Math 21 Module (1stAY1920).pdf - University of the Philippines Diliman MATHEMATICS 21 Elementary Analysis I Course Module Institute of Mathematics

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Unformatted text preview: University of the Philippines Diliman MATHEMATICS 21 Elementary Analysis I Course Module Institute of Mathematics MATHEMATICS 21 Elementary Analysis I Course Module Institute of Mathematics University of the Philippines Diliman iv c 2018 by the Institute of Mathematics, University of the Philippines Diliman. All rights reserved. No part of this document may be distributed in any way, shape, or form, without prior written permission from the Institute of Mathematics, University of the Philippines Diliman. Mathematics 21 Module Writers and Editors: UP In s tit ut e of M at h em at ic s Carlo Francisco Adajar Michael Baysauli Katrina Burdeos Lawrence Fabrero Alip Oropeza Contents UP In s tit ut e of M at h em at ic s 1 Limits and Continuity 1.1 Limit of a Function: An Intuitive Approach . . . . . . . . 1.1.1 An Intuitive Approach to Limits . . . . . . . . . . 1.1.2 Evaluating Limits . . . . . . . . . . . . . . . . . . 1.1.3 Other Techniques in Evaluating Limits . . . . . . . 1.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . 1.2 One-Sided Limits . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . 1.3 Limits Involving Infinity . . . . . . . . . . . . . . . . . . . 1.3.1 Infinite Limits . . . . . . . . . . . . . . . . . . . . 1.3.2 Limits at Infinity . . . . . . . . . . . . . . . . . . . 1.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . 1.4 Limit of a Function: The Formal Definition . . . . . . . . 1.4.1 The Formal Definition of Limits . . . . . . . . . . 1.4.2 Proving Limits using the Definition . . . . . . . . . 1.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . 1.5 Continuity of Functions; The Intermediate Value Theorem 1.5.1 Continuity . . . . . . . . . . . . . . . . . . . . . . 1.5.2 The Intermediate Value Theorem . . . . . . . . . . 1.5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . 1.6 Trigonometric Functions: Limits and Continuity; The Squeeze Theorem . . . . . . . . . . . . . . . . . . . . 1.6.1 The Squeeze Theorem . . . . . . . . . . . . . . . . 1.6.2 Continuity of Trigonometric Functions . . . . . . . 1.6.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . 1.7 New Classes of Functions: Limits and Continuity . . . . . 1.7.1 Inverse Functions . . . . . . . . . . . . . . . . . . . 1.7.2 Exponential and Logarithmic Functions . . . . . . 1.7.3 Inverse Circular Functions . . . . . . . . . . . . . . 1.7.4 Hyperbolic Functions . . . . . . . . . . . . . . . . 1.7.5 Inverse Hyperbolic Functions . . . . . . . . . . . . 1.7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 4 6 8 10 16 19 19 24 29 31 31 33 37 38 38 45 48 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 50 54 57 59 59 60 65 68 72 75 vi CONTENTS UP In s tit ut e of M at h em at ic s 2 Derivatives and Differentiation 2.1 Slopes, the Derivative, and Basic Differentiation Rules . . . . . . . . . . . . 2.1.1 The Tangent Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Definition of the Derivative . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Techniques of Differentiation . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Chain Rule, and more on Differentiability . . . . . . . . . . . . . . . . 2.2.1 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Derivatives from the Left and from the Right . . . . . . . . . . . . . 2.2.3 Differentiability and Continuity . . . . . . . . . . . . . . . . . . . . . 2.2.4 Graphical Consequences of Differentiability and Non-differentiability 2.2.5 Higher Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Derivatives of Exponential and Logarithmic Functions . . . . . . . . . . . . 2.3.1 Derivatives of Logarithmic Functions . . . . . . . . . . . . . . . . . . 2.3.2 Logarithmic Differentiation . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Derivatives of Exponential Functions . . . . . . . . . . . . . . . . . . 2.3.4 Derivative of f (x)g(x) , where f (x) > 0 . . . . . . . . . . . . . . . . . 