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# Lesson 12: Linear Approximation and Differentials (Section 21 slides)

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The line tangent to a curve is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.

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### Lesson 12: Linear Approximation and Differentials (Section 21 slides)

1. 1. Section 2.8 Linear Approximation and Differentials V63.0121.021, Calculus I New York University October 14, 2010Announcements Quiz 2 in recitation this week on §§1.5, 1.6, 2.1, 2.2 Midterm on §§1.1–2.5 . . . . . .
2. 2. Announcements Quiz 2 in recitation this week on §§1.5, 1.6, 2.1, 2.2 Midterm on §§1.1–2.5 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 2 / 36
3. 3. Midterm FAQQuestionWhat sections are covered on the midterm? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 3 / 36
4. 4. Midterm FAQQuestionWhat sections are covered on the midterm?AnswerThe midterm will cover Sections 1.1–2.5 (The Chain Rule). . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 3 / 36
5. 5. Midterm FAQQuestionWhat sections are covered on the midterm?AnswerThe midterm will cover Sections 1.1–2.5 (The Chain Rule).QuestionIs Section 2.6 going to be on the midterm? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 3 / 36
6. 6. Midterm FAQQuestionWhat sections are covered on the midterm?AnswerThe midterm will cover Sections 1.1–2.5 (The Chain Rule).QuestionIs Section 2.6 going to be on the midterm?AnswerThe midterm will cover Sections 1.1–2.5 (The Chain Rule). . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 3 / 36
7. 7. Midterm FAQQuestionWhat sections are covered on the midterm?AnswerThe midterm will cover Sections 1.1–2.5 (The Chain Rule).QuestionIs Section 2.6 going to be on the midterm?AnswerThe midterm will cover Sections 1.1–2.5 (The Chain Rule).QuestionIs Section 2.8 going to be on the midterm?... . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 3 / 36
8. 8. Midterm FAQ, continuedQuestionWhat format will the exam take? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 4 / 36
9. 9. Midterm FAQ, continuedQuestionWhat format will the exam take?AnswerThere will be both fixed-response (e.g., multiple choice) andfree-response questions. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 4 / 36
10. 10. Midterm FAQ, continuedQuestionWhat format will the exam take?AnswerThere will be both fixed-response (e.g., multiple choice) andfree-response questions.QuestionWill explanations be necessary? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 4 / 36
11. 11. Midterm FAQ, continuedQuestionWhat format will the exam take?AnswerThere will be both fixed-response (e.g., multiple choice) andfree-response questions.QuestionWill explanations be necessary?AnswerYes, on free-response problems we will expect you to explain yourself.This is why it was required on written homework. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 4 / 36
12. 12. Midterm FAQ, continuedQuestionIs (topic X) going to be tested? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 5 / 36
13. 13. Midterm FAQ, continuedQuestionIs (topic X) going to be tested?AnswerEverything covered in class or on homework is fair game for the exam.No topic that was not covered in class nor on homework will be on theexam. (This is not the same as saying all exam problems are similar toclass examples or homework problems.) . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 5 / 36
14. 14. Midterm FAQ, continuedQuestionWill there be a review session? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 6 / 36
15. 15. Midterm FAQ, continuedQuestionWill there be a review session?AnswerWe’re working on it. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 6 / 36
16. 16. Midterm FAQ, continuedQuestionWill calculators be allowed? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 7 / 36
17. 17. Midterm FAQ, continuedQuestionWill calculators be allowed?AnswerNo. The exam is designed for pencil and brain. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 7 / 36
18. 18. Midterm FAQ, continuedQuestionHow should I study? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 8 / 36
