end of each chapter. If you find that you do not have time to attempt all of the problems, you should at least attempt a representative selection of them. You will find advice about this on Moodle. You should also work through the Online Tutorals that you will find on Moodle. 8. You will be expected to use the computer algebra package Maple in tests and understand Maple syntax and output for the end of term examination. Note. This version of the Calculus Notes has been prepared by Robert Taggart and Peter Brown. They build on notes first developed by Tony Dooley and subsequently edited by several members of the School of Mathematics and Statistics. The main editors include Mike Banner, Ian Doust and V. Jeyakumar. Copyright is vested in The University of New South Wales, c2020.

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Contents Preface iii Calculus Syllabus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Sets, inequalities and functions 1.1 Sets of numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Solving inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Absolute values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Polynomials and rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 The trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 The elementary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Implicitly defined functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Maple notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 7 9 13 14 17 18 21 22

Problems for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2 Limits 2.1 Limits of functions at infinity . . . . . . . . 2.1.1 Basic rules for limits . . . . . . . . . 2.1.2 The pinching theorem . . . . . . . . 2.1.3 Limits of the form f (x)/g (x) . . . . p p 2.1.4 Limits of the form f (x) − g (x) . 2.1.5 Indeterminate forms . . . . . . . . . 2.2 The definition of lim f (x) . . . . . . . . . . x→∞

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Proving that lim f (x) = L using the limit definition . . . . . . . . . . . . . . . . . . 34 x→∞

2.4 2.5

Proofs of basic limit results (MATH1141 only) . . . . . . . . . . . . . . . . . . . . . 37 Limits of functions at a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.5.1 Left-hand, right-hand and two-sided limits . . . . . . . . . . . . . . . . . . . . 38 2.5.2 Limits and continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.5.3 Rules for limits at a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.6 Maple notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Problems for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 c 2020 School of Mathematics and Statistics, UNSW Sydney

vi 3 Properties of continuous functions 3.1 Combining continuous functions . . 3.2 Continuity on intervals . . . . . . . 3.3 The intermediate value theorem . . 3.4 The maximum-minimum theorem .

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Problems for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4 Differentiable functions 63 4.1 Gradients of tangents and derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 Rules for differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3 Proofs of results in Section 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.4 Implicit differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.5 Differentiation, continuity and split functions . . . . . . . . . . . . . . . . . . . . . . 74 4.6 Derivatives and function approximation . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.7 Derivatives and rates of change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.8 Local maximum, local minimum and stationary points . . . . . . . . . . . . . . . . . 79 4.9 Maple notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Problems for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5 The mean value theorem and applications 85 5.1 The mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2 Proof of the mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.3 Proving inequalities using the mean value theorem . . . . . . . . . . . . . . . . . . . 88 5.4 Error bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.5 The sign of a derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.6 The second derivative and applications . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.7 Critical points, maxima and minima . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.8 Counting zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.9 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.10 L’Hˆopital’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Problems for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6 Inverse functions 111 6.1 Some preliminary examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.2 One-to-one functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.3 Inverse functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.4 The inverse function theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.5 Applications to the trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . 120 6.6 Maple notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Problems for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7 Curve sketching 131 7.1 Curves defined by a Cartesian equation . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.1.1 A checklist for sketching curves . . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.1.2 Oblique asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 c 2020 School of Mathematics and Statistics, UNSW Sydney

vii 7.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Parametrically defined curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.2.1 Parametrisation of conic sections . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.2.2 Calculus and parametric curves . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.2.3 The cycloid and curve of fastest descent . . . . . . . . . . . . . . . . . . . . . 142 7.3 Curves defined by polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 7.3.1 Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 7.3.2 Basic sketches of polar curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.3.3 Sketching polar curves using calculus . . . . . . . . . . . . . . . . . . . . . . . 148 7.4 Maple notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.2

Problems for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 8 Integration 157 8.1 Area and the Riemann Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.1.1 Area of regions with curved boundaries . . . . . . . . . . . . . . . . . . . . . 157 8.1.2 Approximations of area using Riemann sums . . . . . . . . . . . . . . . . . . 160 8.1.3 The definition of area under the graph of a function and the Riemann integral163 8.2 Integration using Riemann sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 8.3 The Riemann integral and signed area . . . . . . . . . . . . . . . . . . . . . . . . . . 168 8.4 Basic properties of the Riemann integral . . . . . . . . . . . . . . . . . . . . . . . . . 169 8.5 The first fundamental theorem of calculus . . . . . . . . . . . . . . . . . . . . . . . . 171 8.6 The second fundamental theorem of calculus . . . . . . . . . . . . . . . . . . . . . . 175 8.7 Indefinite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 8.8 Integration by substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 8.9 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 8.10 Improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 8.11 Comparison tests for improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . 189 8.12 Functions defined by an integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 8.13 Maple notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Problems for Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 9 The 9.1 9.2 9.3 9.4 9.5 9.6 9.7

