Chapter 4 The Exponential and Natural Logarithm Functions

1. Chapter 4The Exponential and Natural Logarithm Functions

2. Chapter Outline • Exponential Functions • The Exponential Function ex • Differentiation of Exponential Functions • The Natural Logarithm Function • The Derivative ln x • Properties of the Natural Logarithm Function

3. §4.1 Exponential Functions

4. Section Outline • Exponential Functions • Properties of Exponential Functions • Simplifying Exponential Expressions • Graphs of Exponential Functions • Solving Exponential Equations

5. Exponential Function

6. Properties of Exponential Functions

7. Simplifying Exponential Expressions EXAMPLE Write each function in the form 2kx or 3kx, for a suitable constant k. SOLUTION (a) We notice that 81 is divisible by 3. And through investigation we recognize that 81 = 34. Therefore, we get (b) We first simplify the denominator and then combine the numerator via the base of the exponents, 2. Therefore, we get

8. Graphs of Exponential Functions Notice that, no matter what b is (except 1), the graph of y = bx has a y-intercept of 1. Also, if 0 < b < 1, the function is decreasing. If b > 1, then the function is increasing.

9. Solving Exponential Equations EXAMPLE Solve the following equation for x. SOLUTION This is the given equation. Factor. Simplify. Since 5x and 6 – 3x are being multiplied, set each factor equal to zero. 5x≠ 0.

10. §4.2 The Exponential Function ex

11. Section Outline • e • The Derivatives of 2x, bx, and ex

12. The Number e

13. The Derivative of 2x

14. Solving Exponential Equations EXAMPLE Calculate. SOLUTION

15. The Derivatives of bx and ex

16. Solving Exponential Equations EXAMPLE Find the equation of the tangent line to the curve at (0, 1). SOLUTION We must first find the derivative function and then find the value of the derivative at (0, 1). Then we can use the point-slope form of a line to find the desired tangent line equation. This is the given function. Differentiate. Use the quotient rule.

17. Solving Exponential Equations CONTINUED Simplify. Factor. Simplify the numerator. Now we evaluate the derivative at x = 0.

18. Solving Exponential Equations CONTINUED Now we know a point on the tangent line, (0, 1), and the slope of that line, -1. We will now use the point-slope form of a line to determine the equation of the desired tangent line. This is the point-slope form of a line. (x1, y1) = (0, 1) and m = -1. Simplify.

19. §4.3 Differentiation of Exponential Functions

20. Section Outline • Chain Rule for eg(x) • Working With Differential Equations • Solving Differential Equations at Initial Values • Functions of the form ekx

21. Chain Rule for eg(x)

22. Chain Rule for eg(x) EXAMPLE Differentiate. SOLUTION This is the given function. Use the chain rule. Remove parentheses. Use the chain rule for exponential functions.

23. Working With Differential Equations Generally speaking, a differential equation is an equation that contains a derivative.

24. Solving Differential Equations EXAMPLE Determine all solutions of the differential equation SOLUTION The equation has the form y΄ = ky with k = 1/3. Therefore, any solution of the equation has the form where C is a constant.

25. Solving Differential Equations at Initial Values EXAMPLE Determine all functions y = f(x) such that y΄ = 3y and f(0) = ½. SOLUTION The equation has the form y΄ = ky with k = 3. Therefore, for some constant C. We also require that f(0) = ½. That is, So C = ½ and

26. Functions of the form ekx

27. §4.4 The Natural Logarithm Function

28. Section Outline • The Natural Logarithm of x • Properties of the Natural Logarithm • Exponential Expressions • Solving Exponential Equations • Solving Logarithmic Equations • Other Exponential and Logarithmic Functions • Common Logarithms • Max’s and Min’s of Exponential Equations

29. The Natural Logarithm of x

30. Properties of the Natural Logarithm

31. Properties of the Natural Logarithm • The point (1, 0) is on the graph of y = ln x [because (0, 1) is on the graph of y = ex]. • ln x is defined only for positive values of x. • ln x is negative for x between 0 and 1. • ln x is positive for x greater than 1. • ln x is an increasing function and concave down.

32. Exponential Expressions EXAMPLE Simplify. SOLUTION Using properties of the exponential function, we have

33. Solving Exponential Equations EXAMPLE Solve the equation for x. SOLUTION This is the given equation. Remove the parentheses. Combine the exponential expressions. Add. Take the logarithm of both sides. Simplify. Finish solving for x.

34. Solving Logarithmic Equations EXAMPLE Solve the equation for x. SOLUTION This is the given equation. Divide both sides by 5. Rewrite in exponential form. Divide both sides by 2.

35. Other Exponential and Logarithmic Functions

36. Common Logarithms

37. Max’s & Min’s of Exponential Equations EXAMPLE The graph of is shown in the figure below. Find the coordinates of the maximum and minimum points.

38. Max’s & Min’s of Exponential Equations CONTINUED At the maximum and minimum points, the graph will have a slope of zero. Therefore, we must determine for what values of x the first derivative is zero. This is the given function. Differentiate using the product rule. Finish differentiating. Factor. Set the derivative equal to 0. Set each factor equal to 0. Simplify.

39. Max’s & Min’s of Exponential Equations CONTINUED Therefore, the slope of the function is 0 when x = 1 or x = -1. By looking at the graph, we can see that the relative maximum will occur when x = -1 and that the relative minimum will occur when x = 1. Now we need only determine the corresponding y-coordinates. Therefore, the relative maximum is at (-1, 0.472) and the relative minimum is at (1, -1).

40. §4.5 The Derivative of ln x

41. Section Outline • Derivatives for Natural Logarithms • Differentiating Logarithmic Expressions

42. Derivative Rules for Natural Logarithms

43. Differentiating Logarithmic Expressions EXAMPLE Differentiate. SOLUTION This is the given expression. Differentiate. Use the power rule. Differentiate ln[g(x)]. Finish.

44. Differentiating Logarithmic Expressions EXAMPLE The function has a relative extreme point for x > 0. Find the coordinates of the point. Is it a relative maximum point? SOLUTION This is the given function. Use the quotient rule to differentiate. Simplify. Set the derivative equal to 0.

45. Differentiating Logarithmic Expressions CONTINUED The derivative will equal 0 when the numerator equals 0 and the denominator does not equal 0. Set the numerator equal to 0. Write in exponential form. To determine whether the function has a relative maximum at x = 1, let’s use the second derivative. This is the first derivative. Differentiate.

46. Differentiating Logarithmic Expressions CONTINUED Simplify. Factor and cancel. Evaluate the second derivative at x = 1. Since the value of the second derivative is negative at x = 1, the function is concave down at x = 1. Therefore, the function does indeed have a relative maximum at x = 1. To find the y-coordinate of this point So, the relative maximum occurs at (1, 1).

47. §4.6 Properties of the Natural Logarithm Function

48. Section Outline • Properties of the Natural Logarithm Function • Simplifying Logarithmic Expressions • Differentiating Logarithmic Expressions • Logarithmic Differentiation

49. Properties of the Natural Logarithm Function

50. Simplifying Logarithmic Expressions EXAMPLE Write as a single logarithm. SOLUTION This is the given expression. Use LIV (this must be done first). Use LIII. Use LI. Simplify.