19. 19. Midterm FAQ, continuedQuestionHow should I study?Answer The exam has problems; study by doing problems. If you get one right, think about how you got it right. If you got it wrong or didn’t get it at all, reread the textbook and do easier problems to build up your understanding. Break up the material into chunks. (related) Don’t put it all off until the night before. Ask questions. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 8 / 36
20. 20. Midterm FAQ, continuedQuestionHow many questions are there? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 9 / 36
21. 21. Midterm FAQ, continuedQuestionHow many questions are there?AnswerDoes this question contribute to your understanding of the material? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 9 / 36
22. 22. Midterm FAQ, continuedQuestionWill there be a curve on the exam? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 10 / 36
23. 23. Midterm FAQ, continuedQuestionWill there be a curve on the exam?AnswerDoes this question contribute to your understanding of the material? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 10 / 36
24. 24. Midterm FAQ, continuedQuestionWhen will you grade my get-to-know-you and photo extra credit? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 11 / 36
25. 25. Midterm FAQ, continuedQuestionWhen will you grade my get-to-know-you and photo extra credit?AnswerDoes this question contribute to your understanding of the material? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 11 / 36
26. 26. Objectives Use tangent lines to make linear approximations to a function. Given a function and a point in the domain, compute the linearization of the function at that point. Use linearization to approximate values of functions Given a function, compute the differential of that function Use the differential notation to estimate error in linear approximations. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 12 / 36
27. 27. OutlineThe linear approximation of a function near a point Examples QuestionsDifferentials Using differentials to estimate errorAdvanced Examples . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 13 / 36
28. 28. The Big IdeaQuestionLet f be differentiable at a. What linear function best approximates fnear a? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 14 / 36
29. 29. The Big IdeaQuestionLet f be differentiable at a. What linear function best approximates fnear a?AnswerThe tangent line, of course! . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 14 / 36
30. 30. The Big IdeaQuestionLet f be differentiable at a. What linear function best approximates fnear a?AnswerThe tangent line, of course!QuestionWhat is the equation for the line tangent to y = f(x) at (a, f(a))? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 14 / 36
31. 31. The Big IdeaQuestionLet f be differentiable at a. What linear function best approximates fnear a?AnswerThe tangent line, of course!QuestionWhat is the equation for the line tangent to y = f(x) at (a, f(a))?Answer L(x) = f(a) + f′ (a)(x − a) . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 14 / 36
32. 32. The tangent line is a linear approximation y . L(x) = f(a) + f′ (a)(x − a)is a decent approximation to f L . (x) .near a. f .(x) . f .(a) . . x−a . x . a . x . . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 15 / 36
33. 33. The tangent line is a linear approximation y . L(x) = f(a) + f′ (a)(x − a)is a decent approximation to f L . (x) .near a. f .(x) .How decent? The closer x is toa, the better the approxmation f .(a) . . x−aL(x) is to f(x) . x . a . x . . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 15 / 36
34. 34. Example.ExampleEstimate sin(61◦ ) = sin(61π/180) by using a linear approximation(i) about a = 0 (ii) about a = 60◦ = π/3.. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
35. 35. Example.ExampleEstimate sin(61◦ ) = sin(61π/180) by using a linear approximation(i) about a = 0 (ii) about a = 60◦ = π/3.Solution (i) If f(x) = sin x, then f(0) = 0 and f′ (0) = 1. So the linear approximation near 0 is L(x) = 0 + 1 · x = x. Thus ( ) 61π 61π sin ≈ ≈ 1.06465 180 180. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
36. 36. Example.ExampleEstimate sin(61◦ ) = sin(61π/180) by using a linear approximation(i) about a = 0 (ii) about a = 60◦ = π/3.Solution (i) Solution (ii) (π) We have f = and If f(x) = sin x, then f(0) = 0 ( ) 3 and f′ (0) = 1. f′ π = . 3 So the linear approximation near 0 is L(x) = 0 + 1 · x = x. Thus ( ) 61π 61π sin ≈ ≈ 1.06465 180 180. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
