MATH-412-TUMANDAY Report in Mat-Math.pptx

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1. EXPONENTIAL AND LOGARITHMIC FUNCTIONS AND INVERSE TRIGONOMETRIC FUNCTIONS MICHELLE C. TUMANDAY, LPT

2. EXPONENTIAL FUNCTIONS OBJECTIVES: Recognize and evaluate exponential functions with base a Graph exponential functions and use the One-to-One property Recognize ,evaluate, and graph exponential functions with base e. Use exponential functions to model and solve real-life problems.

3. Why is it important? : The exponential function is very important in math because it is used to model many real life situations. For example: population growth and decay, compound interest, economics, and more.

4. EXPONENTIAL FUNCTIONS f(x) =   Where a > 0, a ≠1 and x is any real number BASE EXPONENT

5. EVALUATIONG EXPONENTIAL FUNCTIONS FUNCTION VALUE f(x) = x = -3.1 f(x) = x = f(x) = x = FUNCTION VALUE x = -3.1

6. FUNCTION VALUE CALCULATOR KEYSTROKES DISPLAY f(x) = 2^(-)3.1 ENTER 0.1166291 f( ) = 2^(-) ENTER 0.1133147 f( ) = .6^(3 2) ENTER 0.4647580 FUNCTION VALUE CALCULATOR KEYSTROKES DISPLAY 2^(-)3.1 ENTER 0.1166291 0.1133147 0.4647580

7. GRAPHS OF EXPONENTIAL FUNCTIONS Example 1. To graph f(x) = and g(x ) = , begin by constructing a table of values.   Graphs of y =   x -3 -2 -1 1 2 1 2 4 1 4 16 x -3 -2 -1 1 2 1 2 4 1 4 16

8. f(x) =   g(x) =   GRAPHS OF EXPONENTIAL FUNCTIONS

9. Graphs of y = Example 2. The graph of f(x ) = and g(x ) =   Table of Values x -1 1 2 3 2 1 4 Table of Values x -1 1 2 3 2 1 4

10. GRAPHS OF EXPONENTIAL FUNCTIONS f(x) =   g(x) =  

11. Using the One-to-one property Example: 9 = = 2= x+1 x=1   Your turn! = 8 2. 8= 3. = 27  

12. solution = 8 3. = 27 = = -x = 3 -x = 3 x = -3 x = - 3 8 = = 3=2x-1 2x=4 x = 2  

13. Using the One-to-one property Your turn! Use the one-to-one property to solve the equation for x. 1. =27 2. 243 = 3. = 32 4. =  

14. solution 1. =27 = x + 1 = 3 x = 3   3. = 32 = -x = 5 x = -5   2. 243 = = 2x = 5 x =   4. = = x -2 = -3 x = -1  

15. The N atural Base e f(x) =  

16. Evaluating the Natural Exponentia l Function Use a calculator to evaluate the function f(x) = at each value of x x= -2 x =0.25 x = -1 x =-0.3  

17. Function Value Calculator keystrokes Display a. f(-2) = (-) 2 ENTER 0.1353353 b. f ( 0.25 ) = 0.25 ENTER 1.2840254 c. f(-1) = (-) 1 ENTER 0.3678794 d. f(- 0.3 ) = (-) 0.3 ENTER 0.7408182 Function Value Calculator keystrokes Display 0.1353353 1.2840254 0.3678794 0.7408182

18. Application One of the most familiar examples of exponential growth is an investment earning continuously compound interest The formula for interest compounded in times per year is A=P   Balance in the account principal Number of compoundings per year time

19. COMPOUND INTEREST For n compounding per year: A=P 2. For continuous compounding: A=P  

20. Application You invest ₱12,000 at annual interest rate of 3%. Find the balance after 5 years for each type of compounding. Quarterly Monthly Continuously

21. Application a. Quarterly: A=P = 12,000   b. Monthly: A=P = 12,000   Continuous compounding: A=P = 12,000 13,942.40  

22. A pplication You invest ₱6000 at annual rate of 4%.find the balance after 7 years for each type of compounding. Quarterly Monthly continuously Your turn!

