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Elements of the Differential and Integral Calculus

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Elements of the Differential and Integral Calculus Elements of the Differential and Integral Calculus (revised edition) William Anthony Granville1 1-15-2008 1With the editorial cooperation of Percey F. Smith. Contents vi Contents 0 Preface xiii 1 Collection of formulas 1 1.1 Formulas for reference . 1 1.2 Greek alphabet . 4 1.3 Rules for signs of the trigonometric functions . 5 1.4 Natural values of the trigonometric functions . 5 1.5 Logarithms of numbers and trigonometric functions . 6 2 Variables and functions 7 2.1 Variables and constants . 7 2.2 Intervalofavariable.......................... 7 2.3 Continuous variation. 8 2.4 Functions................................ 8 2.5 Independent and dependent variables. 8 2.6 Notation of functions . 9 2.7 Values of the independent variable for which a function is defined 10 2.8 Exercises ............................... 11 3 Theory of limits 13 3.1 Limitofavariable .......................... 13 3.2 Division by zero excluded . 15 3.3 Infinitesimals ............................. 15 3.4 The concept of infinity ( ) ..................... 16 ∞ 3.5 Limiting value of a function . 16 3.6 Continuous and discontinuous functions . 17 3.7 Continuity and discontinuity of functions illustrated by their graphs 18 3.8 Fundamental theorems on limits . 25 3.9 Special limiting values . 28 sin x 3.10 Show that limx 0 =1 ..................... 28 → x 3.11 The number e ............................. 30 3.12 Expressions assuming the form ∞ ................. 31 3.13Exercises ...............................∞ 32 vii CONTENTS 4 Differentiation 35 4.1 Introduction.............................. 35 4.2 Increments .............................. 35 4.3 Comparison of increments . 36 4.4 Derivative of a function of one variable . 37 4.5 Symbols for derivatives . 38 4.6 Differentiable functions . 39 4.7 General rule for differentiation . 39 4.8 Exercises ............................... 41 4.9 Applications of the derivative to Geometry . 43 4.10Exercises ............................... 45 5 Rules for differentiating standard elementary forms 49 5.1 Importance of General Rule . 49 5.2 Differentiation of a constant . 51 5.3 Differentiation of a variable with respect to itself . 51 5.4 Differentiationofasum ....................... 52 5.5 Differentiation of the product of a constant and a function . 52 5.6 Differentiation of the product of two functions . 53 5.7 Differentiation of the product of any finite number of functions . 53 5.8 Differentiation of a function with a constant exponent . 54 5.9 Differentiation of a quotient . 54 5.10Examples ............................... 55 5.11 Differentiation of a function of a function . 59 5.12 Differentiation of inverse functions . 60 5.13 Differentiation of a logarithm . 62 5.14 Differentiation of the simple exponential function . 63 5.15 Differentiation of the general exponential function . 64 5.16 Logarithmic differentiation . 65 5.17Examples ............................... 66 5.18 Differentiation of sin v ........................ 69 5.19 Differentiation of cos v ........................ 70 5.20 Differentiation of tan v ........................ 70 5.21 Differentiation of cot v ........................ 71 5.22 Differentiation of sec v ........................ 71 5.23 Differentiation of csc v ........................ 71 5.24 Differentiation of vers v ....................... 72 5.25Exercises ............................... 72 5.26 Differentiation of arcsin v ...................... 75 5.27 Differentiation of arccos v ...................... 76 5.28 Differentiation of arctan v ...................... 78 5.29 Differentiation of arccotu ...................... 79 5.30 Differentiation of arcsecu ...................... 79 5.31 Differentiation of arccsc v ...................... 80 5.32 Differentiation of arcvers v ...................... 82 5.33Example................................ 82 viii CONTENTS 5.34 Implicit functions . 86 5.35 Differentiation of implicit functions . 86 5.36Exercises ............................... 87 5.37 Miscellaneous Exercises . 88 6 Simple applications of the derivative 91 6.1 Directionofacurve ......................... 91 6.2 Exercises ............................... 94 6.3 Equations of tangent and normal lines . 96 6.4 Exercises ............................... 98 6.5 Parametric equations of a curve . 102 6.6 Exercises ...............................106 6.7 Angle between the radius vector and tangent . 108 6.8 Lengths of polar subtangent and polar subnormal . 112 6.9 Examples ...............................113 6.10 Solution of equations having multiple roots . 115 6.11Examples ............................... 116 6.12 Applications of the derivative in mechanics . 117 6.13 Component velocities. Curvilinear motion. 119 6.14 Acceleration. Rectilinear motion. 