2020_T1_1151_calcNotes_Final.pdf - MATH1131 Mathematics 1A and MATH1141 Higher Mathematics 1A CALCULUS NOTES as additional resource for MATH1151 CRICOS

2020_T1_1151_calcNotes_Final.pdf - MATH1131 Mathematics 1A...

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Unformatted text preview: MATH1131 Mathematics 1A and MATH1141 Higher Mathematics 1A CALCULUS NOTES as additional resource for MATH1151 CRICOS Provider No: 00098G c 2020 School of Mathematics and Statistics, UNSW Sydney iii 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. c Copyright is vested in The University of New South Wales, 2020. iv c 2020 School of Mathematics and Statistics, UNSW Sydney CONTENTS v Contents Preface iii Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Syllabus for MATH1151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Problem schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Revision questions xi 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 p f (x)/g(x)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→∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 27 27 29 30 30 31 31 2.3 Proving that lim f (x) = L using the limit definition . . . . . . . . . . . . . . . . . . 34 2.4 2.5 Proofs of basic limit results (MATH1141 only) . . Limits of functions at a point . . . . . . . . . . . . 2.5.1 Left-hand, right-hand and two-sided limits . 2.5.2 Limits and continuous functions . . . . . . 2.5.3 Rules for limits at a point . . . . . . . . . . Maple notes . . . . . . . . . . . . . . . . . . . . . . 2.6 x→∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 38 38 40 42 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 51 53 54 56 Problems for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4 Differentiable functions 4.1 Gradients of tangents and derivatives . . . . . . 4.2 Rules for differentiation . . . . . . . . . . . . . 4.3 Proofs of results in Section 4.2 . . . . . . . . . 4.4 Implicit differentiation . . . . . . . . . . . . . . 4.5 Differentiation, continuity and split functions . 4.6 Derivatives and function approximation . . . . 4.7 Derivatives and rates of change . . . . . . . . . 4.8 Local maximum, local minimum and stationary 4.9 Maple notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 65 69 71 72 74 77 78 79 81 Problems for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5 The 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 mean value theorem and applications The mean value theorem . . . . . . . . . . . . . . . Proof of the mean value theorem . . . . . . . . . . Proving inequalities using the mean value theorem Error bounds . . . . . . . . . . . . . . . . . . . . . The sign of a derivative . . . . . . . . . . . . . . . The second derivative and applications . . . . . . . Critical points, maxima and minima . . . . . . . . Counting zeros . . . . . . . . . . . . . . . . . . . . Antiderivatives . . . . . . . . . . . . . . . . . . . . L’Hˆopital’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 85 87 88 90 91 93 96 98 100 103 Problems for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6 Inverse functions 6.1 Some preliminary examples . . . 6.2 One-to-one functions . . . . . . . 6.3 Inverse functions . . . . . . . . . 6.4 The inverse function theorem . . 6.5 Applications to the trigonometric 6.6 Maple notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 . 111 . 113 . 115 . 118 . 119 . 124 Problems for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 7 Curve sketching 7.1 Curves defined by a Cartesian equation 7.1.1 A checklist for sketching curves . 7.1.2 Oblique asymptotes . . . . . . . 7.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c 2020 School of Mathematics and Statistics, UNSW Sydney . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 129 130 130 133 vii 7.2 7.3 7.4 Parametrically defined curves . . . . . . . . . . 7.2.1 Parametrisation of conic sections . . . . 7.2.2 Calculus and parametric curves . . . . . 7.2.3 The cycloid and curve of fastest descent Curves defined by polar coordinates . . . . . . 7.3.1 Polar coordinates . . . . . . . . . . . . . 7.3.2 Basic sketches of polar curves . . . . . . 