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MATH1113 Mathematical Foundations for Statistics (Calculus)

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MATH1113 Mathematical Foundations for Statistics (Calculus) MATH1113 Mathematical Foundations for Statistics (Calculus) Lilia Ferrario and David Ridout Last modified: October 31, 2012 Contents 1 Functions, Domains and Ranges 1 1.1 Functions ..................................... 1 1.2 DomainsandRanges ............................... 1 1.3 Domains of Standard Functions . .... 3 1.4 Domains of Functions Given by Formulae . ..... 5 1.5 Domains of Piecewise-Defined Functions . ...... 6 1.6 Composition of Functions . 8 1.7 EvenandOddFunctions ............................. 9 2 Bounds for Functions 11 2.1 Bounds ...................................... 11 2.2 Finding Bounds for Functions . 12 2.3 Finding Bounds for Functions of the Form 1/u(x) ................ 15 3 Limits 17 3.1 An Intuitive Approach to Limits . 17 3.2 Rigorouslimits .................................. 19 3.3 FactsaboutLimits................................ 22 3.4 The“SqueezeTheorem” ............................. 24 3.5 AVeryImportantLimit............................. 25 3.6 Infinitelimits ................................... 26 4 Continuity 30 4.1 ContinuityataPoint ............................... 30 4.2 Algebraic Combinations and Continuity . ..... 32 4.3 Limits and Continuity for Compositions . ..... 33 4.4 ContinuityonanInterval . 34 4.5 The Intermediate Value Theorem . 34 4.6 TheMin-MaxTheorem.............................. 36 4.7 Removable Discontinuities . 38 5 Differentiation 39 5.1 The Derivative of a Function at a Point . ...... 39 5.2 Continuity and Differentiability . ....... 41 5.3 One-SidedDerivatives . 42 5.4 DifferentiationRules . 44 5.5 Chain Rule for Differentiating Compositions . ....... 45 5.6 Some Simple Applications . 46 5.7 Differentials, ∆-notation and Linear Approximation . 47 i 5.8 Implicit Differentiation . ..... 49 5.9 SomeImportantTheorems . 52 6 Inverse Functions 57 6.1 MonotonicFunctions .............................. 57 6.2 One-to-One Functions and Inverses . ...... 58 6.3 Properties of Inverse Functions . ...... 60 6.4 Derivatives of Inverse Functions . ...... 63 6.5 Inverses of Trigonometric Functions . ....... 64 6.6 Logarithms and Exponentials . 66 6.7 Logarithmic Differentiation . ...... 70 6.8 HyperbolicFunctions . 71 6.9 The Inverse Hyperbolic Functions . ..... 72 7 Applications of Derivatives 74 7.1 RelatedRates ................................... 74 7.2 l’Hopital’sRulesˆ ................................. 75 7.3 ExtremeValues .................................. 79 7.4 TheFirstDerivativeTest . 80 7.5 Functions Defined on General Intervals . ...... 82 7.6 Concavity and Points of Inflection . 83 7.7 TheSecondDerivativeTest. 84 7.8 SketchingGraphs ................................. 86 7.9 Optimisation ................................... 87 7.10 Roots of Functions: Newton’s Method . ..... 90 8 Integration 94 8.1 IndefiniteIntegrals .............................. 94 8.2 Summation and Σ Notation............................ 95 8.3 AreasandSums.................................. 96 8.4 DefiniteIntegrals ................................ 98 8.5 Implementing the Definition of Definite Integration . ........... 100 8.6 Properties of the Definite Integral . ....... 101 8.7 The Fundamental Theorem of Calculus . 102 8.8 Definite Integration in Practice . 105 8.9 Integrating Piecewise-Continuous Functions . .......... 106 8.10 ImproperIntegrals .............................. 108 9 Integration Techniques 111 9.1 Integration by Substitution . 111 9.2 Substitution and Definite Integrals . ....... 112 9.3 Trigonometric Integrals . 113 9.4 Inverse Trigonometric Substitutions . ........ 114 9.5 IntegrationbyParts .............................. 116 9.6 Integrating Rational Functions . 118 9.7 Further Partial Fraction Decompositions . ......... 121 ii 10 Taylor Series 124 10.1 TaylorPolynomials . 124 10.2 Lagrange’sRemainderTheorem . 125 10.3TaylorSeries ................................... 127 10.4 Standard Examples of Maclaurin Series . ....... 130 10.5 TheBinomialTheorem .............................. 133 11 Differential Equations 136 11.1 Ordinary Differential Equations and their Solutions . .............. 136 11.2 InitialValueProblems. 137 11.3 Separable Differential Equations . ........ 139 11.4 Applications................................... 140 11.5 First Order Linear Differential Equations . ........... 141 12 Functions of Several Variables 143 12.1 DefinitionsandGeometry. 143 12.2 Domains and Subsets of Rn ............................ 146 12.3 LimitsandContinuity. 148 13 Multivariable Differentiation 153 13.1 PartialDerivatives. 153 13.2 Linear Approximations for Functions of Two Variables . ............ 156 13.3 Differentiability for Multivariable Functions . .............. 158 13.4TheChainRule .................................. 159 13.5 Gradients and Directional Derivatives . ......... 