2.3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Derivatives of Other New Classes of Functions . . . . . . . . . . . . . . . . 2.4.1 Derivatives of Inverse Circular Functions . . . . . . . . . . . . . . . . 2.4.2 Derivatives of Hyperbolic Functions . . . . . . . . . . . . . . . . . . 2.4.3 Derivatives of Inverse Hyperbolic Functions . . . . . . . . . . . . . . 2.4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Rolle’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Relative Extrema of a Function . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Relative Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Critical Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Increasing/Decreasing Functions . . . . . . . . . . . . . . . . . . . . 2.6.4 The First Derivative Test for Relative Extrema . . . . . . . . . . . . 2.6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Concavity and the Second Derivative Test . . . . . . . . . . . . . . . . . . . 2.7.1 Concavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Point of Inflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 The Second Derivative Test for Relative Extrema . . . . . . . . . . . 2.7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Graph Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Graphing Polynomial Funtions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 79 79 82 83 84 87 88 88 90 91 93 93 94 97 100 100 102 103 105 106 107 107 108 109 110 111 111 112 114 116 116 117 118 119 122 124 124 125 127 129 131 131 CONTENTS 2.8.2 2.8.3 2.8.4 2.8.5 vii Review of Asymptotes . . . . . . . . . Graphing Rational Functions . . . . . The Graph of f from the Graph of f 0 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UP In s tit ut e of M at h em at ic s 3 Applications of Differentiation 3.1 Absolute Extrema of a Function on an Interval . . . . . . . . . . . . . . . . 3.1.1 Absolute Extrema on Closed and Bounded Intervals . . . . . . . . . 3.1.2 Absolute Extrema On Open Intervals . . . . . . . . . . . . . . . . . 3.1.3 Optimization: Application of Absolute Extrema on Word Problems . 3.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Rates of Change, Rectilinear Motion, and Related Rates . . . . . . . . . . . 3.2.1 Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Rectilinear Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Local Linear Approximation, Differentials, and Marginals . . . . . . . . . . 3.3.1 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Local Linear Approximation and Approximating ∆y . . . . . . . . . 3.3.3 Marginals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Indeterminate Forms and L’Hˆopital’s Rule . . . . . . . . . . . . . . . . . . . ∞ 0 . . . . . . . . . . . . . . . . 3.4.1 Indeterminate Forms of Type and 0 ∞ 3.4.2 L’Hˆ opital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Indeterminate Forms of Type 0 · ∞ and ∞ − ∞ . . . . . . . . . . . . 3.4.4 Indeterminate Forms of Type 1∞ , 00 and ∞0 . . . . . . . . . . . . . 3.4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Integration and Its Applications 4.1 Antidifferentiation and Indefinite Integrals . . . . . . . . . . . . . . . . . 4.1.1 Antiderivatives or Indefinite Integrals . . . . . . . . . . . . . . . 4.1.2 Particular Antiderivatives . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Integration by Substitution . . . . . . . . . . . . . . . . . . . . . 4.1.4 Rectilinear Motion Revisited . . . . . . . . . . . . . . . . . . . . 1 4.1.5 Antiderivatives of f (x) = and of the other Circular Functions x 4.1.6 Antiderivatives of Exponential Functions . . . . . . . . . . . . . 4.1.7 Antiderivatives Yielding the Inverse Circular Functions . . . . . 4.1.8 Antiderivatives of Hyperbolic Functions . . . . . . . . . . . . . . 4.1.9 *Antiderivatives Yielding Inverse Hyperbolic Functions . . . . . 4.1.10 Summary of Antidifferentiation Rules . . . . . . . . . . . . . . . 4.1.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Area of a Plane Region: The Rectangle Method . . . . . . . . . . 4.2.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 136 137 139 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 . 143 . 144 . 146 . 148 . 154 . 157 . 157 . 