7.3.3 Sketching polar curves using calculus . . Maple notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 137 139 140 142 142 144 146 149 Problems for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 8 Integration 8.1 Area and the Riemann Integral . . . . . . . . . . . . . . . . . . 8.1.1 Area of regions with curved boundaries . . . . . . . . . 8.1.2 Approximations of area using Riemann sums . . . . . . 8.1.3 The definition of area under the graph of a function and 8.2 Integration using Riemann sums . . . . . . . . . . . . . . . . . 8.3 The Riemann integral and signed area . . . . . . . . . . . . . . 8.4 Basic properties of the Riemann integral . . . . . . . . . . . . . 8.5 The first fundamental theorem of calculus . . . . . . . . . . . . 8.6 The second fundamental theorem of calculus . . . . . . . . . . 8.7 Indefinite integrals . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Integration by substitution . . . . . . . . . . . . . . . . . . . . 8.9 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . 8.10 Improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . 8.11 Comparison tests for improper integrals . . . . . . . . . . . . . 8.12 Functions defined by an integral . . . . . . . . . . . . . . . . . 8.13 Maple notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 . . . . . . . . . . . . 155 . . . . . . . . . . . . 155 . . . . . . . . . . . . 158 the Riemann integral161 . . . . . . . . . . . . 163 . . . . . . . . . . . . 166 . . . . . . . . . . . . 167 . . . . . . . . . . . . 169 . . . . . . . . . . . . 173 . . . . . . . . . . . . 176 . . . . . . . . . . . . 178 . . . . . . . . . . . . 182 . . . . . . . . . . . . 184 . . . . . . . . . . . . 187 . . . . . . . . . . . . 193 . . . . . . . . . . . . 196 Problems for Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9 The 9.1 9.2 9.3 9.4 9.5 9.6 9.7 logarithmic and exponential functions Powers and logarithms . . . . . . . . . . . . . The natural logarithm function . . . . . . . . The exponential function . . . . . . . . . . . Exponentials and logarithms with other bases Integration and the ln function . . . . . . . . Logarithmic differentiation . . . . . . . . . . . Indeterminate forms with powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 204 205 207 209 211 212 213 Problems for Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 10 The 10.1 10.2 10.3 10.4 10.5 hyperbolic functions Hyperbolic sine and cosine functions Other hyperbolic functions . . . . . Hyperbolic identities . . . . . . . . . Hyperbolic derivatives and integrals The inverse hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c 2020 School of Mathematics and Statistics, UNSW Sydney . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 217 221 222 224 225 viii 10.6 10.7 10.8 10.9 Integration leading to the inverse hyperbolic functions A summary of important hyperbolic formulae . . . . . (Appendix):The catenary . . . . . . . . . . . . . . . . Maple notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 230 231 234 Problems for Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Answers to selected Chapter 1 . . . . . Chapter 2 . . . . . Chapter 3 . . . . . Chapter 4 . . . . . Chapter 5 . . . . . Chapter 6 . . . . . Chapter 7 . . . . . Chapter 8 . . . . . Chapter 9 . . . . . Chapter 10 . . . . problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 237 238 239 239 240 241 242 243 244 245 246 c 2020 School of Mathematics and Statistics, UNSW Sydney ix CALCULUS SYLLABUS FOR MATH1151 The calculus syllabus for MATH1151 assumes that students are very familiar with the mathematics contained in the NSW HSC Extension 1 course. In particular, it assumes that all students are familiar with the calculus of the exponential and log functions. Whereas the algebra strand of the course contains many results of an algorithmic nature, the calculus strand emphasises an approach to mathematics of a more abstract and conceptual kind. This emphasis is designed to help you cope with more advanced mathematics that you will likely meet in later years. The times given for the various topics are approximate only. A detailed