1. 3 Copyright © Cengage Learning. All rights reserved. Applications of Differentiation

2. Differentials 2014 Copyright © Cengage Learning. All rights reserved. 3.9

3. 3 Understand the concept of a tangent line approximation. Compare the value of the differential (dy) with the actual change in y (Δy). Estimate a propagated error using a differential. Find the differential of a function using differentiation formulas. Objectives

4. 4 Tangent Line Approximations

5. 5 For any function f (x), the tangent is a close approximation of the function for some small distance from the tangent point. We call the equation of the tangent the linearization of the function.

6. 6 The linearization is the equation of the tangent line, and you can use the old formulas if you like. Start with the point/slope equation: linearization of f at c is the standard linear approximation of f at c.

7. 7 Consider a function f that is differentiable at c. The equation for the tangent line at the point (c, f(c)) is given by y – f(c) = f'(c)(x – c) and is called the tangent line approximation (or linear approximation) of f at c. Because c is a constant, y is a linear function of x. Tangent Line Approximations

8. 8 Example 1 – Using a Tangent Line Approximation Find the tangent line approximation of f(x) = 1 + sin x at the point (0, 1).Then use a table to compare the y-values of the linear function with those of f(x) on an open interval containing x = 0. Solution: The derivative of f is f'(x) = cos x. First derivative

9. 9 So, the equation of the tangent line to the graph of f at the point (0, 1) is y – f(0) = f'(0)(x – 0) y – 1 = (1)(x – 0) y = 1 + x. Example 1 – Solution cont'd

10. 10 The table compares the values of y given by this linear approximation with the values of f(x) near x = 0. Notice that the closer x is to 0, the better the approximation is. cont'd Example 1 – Solution

11. 11 This conclusion is reinforced by the graph shown in Figure 3.65. Figure 3.65 cont'd Example 1 – Solution

12. 12 Example 2 – Using a Tangent Line Approximation Consider the curve defined by: a) b) Find the tangent line approximation to the curve at the point (4, – 1).

13. 13 Example 2 – (Con’t) Consider the curve defined by: c) There is a number k such that the point (4.2, k) is on the curve. Using the tangent line found in (b), approximate the value of k. d) Find the actual value of k on the curve. What was the percent of error between actual and approximate values of k?

14. 14 Differentials

15. 15 When the tangent line to the graph of f at the point (c, f(c)) y = f(c) + f'(c)(x – c) is used as an approximation of the graph of f, the quantity x – c is called the change in x, and is denoted by Δx, as shown in Figure 3.66. Differentials

16. 16 When Δx is small, the change in y (denoted by Δy) can be approximated as shown. Δy = f(c + Δx) – f(c) ≈ f'(c)Δx For such an approximation, the quantity Δx is traditionally denoted by dx, and is called the differential of x. The expression f'(x)dx is denoted by dy, and is called the differential of y. Differentials

17. 17 Differentials

18. 18 Example 3 – Comparing Δy and dy Let y = x 2. Find dy when x = 1 and dx = 0.01. Compare this value with Δy for x = 1 and Δx = 0.01. Solution: Because y = f(x) = x 2, you have f'(x) = 2x, and the differential dy is given by dy = f'(x)dx = f'(1)(0.01) = 2(0.01)= 0.02. Now, using Δx = 0.01, the actual change in y is Δy = f(x + Δx) – f(x) = f(1.01) – f(1) = (1.01) 2 – 1 2 = 0.0201. Differential of y Actual change in y

19. 19 Example 2 – Solution Figure 3.67 shows the geometric comparison of dy and Δy. Figure 3.67 cont'd Was the approximation an over-estimate or under-estimate?

20. 20 Error Propagation

21. 21 If you let x represent the measured value of a variable and let x + Δx represent the exact value, then Δx is the error in measurement. Finally, if the measured value x is used to compute another value f(x), the difference between f(x +Δx) and f(x) is the propagated error. (The error in y resulting from an error in x.) Error Propagation

22. 22 Example 3 – Estimation of Error The measured radius of a ball bearing is 0.7 inches, as shown in Figure 3.68. If the measurement is correct to within 0.01 inch, estimate the propagated error in the volume V of the ball bearing. Figure 3.68

23. 23 The formula for the volume of a sphere is, where r is the radius of the sphere. So, you can write and To approximate the propagated error in the volume using the differential: differentiate V to obtain and write dv Example 3 – Solution

24. 24 Example 3 – Solution So, the volume has a propagated error of about 0.06 cubic inch. Is the propagated error in this example large or small? To answer this you need to compare dv with V. This comparison is called relative error.

25. 25 The ratio is called the relative error. This error as a percent is approximately 4.29%. Relative Error

26. 26 Example: Consider a circle of radius 10. If the radius increases by 0.1, use the differential to approximate how much the area will change. very small change in A very small change in r (approximate change in area)

27. 27 (approximate change in area) Compare to actual change: New area: Old area: (actual change in area)

28. 28 Calculating Differentials

29. 29 Each of the differentiation rules can be written in differential form. Calculating Differentials

30. 30 Finding Differentials This notation is called the Leibniz notation for derivatives and differentials, named after the German mathematician Gottfried Wilhelm Leibniz.

31. 31 Example 7 – Finding Differentials Find the differential form of Solution:

32. 32 Example 8– Finding Differentials Find the differential form of Solution:

33. 33 Differentials can be used to approximate function values. To do this for the function given by y = f(x), use the formula which is just the tangent line approximation. It tells us that the change in y can be approximated by dy. Δy = f(x + Δx) – f(x) ≈ dy. The key to using this formula is to choose a value for x that makes the calculations easier. Approximating with Differentials

34. 34 Example 7 – Approximating Function Values Use a tangent line to approximate Solution: Using you can write Now, choosing x = 16 and dx = 0.5, you obtain the following approximation.

35. 35 Example 7 – Solution The tangent line approximation to at x = 16 is the line For x-values near 16, the graphs of f and g are close together, as shown in Figure 3.69. Figure 3.69 cont'd

36. 36 You Try Use a tangent line approximation of At the point (0,2) to approximate

37. 37 Homework Day 1: Pg. 240 #7-35 odds Only do part a on 31 & 33