37. 37. Example.ExampleEstimate sin(61◦ ) = sin(61π/180) by using a linear approximation(i) about a = 0 (ii) about a = 60◦ = π/3.Solution (i) Solution (ii) (π) √ 3 We have f = and If f(x) = sin x, then f(0) = 0 ( ) 3 2 and f′ (0) = 1. f′ π = . 3 So the linear approximation near 0 is L(x) = 0 + 1 · x = x. Thus ( ) 61π 61π sin ≈ ≈ 1.06465 180 180. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
38. 38. Example.ExampleEstimate sin(61◦ ) = sin(61π/180) by using a linear approximation(i) about a = 0 (ii) about a = 60◦ = π/3.Solution (i) Solution (ii) (π) √ 3 We have f = and If f(x) = sin x, then f(0) = 0 ( ) 3 2 and f′ (0) = 1. f′ π = 1 . 3 2 So the linear approximation near 0 is L(x) = 0 + 1 · x = x. Thus ( ) 61π 61π sin ≈ ≈ 1.06465 180 180. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
39. 39. Example.ExampleEstimate sin(61◦ ) = sin(61π/180) by using a linear approximation(i) about a = 0 (ii) about a = 60◦ = π/3.Solution (i) Solution (ii) (π) √ 3 We have f = and If f(x) = sin x, then f(0) = 0 ( ) 3 2 and f′ (0) = 1. f′ π = 1 . 3 2 So the linear approximation So L(x) = near 0 is L(x) = 0 + 1 · x = x. Thus ( ) 61π 61π sin ≈ ≈ 1.06465 180 180. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
40. 40. Example.ExampleEstimate sin(61◦ ) = sin(61π/180) by using a linear approximation(i) about a = 0 (ii) about a = 60◦ = π/3.Solution (i) Solution (ii) (π) √ 3 We have f = and If f(x) = sin x, then f(0) = 0 ( ) 3 2 and f′ (0) = 1. f′ π = 1 . 3 2 √ So the linear approximation 3 1( π) So L(x) = + x− near 0 is L(x) = 0 + 1 · x = x. 2 2 3 Thus ( ) 61π 61π sin ≈ ≈ 1.06465 180 180. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
41. 41. Example.ExampleEstimate sin(61◦ ) = sin(61π/180) by using a linear approximation(i) about a = 0 (ii) about a = 60◦ = π/3.Solution (i) Solution (ii) (π) √ 3 We have f = and If f(x) = sin x, then f(0) = 0 ( ) 3 2 and f′ (0) = 1. f′ π = 1 . 3 2 √ So the linear approximation 3 1( π) So L(x) = + x− near 0 is L(x) = 0 + 1 · x = x. 2 2 3 Thus Thus ( ) ( ) 61π 61π 61π sin ≈ ≈ 1.06465 sin ≈ 180 180 180. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
42. 42. Example.ExampleEstimate sin(61◦ ) = sin(61π/180) by using a linear approximation(i) about a = 0 (ii) about a = 60◦ = π/3.Solution (i) Solution (ii) (π) √ 3 We have f = and If f(x) = sin x, then f(0) = 0 ( ) 3 2 and f′ (0) = 1. f′ π = 1 . 3 2 √ So the linear approximation 3 1( π) So L(x) = + x− near 0 is L(x) = 0 + 1 · x = x. 2 2 3 Thus Thus ( ) ( ) 61π 61π 61π sin ≈ ≈ 1.06465 sin ≈ 0.87475 180 180 180. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
43. 43. Example.ExampleEstimate sin(61◦ ) = sin(61π/180) by using a linear approximation(i) about a = 0 (ii) about a = 60◦ = π/3.Solution (i) Solution (ii) (π) √ 3 We have f = and If f(x) = sin x, then f(0) = 0 ( ) 3 2 and f′ (0) = 1. f′ π = 1 . 3 2 √ So the linear approximation 3 1( π) So L(x) = + x− near 0 is L(x) = 0 + 1 · x = x. 2 2 3 Thus Thus ( ) ( ) 61π 61π 61π sin ≈ ≈ 1.06465 sin ≈ 0.87475 180 180 180Calculator check: sin(61◦ ) ≈. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
44. 44. Example.ExampleEstimate sin(61◦ ) = sin(61π/180) by using a linear approximation(i) about a = 0 (ii) about a = 60◦ = π/3.Solution (i) Solution (ii) (π) √ 3 We have f = and If f(x) = sin x, then f(0) = 0 ( ) 3 2 and f′ (0) = 1. f′ π = 1 . 3 2 √ So the linear approximation 3 1( π) So L(x) = + x− near 0 is L(x) = 0 + 1 · x = x. 2 2 3 Thus Thus ( ) ( ) 61π 61π 61π sin ≈ ≈ 1.06465 sin ≈ 0.87475 180 180 180Calculator check: sin(61◦ ) ≈ 0.87462.. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
45. 45. Illustration y . y . = sin x . x . . 1◦ 6 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 17 / 36
46. 46. Illustration y . y . = L1 (x) = x y . = sin x . x . 0 . . 1◦ 6 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 17 / 36
47. 47. Illustration y . y . = L1 (x) = x b . ig difference! y . = sin x . x . 0 . . 1◦ 6 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 17 / 36
48. 48. Illustration y . y . = L1 (x) = x √ ( ) y . = L2 (x) = 2 3 + 1 2 x− π 3 y . = sin x . . . x . 0 . . π/3 . 1◦ 6 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 17 / 36
49. 49. Illustration y . y . = L1 (x) = x √ ( ) y . = L2 (x) = 2 3 + 1 2 x− π 3 y . = sin x . . ery little difference! v . . x . 0 . . π/3 . 1◦ 6 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 17 / 36