159 . 162 . 166 . 169 . 169 . 170 . 173 . 176 . 179 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 . 191 . 191 . 194 . 195 . 197 . 198 . 200 . 200 . 203 . 204 . 206 . 207 . 211 . 211 . 215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 180 183 185 187 viii CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . em at ic s . . . . . . . . . . . . . . . at h of M e 4.7 ut 4.6 tit 4.5 In s 4.4 UP 4.3 4.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . The Fundamental Theorem of the Calculus . . . . . . 4.3.1 First Fundamental Theorem of the Calculus . . 4.3.2 The Second Fundamental Theorem of Calculus 4.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . Generalization of the Area of a Plane Region . . . . . 4.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . Arc Length of Plane Curves . . . . . . . . . . . . . . . 4.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . Volumes of Solids . . . . . . . . . . . . . . . . . . . . . 4.6.1 Volumes of Solids of Revolution . . . . . . . . . 4.6.2 Volume of Solids by Slicing . . . . . . . . . . . 4.6.3 Exercises . . . . . . . . . . . . . . . . . . . . . Mean Value Theorem for Integrals . . . . . . . . . . . 4.7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 221 221 224 227 230 245 248 252 254 254 272 276 280 285 Chapter 1 Limits and Continuity 1.1 Limit of a Function: An Intuitive Approach em at ic s We begin this course with an introduction to the core concept needed in studying calculus: the limit of a function. We start studying the notion of limits in an informal, intuitive way. We treat limits using a descriptive, graphical, and numerical approach. We then develop computational methods in evaluating limits of algebraic expressions. at h At the end of this section, the student will be able to: of M • interpret the limit of a function through graphs and tables of values; ut e • compute the limit of polynomial and rational functions using limit theorems; and 1.1.1 4 UP In s tit • evaluate limits of functions using substitution, cancellation of common factors, and rationalization of radical expressions (for indeterminate forms 0/0). An Intuitive Approach to Limits f (x) = 3x − 1 4 3x2 − 4x + 1 g(x) = x−1 ( 4 3 3 3 2 2 2 1 1 1 0 −1 1 2 0 −1 1 2 3 0 −1 h(x) = 1 3x − 1, 0, 2 x 6= 1 x=1 3 Figure 1.1.1: Graphs of y = f (x), y = g(x) and y = h(x) in Illustration 1.1.1. In this subsection, we use graphs of functions in order to develop an intuitive notion of the basic concept of limits. We make a distinction between the value of a function at a real number a and 1 2 CHAPTER 1. LIMITS AND CONTINUITY the function’s behavior for values very near a. A function f may be undefined at a, but it can be described by studying the values of f when x is very close to a, but not equal to a. To illustrate our point, let us consider the following functions: Illustration 1.1.1. 1. Let f (x) = 3x − 1 and consider the tables below. x 0 0.5 0.9 0.99 0.99999 f (x) −1 0.5 1.7 1.97 1.99997 x 2 1.5 1.1 1.001 1.00001 f (x) 5 3.5 2.3 2.003 2.00003 em at ic s In the tables above, we evaluated f at values of x very close to 1. Observe that as the values of x get closer and closer to 1, the values of f (x) get closer and closer to 2. If we continue replacing x with values even closer to 1, the value of f (x) will get even closer to 2. 3x2 − 4x + 1 (3x − 1)(x − 1) = . Note that g(x) is undefined at x = 1. Observe x−1 x−1 though that if x 6= 1, then g(x) = 3x − 1 = f (x). Thus, g is identical to f except only at x = 1. Hence, as in the first item, if x assumes values going closer and closer to 1 but not reaching 1, then the values of g(x) go closer and closer to 2. ( 3x − 1, x 6= 1 3. Let h(x) = . Here, h(1) = 0. If x 6= 1, then h(x) = f (x) and as in above, 0, x=1 h(x) goes closer and closer to 2 as x goes closer and closer to 1. (See Figure 1.1.1 for a comparison of f , g and h.) UP In s tit ut e of M at h 2. Let g(x) = In each of the above examples, we saw that as x got closer and closer to a certain number a, the value of the function approached a particular number. This does not always happen, but in the case that it does, the number to which the function