Top-down Calculus. S. Gill Williamson

Transcription

1 Top-down Calculus Chapter 2 Computing Derivatives S. Gill Williamson Gill Williamson Home Page CC Google books

2

3 Preface This chapter, Chapter 2 of Top-down Calculus, is devoted to developing the technical skills needed to compute derivatives and understand what has been computed. Software systems have been developed to compute derivatives and express the answers in terms of standard functions (when possible). Some of these products are available for free on the web. Others are for sale, sometimes with special discounts to students. Check them out, but these programs are no substitute for understanding what your are doing. Top-down Calculus was developed in the 1980 s for a summer session program to train high school teachers in San Diego County to teach calculus. These teachers had all taken calculus themselves, but they were wary of standing before a class and fielding questions math anxiety of the second kind. They knew about Newton, Leibniz, the falling apple, etc. What they didn t know was how to respond quickly when a student asked, Hey teacher, how do I work this one? My approach in this book is to emphasize intuition and technique. The important chain rule is presented intuitively on page 11 (instead of page 100+ as in many standard calculus books). Exercises are presented as follows: prototypical exercise set, solutions and discussion, numerous exercise sets that are variations on the prototypes. My students (the teachers) were encouraged to work one or two of the variation exercise sets in detail and then scan the remaining variations noting the techniques required for each problem. The idea was that this scanning process would prepare them to deal with their own students questions. They would be able to say with some confidence, Well, Johnny, why don t you try this approach. At least this would buy time for them to think about the question more carefully. Subsequent to the summer session program for high school teachers, I used this material for the one-quarter calculus course that I regularly taught in the Department of Mathematics at the University of California, San Diego. It seemed to work well for that purpose, but it is no competitor for the magnificent (but very expensive) standard calculus books. I also used this material for calculus taught in summer session. There, the concise nature of this material worked very well. iii

4 iv

5 Table of Contents Chapter 2 Enlarging the class of functions you can differentiate...39 Proving that (x r ) = rx r 1 for any real number r The quotient rule...42 Summary: chain rule, product rule, quotient rule...44 Derivative of sin(x) is cos(x)...45 The number e...49 Exponential and logarithmic derivatives...47 Graphical properties of a x and log a (x)...50 Derivatives of a x and log a (x)...52 Examples of computing deriviatives...55 Derivatives of trigonometric functions...57 Derivatives of the loglog functions...58 Derivative of x x...59 Higher order derivatives...60 Logarithmic differentiation...62 Implicit differentiation...64 Basic logarithmic facts...65 Basic trigonometric facts...66 Exercises Summary of trig and log differentiation rules...68 Solutions to Exercise Variations on Exercises Hyperbolic and inverse trigonometric functions...86 Differentiation rules for inverse trig functions...93 Hyperbolic functions...95 Index...Index 1 v

6 vi

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66 INDEX Chapter 2 chain rule, 44 compositional inverse, 49, 86, 89 derivative of absolute value, 58 differential notation, 42 domain of function, 86 exponential and logarithmic derivatives, 55 exponential function, 47, 53 function terminology, 86 functional inverse, 89 graph of a function defined, 87 hyperbolic function, 95 image of a function, 87 implicit differentation, 64 injective function, 88 inverse hyperbolic functions, 97 inverse trigonometric functions, 86, 92, 93 logarithmic functions, 51, 54 logarithmic, exponential summary, 65, 68 number e, 49 natural or base e logarithm, 52 one-to-one function, 88 onto function, 87 product rule, 44, 62 quotient rule, 42, 44 radians, 45 range of function, 86 rational, irrational numbers, 41 Sally, 50, 89 secant and cosecant functions, 56, 57 surjective function, 87 tangent and cotangent functions, 57 trig and log differentiation rules, 68 trigonmometric derivatives, 46, 68 trigonmometric identities, 66 Index 1