Math 137 Calculus 1 for Honours Mathematics Course Notes Barbara A. Forrest and Brian E. Forrest Version 1.61 Copyright c Barbara A. Forrest and Brian E. Forrest. All rights reserved. August 1, 2021 All rights, including copyright and images in the content of these course notes, are owned by the course authors Barbara Forrest and Brian Forrest. By accessing these course notes, you agree that you may only use the content for your own personal, non-commercial use. You are not permitted to copy, transmit, adapt, or change in any way the content of these course notes for any other purpose whatsoever without the prior written permission of the course authors. Author Contact Information: Barbara Forrest ([email protected]) Brian Forrest ([email protected]) i QUICK REFERENCE PAGE 1 Right Angle Trigonometry opposite ad jacent opposite sin θ = hypotenuse cos θ = hypotenuse tan θ = ad jacent 1 1 1 csc θ = sin θ sec θ = cos θ cot θ = tan θ Radians Definition of Sine and Cosine The angle θ in For any θ, cos θ and sin θ are radians equals the defined to be the x− and y− length of the directed coordinates of the point P on the arc BP, taken positive unit circle such that the radius counter-clockwise and OP makes an angle of θ radians negative clockwise. with the positive x− axis. Thus Thus, π radians = 180◦ sin θ = AP, and cos θ = OA. 180 or 1 rad = π . The Unit Circle ii QUICK REFERENCE PAGE 2 Trigonometric Identities Pythagorean cos2 θ + sin2 θ = 1 Identity Range −1 ≤ cos θ ≤ 1 −1 ≤ sin θ ≤ 1 Periodicity cos(θ ± 2π) = cos θ sin(θ ± 2π) = sin θ Symmetry cos(−θ) = cos θ sin(−θ) = − sin θ Sum and Difference Identities cos(A + B) = cos A cos B − sin A sin B cos(A − B) = cos A cos B + sin A sin B sin(A + B) = sin A cos B + cos A sin B sin(A − B) = sin A cos B − cos A sin B Complementary Angle Identities π − cos( 2 A) = sin A π − sin( 2 A) = cos A Double-Angle cos 2A = cos2 A − sin2 A Identities sin 2A = 2 sin A cos A 2 1+cos 2θ Half-Angle cos θ = 2 2 1−cos 2θ Identities sin θ = 2 Other 1 + tan2 A = sec2 A iii QUICK REFERENCE PAGE 3 f (x) = x2 f (x) = x3 f (x) = jxj f (x) = cos(x) f (x) = sin(x) f (x) = tan(x) f (x) = sec(x) f (x) = csc(x) f (x) = cot(x) 1 1 f (x) = ex f (x) = ln(x) iv QUICK REFERENCE PAGE 4 Table of Antiderivatives Differentiation Rules R xn+1 xn dx = + C Function Derivative n + 1 R 1 a 2 R 0 a−1 dx = ln(j x j) + C f (x) = cx , a , 0, c f (x) = cax x R f (x) = sin(x) f 0(x) = cos(x) ex dx = ex + C R f (x) = cos(x) f 0(x) = − sin(x) sin(x) dx = − cos(x) + C R f (x) = tan(x) f 0(x) = sec2(x) cos(x) dx = sin(x) + C R f (x) = sec(x) f 0(x) = sec(x) tan(x) sec2(x) dx = tan(x) + C 1 R 1 0 dx = arctan(x) + C f (x) = arcsin(x) f (x) = p 2 1 − x2 1 + x 1 f (x) = arccos(x) f 0(x) = − p R 1 2 p dx = arcsin(x) + C 1 − x 1 − x2 0 1 f (x) = arctan(x) f (x) = R −1 1 + x2 p dx = arccos(x) + C − x2 f (x) = ex f 0(x) = ex 1 R sec(x) tan(x) dx = sec(x) + C f (x) = ax with a > 0 f 0(x) = ax ln(a) x 1 R x a f (x) = ln(x) for x > 0 f 0(x) = a dx = + C x ln(a) n-th degree Taylor polynomial for f centered at x = a n (k) P f (a) − k Tn;a(x) = k! (x a) k=0 00 (n) 0 − f (a) − 2 ··· f (a) − n = f (a) + f (a)(x a) + 2! (x a) + + n! (x a) Linear Approximations (L0(x)) and Taylor Polynomials (Tn;0(x)) x 0 0 0 f (x) = e L0(x) = T1;0(x) = f (0) + f (0)(x − 0) = e + e (x) = 1 + x 00 0 f (0) 2 0 0 e0 2 x2 T2;0(x) = f (0) + f (0)(x − 0) + 2! (x − 0) = e + e (x) + 2! (x − 0) = 1 + x + 2 x2 x3 T3;0(x) = 1 + x + 2 + 6 x2 x3 x4 T4;0(x) = 1 + x + 2 + 6 + 24 f (x) = sin(x) L0(x) = T1;0(x) = x T2;0(x) = x x3 T3;0(x) = x − 6 x3 T4;0(x) = x − 6 f (x) = cos(x) L0(x) = T1;0(x) = 1 x2 T2;0(x) = 1 − 2 x2 T3;0(x) = 1 − 2 x2 x4 T4;0(x) = 1 − 2 + 24 v Table of Contents Page 1 Sequences and Convergence 1 1.1 Absolute Values ........................... 1 1.1.1 Inequalities Involving Absolute Values ........... 3 1.2 Sequences and Their Limits ..................... 7 1.2.1 Introduction to Sequences ................. 7 1.2.2 Recursively Defined Sequences .............. 10 1.2.3 Subsequences and Tails .................. 16 1.2.4 Limits of Sequences ..................... 17 1.2.5 Divergence to ±∞ ...................... 27 1.2.6 Arithmetic for Limits of Sequences ............. 28 1.3 Squeeze Theorem .......................... 37 1.4 Monotone Convergence Theorem ................. 39 1.5 Introduction to Series ........................ 45 1.5.1 Geometric Series ...................... 49 1.5.2 Divergence Test ....................... 51 2 Limits and Continuity 56 2.1 Introduction to Limits for Functions ................. 56 2.2 Sequential Characterization of Limits ................ 