syllabus and lecture schedule will be uploaded to Moodle. PROBLEM SETS The problems in the MATH1151 Calculus Problems booklet come in three varieties: really challenging problems, marked with **; slightly harder than normal questions, marked with * and standard level questions with no additional markings at all. All students should make sure that they attempt and can do these standard questions and make serious attempts at the * and ** questions. Remember that working through a wide range of problems is the key to success in mathematics. PROBLEM SCHEDULE The main reason for having tutorials is to give you a chance to get help with problems which you find difficult and with parts of the lectures or textbook which you don’t understand. To get real benefit from tutorials, you need to try the relevant problems before the tutorial so that you can find out the areas in which you need help. A problem schedule will be uploaded to Moodle. c 2020 School of Mathematics and Statistics, UNSW Sydney x c 2020 School of Mathematics and Statistics, UNSW Sydney xi Revision questions Inequalities and Absolute Values 1. Sketch the set of points (x, y) which satisfy the following relations. a) 0 ≤ y ≤ 2x and 0≤x≤2 b) y/2 ≤ x ≤ 2 and 0≤y≤4 2. Solve a) x(x − 1) > 0 3. Solve a) x + 1 < 3 b) b) Trigonometry (x − 1)(x − 2) < 0 x + 2 > 3 c) > − 21 3x + 2 < 1 c) 4. Find the exact value of each of the following:     π 5π 7π a) cos b) sin c) tan 12 12 12 5. If A and B are acute with sin(A) = 1 x sec  11π 12 cos(A) b) tan(A) c) sin(B) d) cos(B) e) sin(A + B) f) cos(A − B) g) sin(2A) h) tan(2B) 6. If A and B are acute with sin(A) = cos(2A) b) sin(A − B) > 1 2  12 3 and tan(B) = find (without the use of a calculator): 5 5 a) a) 1 1−x x−1 x+1 < 1 d) d) d) c) 24 8 and cos(B) = find (without finding A and B): 25 17 tan(A + B) 7. Find the period and amplitude for each of the following functions. x π   π b) y = −2 cos a) y = 3 sin 2x − + 4 3 2 c 2020 School of Mathematics and Statistics, UNSW Sydney xii 8. Express each of the following in terms of a single sine function in the form R sin(x ± α), where R > 0 and α is acute. a) c) sin(x) + cos(x) √ 3 sin(x) − cos(x) √ 2 sin(x) + 2 3 cos(x) √ √ 8 sin(x) − 8 cos(x) b) d) Functions 9. What is the (maximal) domain √ a) f (x) = 5 − x2 b) √ c) f (x) = 1 − 2 sin x d) √ f) e) f (x) = x − 1 √ g) f (x) = sin x h) i) f (x) = 1 + tan2 x and range of the following functions? √ f (x) = x2 − 5 f (x) = (x − 8)−1/3 f (x) = √1 x−1 x cos √ f (x) = 1−x |x| if if if x<0 0≤x≤1 x>1 10. Sketch the graph of each of the functions in Problem 9. 11. Sketch each of the following functions without using calculus. a) An odd function, f (x), defined on [−2, 2] such that f (x) = x2 (1 − x) when 0 ≤ x ≤ 2. b) An even function, f (x), defined on [−3, 3] such that f (x) = (x − 1)2 (x − 2) when 0 ≤ x ≤ 3. 12. If f (x) = x + 5 and g(x) = x2 − 3 find a) g(f (0)) b) g(f (x)) c) 13. If f (x) = x − 1 and g(x) = √ a) f (x) + g(x) b) f (g(2)) d) f (g(x)) 1 , give the explicit forms of x−1 f (x)g(x) c) f (x) g(x) d) f (g(x)) Limits of some Rational Functions 14. Find a) d) x−2 x→2 x2 − 5x + 6 1 − x4 lim x→1 1 − x lim b) e) x2 − 5x + 6 x→2 2x2 − 3x − 2 2x2 − 3x + 7 lim x→∞ 3x2 + x − 1 lim c) f) λ2 − 0.8λ − 0.2 λ→1 λ−1 3 2x + 3x + 2 lim x→∞ −5x3 + 4x − 1 lim c 2020 School of Mathematics and Statistics, UNSW Sydney xiii Simple Differentiation 15. Find the derivative of each of the following functions. √ a) f (x) = (2x + 5)3 b) g(t) = t2 − 4 c) h(x) = d) f (x) = sin3 x g) f (x) = e−x j) f (x) = x cos 2x x+e f (x) = x+π sin x f (x) = 2x + 5 m) p) 2 /2 1 (2x + 3)3/2 e) g(x) = cos(x3 ) f) h(x) = sec(2x2 + 3) h) g(x) = x2 (2x − 1)4 i) h(θ) = θ tan θ k) g(x) = x3 sin x 2x2 + 3 g(x) = 3x − 2 l) h(x) = x ln x t h(t) = √ 2 t −4 n) o) Tangents and Normals 16. Find the equation of the tangent and the equation of the normal to each of the following curves. 1 a) y = 4x + at the point (1, 5) x 1 b) y = x3 − 1 + 2 at the point (1, 1) x cos x π c) y = at the point where x = 1 − sin x 6 Stationary Points 17. Locate and identify the stationary points for x 1+x...
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