logarithmic and exponential functions 207 Powers and logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 The natural logarithm function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 The exponential function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Exponentials and logarithms with other bases . . . . . . . . . . . . . . . . . . . . . . 214 Integration and the ln function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Logarithmic differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Indeterminate forms with powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Problems for Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 10 The 10.1 10.2 10.3

hyperbolic functions 223 Hyperbolic sine and cosine functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Other hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Hyperbolic identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 c 2020 School of Mathematics and Statistics, UNSW Sydney

viii 10.4 10.5 10.6 10.7 10.8 10.9

Hyperbolic derivatives and integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 The inverse hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Integration leading to the inverse hyperbolic functions . . . . . . . . . . . . . . . . . 234 A summary of important hyperbolic formulae . . . . . . . . . . . . . . . . . . . . . . 236 (Appendix):The catenary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Maple notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

Problems for Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Answers to selected Chapter 1 . . . . . Chapter 2 . . . . . Chapter 3 . . . . . Chapter 4 . . . . . Chapter 5 . . . . . Chapter 6 . . . . . Chapter 7 . . . . . Chapter 8 . . . . . Chapter 9 . . . . . Chapter 10 . . . .

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Index

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243 . 243 . 244 . 245 . 245 . 246 . 247 . 248 . 249 . 250 . 251 252

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CALCULUS SYLLABUS The calculus course for both MATH1131 and MATH1141 is based on these MATH1131/MATH1141 Calculus Notes that are included in the Course Pack. A detailed syllabus and lecture schedule will be posted on Moodle. The computer package Maple will be used in the calculus course. An introduction to Maple is included in the booklet titled First Year Maple Notes. CALCULUS PROBLEM SETS The Calculus problems are located at the end of each chapter of the Calculus Notes booklet. They are also available from the course module on the UNSW Moodle server. Some of the problems are very easy, some are less easy but still routine and some are quite hard. To help you decide which problems to try first, each problem is marked with an [R], an [H] or an [X]. The problems marked [R] form a basic set of problems which you should try first. Problems marked [H] are harder and can be left until you have done the problems marked [R]. Problems marked [V] have a video solution available on Moodle. You do need to make an attempt at the [H] problems because problems of this type will occur on tests and in the exam. If you have difficulty with the [H] problems, ask for help in your tutorial. The problems marked [X] are intended for students in MATH1141 – they relate to topics which are only covered in MATH1141. Extra problem sheets for MATH1141 may be issued in lectures. Remember that working through a wide range of problems is the key to success in mathematics. However, solving problems and writing mathematics clearly are two separate skills that need to be developed through practice. We recommend that you keep a workbook to practice writing solutions to mathematical problems.

c 2020 School of Mathematics and Statistics, UNSW Sydney

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c 2020 School of Mathematics and Statistics, UNSW Sydney

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Chapter 1

Sets, inequalities and functions In the days of the Roman empire, the word ‘calculus’ denoted a pebble that was used for counting and gambling. As time progressed, the word ‘calculare’ came to mean ‘to compute.’ In the second half of the seventeenth century, two mathematicians, the Englishman Isaac Newton and the German Gottfried Leibniz, independently invented methods for • calculating gradients of tangents to curves, • calculating instantaneous acceleration and velocity, • calculating the area of regions with a curved boundary, • calculating the volume of solids with curved boundary, • calculating the length of a curve, • calculating the work done by a force, and • calculating the centre of mass of a general solid. These methods were developed by combining algebra, geometry and trigonometry with the limiting process and became known as the calculus. Calculus (as it is known today) has many applications to engineering, physics, chemistry, biology, geology, surveying, sociology, economics and statistics. It includes two major branches: the differential calculus (introduced in Chapter 4) and the integral calculus (introduced in Chapter 8). These two branches are related by the fundamental theorem of calculus (see Theorem 8.5.1). The underlying tool used to develop these branches is the concept of the limit (introduced in Chapter 2), which is applied to functions (introduced in Chapter 1). The main goal of this chapter is to introduce functions. Although functions will be familiar to students from high school, important points that deserve students’ attention are emphasised here. Since the functions we study in this course take real numbers as inputs and give real numbers of outputs, we begin the chapter with a review of different sets that consist of real numbers. We also devote a little time to revising inequalities and absolute values, in preparation for our discussion of limits in Chapter 2.

1.1

Sets of numbers

(Ref: SH10 §1.2) c 2020 School of Mathematics and Statistics, UNSW Sydney

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CHAPTER 1. SETS, INEQUALITIES AND FUNCTIONS...