Chapter 4 Exponential and Logarithmic Functions

1. Chapter 4Exponential and Logarithmic Functions

2. General Exponential Function • Definition: If a > 0 and a≠1, then the generalexponential functionwithbase a is given by f(x)=ax. • Example:

3. Properties of Exponents • Let a and b be positive numbers.

4. Example • Sketch the graph of the following exponential functions. (a) (b) (c)

5. general exponential functionf(x)=ax具有下列的特性： 1.domain為(-,)。 2.它的圖形必通過點 (0,1)。 3.若 a>1，則函數為遞增，稱為exponential growth function。 4.若 0 < a<1，則函數為遞減，稱為exponential decay function。 ；若 ， 則函數在 上為遞減。

6. 若 a>1，則函數為遞增，稱為exponential growth function。 4.若 0 < a<1，則函數為遞減，稱為exponential decay function。

7. 所謂利息(interest)，就是將錢存入銀行，經雙方約定每年按所存金額，即本金(principal)，乘以固定的比率，即年利率(annual interest rate)，結算而得。即 每年利息 = 本金 × 年利率 • 我們依各期末所衍生的利息是否併入下期之本金去衍生利息，來區分單利法(simple interest)及複利法(compound interest)。

8. 單利 • 所謂單利法，即每期利息之計算，都是以最初儲蓄的本金來計算，而它的利息不併入下期的本金去衍生利息。故 • 設P為本金，年利率為 r，若依單利法以每年分 n期計息，則 t年末的利息 I為

9. 複利 • 所謂複利法，即計算每期末之利息，是以前一期末所得之本利和作為下一期的本金。 • 設P為本金、年利率為 r，若依複利法以每年分 n期計息，則t年末之本利和為

10. 連續複利 • 若令P為本金r為年利率，當時間間距繼續縮短，則一年後本金和會趨近於 令x=r/n 令

11. Limit Definition of e • The irrational number e is defined to be limit of as . That is , • e 2.71828182846

12. Example • The graph of exponential functionf(x) = ex.

13. Exercise 4.1 • 9,12,17

14. 4.2 Logarithmic Functions

15. one-to-one function • A function is one-to-one if, for elements a and b in the domain of f,

16. Inverse function • Let f and g are two one-to-one functions such that f(g(x))=x for each x in the domain of g and g(f(x))=x for each x in the domain of f. • The function g is the inverse function of f. • The function g is denoted by f -1.

17. Example function inverse function f(x)=2x

18. Logarithmic Function(對數函數) • If a > 0 and a≠1, the Logarithm of x to the base ais the functiony = logax, is defined as logax = y if and only if ay = x. • Domain oflogax : (0,∞) • TheCommon logarithm (常用對數) is defined as log10x = b if and only if 10b = x.

19. Properties of logarithms • Product Rule: • Quotient Rule: • Power Rule: • Inverse Properties: • General Properties: • One-to-one Property:

20. Natural Logarithmic Function • Thenatural logarithm(自然對數) ln xis defined as ln x = logex. • ln x is read as “the natural log of x”. Logarithmic form Exponential form

22. Inverse Properties of Logarithms and Exponents. Example:

23. Properties of Logarithms • ln xy =ln x +ln y

24. Example • Assume x > 0 and y > 0.

25. Example • Assume x > 0 and y > 0.

26. Example • Solve the following equations.

27. Example • 一筆存款以連續複利計算。如果六年後倍增，則年利率為何。 • Solution: • 連續複利計算的公式為 A = Pert，其中P為原存款總額； A為t年後存款總額；則r表年利率。 • 2P = Per(6) e6r = 2 • lne6r = ln 2  6r= ln 2 • So

28. Example • 一筆存款以連續複利計算。若年利率為r，則幾年後倍增。 • Solution: • 連續複利計算的公式為 A = Pert，其中P為原存款總額； A為t年後存款總額；則r表年利率。 • 2P = Pert ert = 2 • lnert = ln 2  rt= ln 2 • So

29. Exercise 4.2 • 19,22,27,32,35,40,41,42,45

30. 4.3  Derivatives of Exponential Functions

31. Derivatives of Natural Exponential Functions Let u be a differentiable function of x.

32. Example • Differentiate the following functions.

33. Example • Find the slope of the tangent line of at the points (0,1) and (1.e). • Solution: • f ’(x)=ex • The slope of the tangent line at the points (0,1) is f ’(0)= e0= 1 • The slope of the tangent line at the points (1,e) is f ’(1)= e1 = e

34. Example • Differentiate the following functions.

35. 練習 • Differentiate the following functions.

36. Example • Differentiate the following functions.

37. Other Bases and Differentiation • Let u be a differentiable function of x.

38. Exercise 4.3 • 7,10,11,17,20,25,28,30,31,34,35,39

39. 4.4  Derivatives of Logarithmic Functions

40. Derivative of the Natural Logarithmic Function Let u be a differentiable function of x. 1. 2.

41. Example • Find the derivative of f (x) = ln 2x • Solution:

42. Example • Find the derivatives of the functions.

43. Example • Find the derivative of • Solution:

44. Example • Find the derivative of • Solution:

45. Example • Find the derivative of • Solution:

46. 練習 • Find the derivative of • Solution:

47. Change of base formula Example:

48. Other Bases and Differentiation • Let u be a differentiable function of x.

49. Example • Find the derivative of • Solution:

50. Exercise 4.4 • 7,11,12,14,19,24,27,30,31,36,38,39,43,48,57