23. solution a. Quarterly: A=P = 6 ,000   b . Monthly: A=P = 6 ,000   Continuous compounding: A=P = 6 ,000 ,938.78  

24. Complete the table by finding the balance A when P dollars is invested at rate r for t years, compounded n times per year. n 1 2 4 12 365 continuous A P = $1500, r =2%, t = 10 years P = $2500 , r =3.5%, t = 10 years P = $2500 , r =4%, t = 20 years P = $ 1000 , r =6%, t = 40 years

25. ANSWERS n 1 2 4 12 365 continuous $1500 $1,828.5 $1,830.3 $1,831.2 $ 1,831.8 $ 1,832.1 $ 1,832.1 $2,500 $3,526.5 $3,537 $3,542.3 $3,545.9 $3,547.6 $3,547.7 $2,500 $5,477.8 $5,520.1 $5,541.8 $5,556.5 $5,563.6 $5,563.9 $1,000 $10,285.7 $10,640.9 $10,828.5 $10,957.5 $11,021 $11,023.2

26. LOGARITHMIC FUNCTIONS recognize and evaluate logarithmic functions with base a graph logarithmic functions Recognize, evaluate, and graph natural logarithmic functions Use logarithmic functions to model and solve real-life problems

27. Why is it important? Logarithmic functions are used in finance and economics to model exponential growth or decay. For example, compound interest, which is a fundamental concept in finance, involves the use of logarithmic functions.

28. LOGARITHMIC FUNCTIONS For x > 0, a>0, and a 1, y = log a x if and only if x = . The function f(x) = log a x is called logarithmic function with base a.  

29. EVALUATING LOGARITHMS

30. EVALUATING LOGARITHMS Evaluate each logarithm at the given value of x. f(x) = log 2 x ; x = 32 f(32) = log 2 32=? This is asking for exponent. What exponent do you put on the base of 2 to get 32? f(32) = log 2 32 = 5 because = 32  

31. EVALUATING LOGARITHMS Solve for x:log 6 x = 2 Rewrite the problem in exponential form = x x = 36  

32. EVALUATING LOGARITHMS Solve for y:log 5 = y log 5 = y = y = -2  

33. EVALUATING LOGARITHMS Evaluate each logarithm at the given value of x. f(x ) = log 4 x; x=2 f(x) = log 3 x; x=1 f(x) = log 10 x; x=   Your turn!

34. solution 2. f(x) = log 3 x; x=1 f(1) = log 3 1 = 0 = 1   1. f(x ) = log 4 x; x=2 f(2) = log 4 2 = 3.f(x ) = log 10 x; x= f( )= log 10 = -10 =  

35. Properties of logarithm log a 1 = 0 because = 1. log a a = 1 because = a. log a = = x. (inverse properties) If log a x = log a y, then x = y (One-to-One property )  

36. Using properties of logarithms To simplify log 4 1, use property 1 to write log 4 1 = 0 To simplify log 3 x = log 3 12, use the One-to-One Property. log 3 x = log 3 12 x = 12

37. To solve log (2x+1) = log 3x for x, use the One-to-One Property log ( 2x+1 ) = log 3x 2x+1 = 3x 1 = x x = 1

38. Use the one-to-one property to solve the equation for x. log 5 (x+1) = log 5 6 l og 11 = log ( +7) log 2 (x-3) = log 2 9 log ( +6x) = log 27   Your turn!

39. solution log 5 (x+1) = log 5 6 x + 1 = 6; x = 5 log 11 = log ( +7 ) 11 = +7 = 4 x = 2   log 2 (x-3) = log 2 9 x – 3 = 9 x = 6 4. log ( +6x) = log 27 + 6x = 27 + 6x - 27 = 0 x = 3; x = -9  

40. Graphs of logarithmic functions h(x) = 2 + log x f (x) =log x

41. Natural logarithmic functions f(x) = x = ln x, x > 0 Use calculator to evaluate the function f(x) = ln x at x = 2   Function value Calculator keystrokes display f(2) = ln 2 Ln 2 ENTER 0.6931472

42. INVERSE TRIGONOMETRIC FUNCTIONS Evaluate and graph the inverse sine function Evaluate and graph other inverse trigonometric functions

43. Inverse of sine function y = sin x Sin x has an inverse function on this interval

44. Inverse of sine function

45. Inverse of sine function On the interval [- the function y = sin x is increasing On the interval [- the function y = sin x takes on its full range of values, -1 On the interval [- the function y = sin x is one-to-one.  