66 2.3 Arithmetic Rules for Limits of Functions .............. 70 2.4 One-sided Limits ........................... 76 2.5 The Squeeze Theorem ....................... 78 2.6 The Fundamental Trigonometric Limit ............... 82 2.7 Limits at Infinity and Asymptotes .................. 86 2.7.1 Asymptotes and Limits at Infinity .............. 87 2.7.2 Fundamental Log Limit ................... 93 2.7.3 Vertical Asymptotes and Infinite Limits ........... 98 2.8 Continuity ............................... 103 2.8.1 Types of Discontinuities ................... 105 2.8.2 Continuity of Polynomials, sin(x), cos(x), ex and ln(x)... 107 2.8.3 Arithmetic Rules for Continuous Functions ........ 110 2.8.4 Continuity on an Interval .................. 113 2.9 Intermediate Value Theorem .................... 115 2.9.1 Approximate Solutions of Equations ............ 119 2.9.2 The Bisection Method .................... 123 2.10 Extreme Value Theorem ....................... 126 2.11 Curve Sketching: Part 1 ....................... 130 vi 3 Derivatives 133 3.1 Instantaneous Velocity ........................ 133 3.2 Definition of the Derivative ..................... 135 3.2.1 The Tangent Line ...................... 137 3.2.2 Differentiability versus Continuity .............. 138 3.3 The Derivative Function ....................... 141 3.4 Derivatives of Elementary Functions ................ 143 3.4.1 The Derivative of sin(x) and cos(x)............. 145 3.4.2 The Derivative of ex ..................... 147 3.5 Tangent Lines and Linear Approximation .............. 150 3.5.1 The Error in Linear Approximation ............. 154 3.5.2 Applications of Linear Approximation ............ 157 3.6 Newton’s Method ........................... 161 3.7 Arithmetic Rules of Differentiation ................. 166 3.8 The Chain Rule ........................... 170 3.9 Derivatives of Other Trigonometric Functions ........... 175 3.10 Derivatives of Inverse Functions .................. 177 3.11 Derivatives of Inverse Trigonometric Functions .......... 183 3.12 Implicit Differentiation ........................ 189 3.13 Local Extrema ............................ 196 3.13.1 The Local Extrema Theorem ................ 199 3.14 Related Rates ............................ 202 4 The Mean Value Theorem 208 4.1 The Mean Value Theorem ...................... 208 4.2 Applications of the Mean Value Theorem ............. 213 4.2.1 Antiderivatives ........................ 213 4.2.2 Increasing Function Theorem ................ 218 4.2.3 Functions with Bounded Derivatives ............ 221 4.2.4 Comparing Functions Using Their Derivatives ....... 223 4.2.5 Interpreting the Second Derivative ............. 226 4.2.6 Formal Definition of Concavity ............... 228 4.2.7 Classifying Critical Points: The First and Second Derivative Tests ........... 232 4.2.8 Finding Maxima and Minima on [a; b]........... 236 4.3 L’Hoˆpital’s Rule ............................ 239 4.4 Curve Sketching: Part 2 ....................... 247 5 Taylor Polynomials and Taylor’s Theorem 259 5.1 Introduction to Taylor Polynomials and Approximation ....... 259 5.2 Taylor’s Theorem and Errors in Approximations .......... 271 5.3 Big-O ................................. 279 5.3.1 Calculating Taylor Polynomials ............... 286 vii Chapter 1 Sequences and Convergence It is often the case that in order to solve complex mathematical problems we must first replace the problem with a simpler version for which we have appropriate tools to find a solution. In doing so our solution to the simplified problem may not work for the original question, but it may be close enough to provide us with useful in- formation. Alternatively, we may be able to design an algorithm that will generate successive approximate solutions to the full problem in such a manner that if we ap- ply the process enough times, the result will eventually be as close as we would like to the actual solution. For example, there is no algebraic method to solve the equation ex = x + 2: However, we can graphically show that there are two distinct solutions for this equa- tion and that the two solutions are close to -2 and 1, respectively. One process we could use to solve this equation is a type of binary search algorithm that is based on the fact that the function f (x) = ex − (x + 2) is continuous. We could also use an al- ternate process which relies on the very useful fact that if a function is differentiable at a point x = a, then its tangent line is a very good approximation to the graph of a function near x = a.