120 6.15 Component accelerations. Curvilinear motion. 120 6.16Examples ............................... 121 6.17 Application: Newton’s method . 126 6.17.1 Description of the method . 126 6.17.2 Analysis............................ 127 6.17.3 Fractals ............................ 127 7 Successive differentiation 129 7.1 Definition of successive derivatives . 129 7.2 Notation................................ 129 7.3 The n-thderivative.......................... 130 7.4 Leibnitz’s Formula for the n-th derivative of a product . 130 7.5 Successive differentiation of implicit functions . 132 7.6 Exercises ...............................133 8 Maxima, minima and inflection points. 137 8.1 Introduction.............................. 137 8.2 Increasing and decreasing functions . 141 8.3 Tests for determining when a function is increasing or decreasing 143 8.4 Maximum and minimum values of a function . 144 8.5 Examining a function for extremal values: first method . 146 8.6 Examining a function for extremal values: second method . 147 8.7 Problems ...............................150 8.8 Points of inflection . 165 8.9 Examples ...............................166 8.10Curvetracing ............................. 167 ix CONTENTS 8.11Exercises ............................... 168 9 Differentials 173 9.1 Introduction.............................. 173 9.2 Definitions............................... 173 9.3 Infinitesimals ............................. 175 9.4 Derivative of the arc in rectangular coordinates . 176 9.5 Derivative of the arc in polar coordinates . 177 9.6 Exercises ...............................179 9.7 Formulas for finding the differentials of functions . 180 9.8 Successive differentials . 181 9.9 Examples ...............................182 10 Rates 185 10.1 The derivative considered as the ratio of two rates . 185 10.2Exercises ............................... 186 11 Change of variable 193 11.1 Interchange of dependent and independent variables . 193 11.2 Change of the dependent variable . 194 11.3 Change of the independent variable . 195 11.4 Simultaneous change of both independent and dependent variables196 11.5Exercises ............................... 198 12 Curvature; radius of curvature 201 12.1Curvature ............................... 201 12.2 Curvatureofacircle . 201 12.3 Curvatureatapoint . 202 12.4 Formulas for curvature . 203 12.5 Radius of curvature . 206 12.6 Circleofcurvature .......................... 208 12.7Exercises ............................... 210 13 Theorem of mean value; indeterminant forms 215 13.1Rolle’sTheorem ........................... 215 13.2 The Mean-value Theorem . 216 13.3 The Extended Mean Value Theorem . 218 13.4Exercises ............................... 219 13.5 Application: Using Taylor’s Theorem to Approximate Functions. 219 13.6 Example/Application: Finite Difference Schemes . 224 13.7 Application: L’Hospital’s Rule . 226 13.8Exercises ............................... 230 14 References 233 x CONTENTS xii Chapter 0 Preface Figure 1: Sir Isaac Newton. That teachers and students of the Calculus have shown such a generous appre- ciation of Granville’s “Elements of the Differential and Integral Calculus” has been very gratifying to the author. In the last few years considerable progress has been made in the teaching of the elements of the Calculus, and in this revised edition of Granville’s “Calculus” the latest and best methods are exhib- ited,methods that have stood the test of actual classroom work. Those features of the first edition which contributed so much to its usefulness and popularity have been retained. The introductory matter has been cut down somewhat in order to get down to the real business of the Calculus sooner. As this is designed essentially for a drill book, the pedagogic principle that each result should be made intuitionally as well as analytically evident to the student has been kept constantly in mind. The object is not to teach the student to rely on his intuition, but, in some cases, to use this faculty in advance of analytical xiii investigation. Graphical illustration has been drawn on very liberally. This Calculus is based on the method of limits and is divided into two main parts,Differential Calculus and Integral Calculus. As special features, attention may be called to the effort to make perfectly clear the nature and extent of each new theorem, the large number of carefully graded exercises, and the summa- rizing into working rules of the methods of solving problems. In the Integral Calculus the notion of integration over a plane area has been much enlarged upon, and integration as the limit of a summation is constantly emphasized. The existence of the limit e has been assumed and its approximate value calcu- lated from its graph. A large number of new examples have been added, both with and without answers. At the end of almost every chapter will be found a collection
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