46. Inverse of sine function y = arcsin if and only if sin y = x Where -1 The domain of y = arcsin x is [-1,1], and the range is [ , ]  

47. Evaluating the inverse Sine Function If possible, find the exact value of each expression Arcsin 2  

48. solution You know that sin(- ) = - and - lies in [- ], so arcsin = - ( angle whose sine is ) You know that sin = and lies in [- ], so = ( angle whose sine is )  

49. solution c. It is not possible to evaluate when x = 2 because there is no angle whose sine is 2. Remember that the domain of the inverse sine function is [-1,1]  

50. Graphing the arcsin function By definition, the equations y = arcsin x and sin y = x are equivalent for . So, their graph are the same. From the interval [ ], assign values to y in the equation sin y = x to make a table of values.   y x = sin y -1 - - 1 y x = sin y -1 1

51. y = arcsin x

52. Other Inverse Trigonometric Functions y = cos x Cos x has an inverse function on this interval

53. Inverse C osine F unction y = arccos x or y =  

54. Inverse Cosine Function

55. Inverse Tangent Function

56. Inverse Tangent Function

57. Graphs of three inverse trigonometric function y = arcsin x y = arccos x y = arctan x

58. Definitions of the Inverse Trigonometric Functions FUNCTION DOMAIN RANGE y= arcsinx if and only if sin y = x -1 1 y= arccos x if and only if cos y = x -1 1 y= arctan x if and only if tan y = x Definitions of the Inverse Trigonometric Functions FUNCTION DOMAIN RANGE y= arcsinx if and only if sin y = x y= arccos x if and only if cos y = x y= arctan x if and only if tan y = x

59. Find the exact value of each expression: arccos a rctan ) a rctan  

60. solution arccos This represents the angle whose cosine is ​​. Since cosine is positive in the first and fourth quadrants, we're looking for an angle in those quadrants whose cosine is ​​. This angle is in radians .  

61. b. This represents the angle whose cosine is -1. Cosine is -1 at π radians ( 180 degrees) because it's the angle where cosine is at its minimum value . c. arctan This represents the angle whose tangent is 0. Tangent is 0 at 0 radians ( 0 degrees) because it's where the sine is 0.  

62. d. This represents the angle whose tangent is -1. Tangent is -1 at − radians (-45 degrees) because it's in the fourth quadrant where tangent is negative and it forms a 45-45-90 triangle.  

63. e . ) This represents the angle whose cosine is ​​. Cosine is negative in the second and third quadrants. The angle in the second quadrant with as its cosine is  

64. f. a rctan This represents the angle whose tangent is ​. Tangent is at ​ radians (60 degrees) because it's where the tangent function reaches its maximum value in the first quadrant.  

65. Integrals Involving Exponential and Logarithmic Functions

66. RULE: INTEGRAL OF EXPONENTIAL FUNCTIONS = e x + C = + C  

67. Examples 1. Find the . = 2 = 2e x + C 2. Find the ∫ dx . ∫ dx = 3∫ dx = 3 ln /x/ + C  

68. Finding an Antiderivative of an Exponential Function Find the antiderivative of the exponential function e -x solution: U se substitution: Setting u = -x, and then du = -1dx. Multiply the du equation by -1, so you now have –du = dx. Then, = - = - e u + C = - e u + C = - e -x + C  

69. Find the antiderivative of the exponential function   Your turn!

70. Find the antiderivative of the exponential function First rewrite the problem using a rational exponent = ∫ e x (1+e x ) 1/2 u = 1+e x ; du = e x dx . ∫e x (1+e x ) 1/2 dx = ∫u 1/2 du  

71. ∫u 1/2 du = +C = + C = + C  

72. Integrals Involving Logarithmic Functions Integrating functions of the form  f ( x )= or  f ( x ) = result in the absolute value of the natural log function  

73. Rule: The Basic Integral Resulting in the natural Logarithmic Function ∫ dx = ln|x |+ C ∫ dx=ln|u(x )|+C  

74. EXAMPLES 1.Find the antiderivative of . ∫ dx = ln|x+2 |+ C 2. Find the antiderivative of ∫ dx=3 ∫ dx =3 ∫ =3ln|u|+ C = 3ln|x−10|+C,x≠10.