value gets closer and closer is what we will call the limit of the function as x approaches a. Let f be a function defined on some open interval I containing a, except possibly at a. We say that the limit of f (x) as x approaches a is L, where L ∈ , denoted R lim f (x) = L, x→a if we can make f (x) as close to L as we like by taking values of x sufficiently close to a (but not necessarily equal to a). Remark 1.1.2. Alternatively, lim f (x) = L if the values of f (x) get closer and closer to L as x x→a assumes values going closer and closer to a but not reaching a. 1.1. LIMIT OF A FUNCTION: AN INTUITIVE APPROACH 3 Example 1.1.3. Since the value of 3x − 1 goes closer and closer to 2 as x goes closer and closer to 1 as shown in Illustration 1.1.1, we now write lim (3x − 1) = 2. x→1 Remark 1.1.4. Note that in finding the limit of f (x) as x tends to a, we only need to consider values of x that are very close to a but not exactly a. This means that the limit may exist even if f (a) is undefined. 3x2 − 4x + 1 is undefined at x = 1. x−1 However, since x only approaches 1 and is not equal to 1, we conclude that x − 1 6= 0. Hence, Example 1.1.5. In Illustration 1.1.1, we see that g(x) = (3x − 1)(x − 1) = lim (3x − 1) = 2. x→1 x→1 x−1 lim g(x) = lim x→1 s Remark 1.1.6. If lim f (x) and f (a) both exist, their values may not be equal. In other words, it em at ic x→a is possible that f (a) 6= lim f (x). x→a Example 1.1.7. Recall that 3x − 1, x 6= 1 0, x=1 of M h(x) = at h ( from Illustration 1.1.1. Here, h(1) = 0 but lim h(x) = 2. e x→1 In s tit ut Remark 1.1.8. If f (x) does not approach a real number as x tends to a, then we say that the limit of f (x) as x approaches a does not exist (dne). UP Example 1.1.9. Let H(x) be defined by H(x) = 1, x≥0 0, x<0 This function is called the Heaviside step function. The graph of the function is given below: 1 −3 −2 −1 −1 0 1 2 3 From the graph, we see that there is no particular value to which H(x) approaches as x approaches 0. We cannot say that the limit is 0 because if x approaches 0 through values greater than 0, the value of H(x) approaches 1. In the same way, we cannot say that the limit is 1 because if x approaches 0 through values less than 0, the value of H(x) approaches 0. In this case, lim H(x) x→0 does not exist. 4 CHAPTER 1. LIMITS AND CONTINUITY 1.1.2 Evaluating Limits In the previous subsection, we tried to compute the limit of a given function using tables of values. However, this method only gives us an estimate of the limit, not guaranteeing that the value which the function seems to approach is indeed the limit. In this section, we will compute limits not by making tables of values or by graphing, but by applying the theorem below. Theorem 1.1.10. Let f (x) and g(x) be functions defined on some open interval containing a, except possibly at a. 1. If lim f (x) exists, then it is unique. x→a 2. If c ∈ R, then x→a lim c = c. 3. lim x = a x→a 4. Suppose lim f (x) = L1 and lim g(x) = L2 where L1 , L2 ∈ R, and c ∈ R. s x→a em at ic x→a (a) lim [f (x) ± g(x)] = L1 ± L2 x→a (b) lim [cf (x)] = cL1 at h x→a (c) lim [f (x)g(x)] = L1 L2 of M x→a L1 f (x) = , provided that g(x) 6= 0 on some open interval containing a, except g(x) L2 possibly at a, and L2 6= 0. (d) lim x→a p n f (x) = p n L1 for n ∈ UP (f ) lim N. In s x→a tit (e) lim [f (x)]n = (L1 )n for n ∈ ut e x→a N, n > 1 and provided that L1 > 0 when n is even. Example 1.1.11. Determine lim (2x3 − 4x2 + 1). x→−1 Solution. From the theorem above, lim (2x3 − 4x2 + 1) = lim 2x3 − lim 4x2 + lim 1 x→−1 x→−1 x→−1 x→−1 = 2 lim x3 − 4 lim x2 + 1 x→−1 x→−1 = 2(−1)3 − 4(−1)2 + 1 = −5. (x − 3)(x2 − 2) . x→1 x2 + 1 Example 1.1.12. Evaluate lim 1.1. LIMIT OF A FUNCTION: AN INTUITIVE APPROACH 5 Solution. First, note that lim (x2 + 1) = lim x2 + lim 1 = 1 + 1 = 2 6= 0. x→1 x→1 x→1 Using the theorem, lim (x − 3)(x2 − 2) (x − 3)(x2 − 2) lim x→1 x2 + 1 = x→1 lim (x2 + 1) x→1 lim (x − 3) · lim (x2 − 2) = x→1 x→1 = = x→1 lim (x2 + 1) x→1    lim x − lim 3 lim x2 − lim 2 x→1 x→1 x→1 2 lim (x + 1) (1 − 3)(12 x→1 − 2) 2 em at ic s = 1. √ Example 1.1.13. Evaluate: lim at h x...
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