1. MATH 31 LESSONS Chapter 8: Exponential and Log Functions Exponential Functions

2. Laws of Exponents

3. If the bases are the same... Keep the base the same and add the exponents

4. Similarly,

6. Multiply the exponents

7. Similarly,

8. Note: This works only for multiplication and division. It does NOT work for addition or subtraction.

10. Anything raised to the zeroeth power is equal to 1.

12. Similarly,

14. The root is on the bottom.

15. Sketching Exponential Functions Exponential functions are of the form where b is a positive constant

16. Case 1: b > 1 “Exponential Growth”

18. = 

21. = 0 HA: y = 0

22. Case 2: 0 < b < 1 “Exponential Decay”

24. = 0 HA: y = 0

27. = 

28. Ex. 1 (include asymptote) Try this question yourself first. Then, check the answer on the following pages.

29. Basic function: (decay)

30. Reflect about the x-axis

31. Translate down 4 units HA: y =  4 44

32. Ex. 2Evaluate: Try this question yourself first. Then, check the answer on the following pages.

34. Test a number that is on the left side of 2 (i.e. x = 1.9999) = 0

35. Derivatives of Exponential Functions Using first principles,

36. Using the exponential law:

38. Mathematicians could not determine this directly. So, they chose an indirect path. But what is?

39. Mathematicians realized that if this would lead to a simple derivative:

40. Finding the Ideal Base Mathematicians began to search for a base b that would satisfy

41. For inspiration, consider bases 2 and 3 : h 0.1 0.01 0.001 0 0.7177 0.6956 0.6934 0.693 1.1612 1.1047 1.0992 1.099

42. h 0.1 0.01 0.001 0 0.7177 0.6956 0.6934 0.693 1.1612 1.1047 1.0992 1.099 Clearly, the ideal base must be between 2 and 3 (and more specifically, closer to 3).

43. In fact, the ideal base is known as Euler’s constant, e e  2.71828 Mathematicians showed that

44. Derivative of y = e x Recall, the derivative of y = b x was

45. However, if we introduce b = e :

46. Since, it follows that

47. Thus, the derivative of e x is itself !

48. For a composite function, Write the exponential function again Don’t forget the derivative of the inside function

49. e.g.

50. Write the exponential function again

51. e.g. Write the exponential function again Don’t forget the derivative of the inside function

52. Understanding the Graph for y = e x Does the function y = e x represent exponential growth or decay?

53. Since e  2.71828, this would be exponential growth.

54. Recall, Since y = y, it follows that the y-coordinate also represents the tangent slope

55. The y-coordinate is the tangent slope (0, 1) m t = 1

56. The y-coordinate is the tangent slope (2, e 2 ) m t = e 2

57. Ex. 3Differentiate Try this question yourself first. Then, check the answer on the following pages.

60. Ex. 4 Find y. Try this question yourself first. Then, check the answer on the following pages.

61. Product rule e is just a constant. So, you ignore the constant coefficient and take the derivative of x.

63. Ex. 5Where (x-values) does the function have a horizontal tangent? Try this question yourself first. Then, check the answer on the following pages.

64. Quotient Rule

65. A function has a horizontal tangent when its tangent slope (i.e. derivative) is zero. when Note: e 2x > 0 for all x  , so it can never equal zero.