162 13.6 Extrema and Optimisation . 165 14 Multiple Integration 172 14.1 Doubleintegrals................................ 172 14.2 Evaluating Double Integrals by Iteration . .......... 174 14.3 AreaIntegrals.................................. 179 14.4 Changing the Order of Integration . 180 14.5 PolarCoordinates................................ 182 iii Chapter 1 Functions, Domains and Ranges 1.1 Functions A function f is a rule (or mapping) that assigns to each element x in a set A one and only one element y in a set B. To fix ideas, think of functions that you have met previously, for example where A = B = R and f (x)= 4x + 3. Here, the symbol R denotes the set of all real numbers. Figures 1.1 and 1.2 illustrate these concepts. When we write an expression y = f (x), we say that y is the dependent variable, x is the independent variable (the argument of f ), and f is the function, mapping or rule. Strictly speaking, a function is a symbol such as f and its value at the point x is f (x). It is common (and convenient!), however, to blur the distinction between the two. Common ways of writing a function include f (x)= x2, f : x x2, or just x x2, 7−→ 7−→ where f , in this example, is the function from R to R that squares its argument. Until further notice, we will only consider functions for which both the independent and dependent variables are real numbers. 1.2 Domains and Ranges The domain of a function f consists of all the real numbers x that f will accept. If D is the domain of the function f , then the set f (D)= f (x) : x D { ∈ } consisting of all the values that f takes is called the range or image of f . Simply put, the range is the set of all numbers that the function can produce if you stick in a number from the domain. Thus, the function f can be seen as some sort of machine (see Figure 1.3), producing an output value f (x) in its range whenever it is fed an input value x from its domain. When a function is given an input value outside its domain, the result is not defined. 1 3 Example 1.1. Let f (x)= 2x + 1 on the domain [0,1]. What is f ( 2 ) and f ( 2 )? 1 1 3 Solution. As 2 is in the domain of f , we compute that f ( 2 )= 2. However, 2 is not in the domain of f , and so f (1.5) is not defined. 1 4 3.2 2.4 1.6 0.8 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 -0.8 Figure 1.1: The curve shown in this figure can be represented by a function, since each element x corresponds to one and only one element y. 2.4 1.6 0.8 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.8 -1.6 -2.4 Figure 1.2: The curve shown in this figure cannot be represented by a function, because a function cannot assign two values to each element x. 2 Figure 1.3: This function machine has the rule “double the number and then add seven”. You could compare this to a soft drink machine: If you put in $1 worth of coins, you get a drink. But, some machines don’t accept 5¢ coins — they are legal tender but are not in the “domain” of the machine. If we neglect to specify the domain for a given function, then it is natural to assume that the domain consists of all the real numbers for which the function is defined. This is called the implicit domain rule. Example 1.2. Let f (x)= 2x + 1 on the domain D =[0,1] (as in Example 1.1). What is the range of f ? Solution. In this case, it suffices to consider the endpoints 0 and 1 of the domain D. Since f (0)= 1 and f (1)= 3, the range of f is R =[1,3]. Example 1.3. Let f (x)= 2x2 + 1 on the domain D =[ 2,1]. What is the range of f ? − Solution. We again consider the endpoints 2 and 1 of the domain D: f ( 2)= 9 and f (1)= 3. However, this does not suffice in this case because− f (0)= 1. In fact, the range− of f is R =[1,9] as you should check (sketch the graph). Example 1.4. Let f (x)= 1 √x. What is the range of f ? − Solution. As the domain has not been specified, we should assume the implicit domain rule and conclude that the natural domain of f is [0,∞), because negative numbers do not have (real) square roots. Since the square root function takes all values in [0,∞), the range of f is R =( ∞,1]. − 1.3 Domains of Standard Functions We give the domains of many standard functions in Table 1.1, noting that these may be helpful for determining the domains of more complicated functions. You probably already know most of these functions — any that you don’t will be introduced later on in the course. 3 Constant functions f (x)= k Defined for all x. The identity function f (x)= x Defined for all x. Powers f (x)= xn If n > 0, defined for all x. (n Z) If n < 0, defined for all x = 0. ∈ 6 Roots f (x)= √n x = x1/n If n > 0 is odd, defined for all x. (n Z) If n > 0 is even, defined for all x > 0. ∈ If n < 0 is odd, defined for all x = 0. 6 If n < 0 is even, defined for all x > 0. Exponentials f (x)= eax Defined for all x and all a. Logarithms f (x)= lnx Defined for all x > 0. f (x)= log x Defined for all x > 0 if b > 0 (and b = 1). b 6 Trig. functions cosx, sinx Defined for all x.
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