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ESSENTIAL CALCULUS CH05 Inverse functions

ESSENTIAL CALCULUS CH05 Inverse functions. In this Chapter:. 5.1 Inverse Functions 5.2 The Natural Logarithmic Function 5.3 The Natural Exponential Function 5.4 General Logarithmic and Exponential Functions 5.5 Exponential Growth and Decay 5.6 Inverse Trigonometric Functions

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ESSENTIAL CALCULUS CH05 Inverse functions

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  1. ESSENTIAL CALCULUSCH05 Inverse functions

  2. In this Chapter: • 5.1 Inverse Functions • 5.2 The Natural Logarithmic Function • 5.3 The Natural Exponential Function • 5.4 General Logarithmic and Exponential Functions • 5.5 Exponential Growth and Decay • 5.6 Inverse Trigonometric Functions • 5.7 Hyperbolic Functions • 5.8 Indeterminate Forms and 1’Hospital’s Rule Review

  3. Chapter 5, 5.1, P246

  4. Chapter 5, 5.1, P246

  5. ▓In the language of inputs and outputs, Definition 1 says that f is one-to-one if each output corresponds to only one input. FIGURE 2 This function is not one-to-one because f(x1)=f(x2). Chapter 5, 5.1, P247

  6. FIGURE 3 ƒ=(x)=x2 is one-to-one. Chapter 5, 5.1, P247

  7. FIGURE 4 g(x)=x2 is not one-to-one. Chapter 5, 5.1, P247

  8. 1. DEFINITION A function f is called a one-to-one function if it never takes on the same value twice; that is, F(x1)≠f(x2) whenever x1≠x2 Chapter 5, 5.1, P247

  9. HORIZONTAL LINE TEST A function is one-to-one if and only if no horizontal line intersects its graph more than once. Chapter 5, 5.1, P247

  10. FIGURE 6 The inverse function reverses inputs and outputs. Chapter 5, 5.1, P248

  11. domain of f-1=range of f range of f-1=domain of f Chapter 5, 5.1, P248

  12. Do not mistake the -1 in f-1 for an exponent. Thus f-1(x) does not mean Chapter 5, 5.1, P248

  13. Chapter 5, 5.1, P248

  14. f-1(f(x))=x for every x in A f(f-1(x))=x for every x in B Chapter 5, 5.1, P248

  15. Chapter 5, 5.1, P249

  16. 5.HOW TO FIND THE INVERSE FUNCTION OF A ONE-TO-ONE FUNCTION f STEP 1 Write =f(x). STEP 2 Solve this equation for x in terms of y (if possible). STEP 3 To express f-1 as a function of x, interchange x and y. The resulting equation is y=f-1(x). Chapter 5, 5.1, P249

  17. Chapter 5, 5.1, P250

  18. Chapter 5, 5.1, P250

  19. The graph of f-1 is obtained by reflecting the graph of f about the line y=x. Chapter 5, 5.1, P250

  20. Chapter 5, 5.1, P250

  21. 6.THEOREM If f is a one-to-one continuous function defined on an interval, then its inverse function f-1 is also continuous. Chapter 5, 5.1, P251

  22. Chapter 5, 5.1, P251

  23. 7. THEOREM If f is a one-to-one differentiable function with inverse function f-1 and f’(f-1(a))≠0, then the inverse function is differentiable at a and Chapter 5, 5.1, P251

  24. Chapter 5, 5.1, P253

  25. 18. The graph of f is given. (a) Why is f one-to-one? (b) What are the domain and range of f-1? (c) What is the value of f-1(2)? (d) Estimate the value of f-1(0) . Chapter 5, 5.1, P253

  26. 29–30 ■ Use the given graph of f to sketch the graph of f-1. Chapter 5, 5.1, P253

  27. Chapter 5, 5.2, P254

  28. Chapter 5, 5.2, P254

  29. Chapter 5, 5.2, P254

  30. 1.DEFINITION The natural logarithmic function is the function defined by ln x>0 Chapter 5, 5.2, P254

  31. Chapter 5, 5.2, P255

  32. 3.LAWS OF LOGARITHMS If x and y are positive numbers and r is a rational number, then 1. ln(xy)=ln x+ ln y 2. ln( )=ln x-ln y 3. ln (xr)=r ln x Chapter 5, 5.2, P255

  33. Chapter 5, 5.2, P256

  34. (a) ln x=∞ (b) ln x=-∞ Chapter 5, 5.2, P256

  35. 5. DEFINITION eis the number such that . ln e=1. Chapter 5, 5.2, P257

  36. Chapter 5, 5.2, P257

  37. Chapter 5, 5.2, P257

  38. Chapter 5, 5.2, P258

  39. Chapter 5, 5.2, P259

  40. Chapter 5, 5.2, P260

  41. STEPS IN LOGARITHMIC IFFERENTIATION • Take natural logarithms of both sides of an • equation y=f(x) and use the Laws of Logarithms to simplify. • 2. Differentiate implicitly with respect to x. • 3. Solve the resulting equation for y’. Chapter 5, 5.2, P260

  42. Chapter 5, 5.3, P262

  43. Chapter 5, 5.3, P262

  44. and Chapter 5, 5.3, P262

  45. Chapter 5, 5.3, P263

  46. Chapter 5, 5.3, P263

  47. x>0 Chapter 5, 5.3, P263

  48. for all x Chapter 5, 5.3, P263

  49. Chapter 5, 5.3, P264

  50. 6. PROPERTIES OF THE NATURAL EXPONENTIAL FUNCTION The exponential function f(x)=ex is an increasing continuous function with domain R and range(0,∞) . Thus, ex>0 for all x. Also So the x-axis is a horizontal asymptote of f(x)=ex Chapter 5, 5.3, P264

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