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Practice Makes Perfect Calculus
Dr. William Clark, Sandra McCune
英文版
Contents
Preface ix
I LIMITS 1
1 The limit concept 3
Limit definition and intuition 3
Properties of limits 4
2 Special limits 7
Zero denominator limits 7
Infinite limits and limits involving infinity 8
Left-hand and right-hand limits 9
3 Continuity 11
Definition of continuity 11
Properties of continuity 12
Intermediate Value Theorem (IVT) 13
II DIFFERENTIATION 15
4 Definition of the derivative and derivatives
of some simple functions 17
Definition of the derivative 17
Derivative of a constant function 18
Derivative of a linear function 19
Derivative of a power function 19
Numerical derivatives 20
5 Rules of differentiation 23
Constant multiple of a function rule 23
Rule for sums and differences 24
Product rule 25
Quotient rule 26
Chain rule 28
Implicit differentiation 29
6 Additional derivatives 33
Derivative of the natural exponential function ex 33
Derivative of the natural logarithmic function lnx 34
Derivatives of exponential functions for bases other than e 34
Derivatives of logarithmic functions for bases other than e 35
Derivatives of trigonometric functions 36
Derivatives of inverse trigonometric functions 37
Higher-order derivatives 39
III INTEGRATION 41
7 Indefinite integral and basic integration
formulas and rules 43
Antiderivatives and the indefinite integral 43
Integration of constant functions 44
Integration of power functions 45
Integration of exponential functions 46
Integration of derivatives of trigonometric functions 47
Integration of derivatives of inverse trigonometric functions 48
Two useful integration rules 49
8 Basic integration techniques 53
Integration by substitution 53
Integration by parts 55
Integration by using tables of integral formulas 57
9 The definite integral 61
Definition of the definite integral and the
First Fundamental Theorem of Calculus 61
Useful properties of the definite integral 62
Second Fundamental Theorem of Calculus 64
Mean Value Theorem for Integrals 65
IV APPLICATIONS OF THE DERIVATIVE
AND THE DEFINITE INTEGRAL 67
10 Applications of the derivative 69
Slope of the tangent line at a point 69
Instantaneous rate of change 70
Differentiability and continuity 72
Increasing and decreasing functions, extrema, and critical points 73
Concavity and points of inflection 77
Mean Value Theorem 79
11 Applications of the definite integral 83
Area of a region under one curve 83
Area of a region between two curves 84
Length of an arc 86
Appendix A: Basic functions and their graphs 89
Appendix B: Basic differentiation formulas and rules 97
Appendix C: Integral formulas 99
Answer key 103
Worked solutions 117
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