50. 50. Another ExampleExample √Estimate 10 using the fact that 10 = 9 + 1. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 18 / 36
51. 51. Another ExampleExample √Estimate 10 using the fact that 10 = 9 + 1.Solution √The key step is to use a linear approximation to f(x) = √ x near a = 9to estimate f(10) = 10. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 18 / 36
52. 52. Another ExampleExample √Estimate 10 using the fact that 10 = 9 + 1.Solution √The key step is to use a linear approximation to f(x) = √ x near a = 9to estimate f(10) = 10. f(10) ≈ L(10) = f(9) + f′ (9)(10 − 9) 1 19 =3+ (1) = ≈ 3.167 2·3 6 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 18 / 36
53. 53. Another ExampleExample √Estimate 10 using the fact that 10 = 9 + 1.Solution √The key step is to use a linear approximation to f(x) = √ x near a = 9to estimate f(10) = 10. f(10) ≈ L(10) = f(9) + f′ (9)(10 − 9) 1 19 =3+ (1) = ≈ 3.167 2·3 6 ( )2 19Check: = 6 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 18 / 36
54. 54. Another ExampleExample √Estimate 10 using the fact that 10 = 9 + 1.Solution √The key step is to use a linear approximation to f(x) = √ x near a = 9to estimate f(10) = 10. f(10) ≈ L(10) = f(9) + f′ (9)(10 − 9) 1 19 =3+ (1) = ≈ 3.167 2·3 6 ( )2 19 361Check: = . 6 36 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 18 / 36
55. 55. Dividing without dividing?ExampleA student has an irrational fear of long division and needs to estimate577 ÷ 408. He writes 577 1 1 1 = 1 + 169 = 1 + 169 × × . 408 408 4 102 1Help the student estimate . 102 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 19 / 36
56. 56. Dividing without dividing?ExampleA student has an irrational fear of long division and needs to estimate577 ÷ 408. He writes 577 1 1 1 = 1 + 169 = 1 + 169 × × . 408 408 4 102 1Help the student estimate . 102Solution 1Let f(x) = . We know f(100) and we want to estimate f(102). x 1 1 f(102) ≈ f(100) + f′ (100)(2) = − (2) = 0.0098 100 1002 577 =⇒ ≈ 1.41405 408 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 19 / 36
57. 57. QuestionsExampleSuppose we are traveling in a car and at noon our speed is 50 mi/hr.How far will we have traveled by 2:00pm? by 3:00pm? By midnight? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 20 / 36
58. 58. AnswersExampleSuppose we are traveling in a car and at noon our speed is 50 mi/hr.How far will we have traveled by 2:00pm? by 3:00pm? By midnight? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 21 / 36
59. 59. AnswersExampleSuppose we are traveling in a car and at noon our speed is 50 mi/hr.How far will we have traveled by 2:00pm? by 3:00pm? By midnight?Answer 100 mi 150 mi 600 mi (?) (Is it reasonable to assume 12 hours at the same speed?) . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 21 / 36
60. 60. QuestionsExampleSuppose we are traveling in a car and at noon our speed is 50 mi/hr.How far will we have traveled by 2:00pm? by 3:00pm? By midnight?ExampleSuppose our factory makes MP3 players and the marginal cost iscurrently \$50/lot. How much will it cost to make 2 more lots? 3 morelots? 12 more lots? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 22 / 36
61. 61. AnswersExampleSuppose our factory makes MP3 players and the marginal cost iscurrently \$50/lot. How much will it cost to make 2 more lots? 3 morelots? 12 more lots?Answer \$100 \$150 \$600 (?) . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 23 / 36
62. 62. QuestionsExampleSuppose we are traveling in a car and at noon our speed is 50 mi/hr.How far will we have traveled by 2:00pm? by 3:00pm? By midnight?ExampleSuppose our factory makes MP3 players and the marginal cost iscurrently \$50/lot. How much will it cost to make 2 more lots? 3 morelots? 12 more lots?ExampleSuppose a line goes through the point (x0 , y0 ) and has slope m. If thepoint is moved horizontally by dx, while staying on the line, what is thecorresponding vertical movement? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 24 / 36
63. 63. AnswersExampleSuppose a line goes through the point (x0 , y0 ) and has slope m. If thepoint is moved horizontally by dx, while staying on the line, what is thecorresponding vertical movement? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 25 / 36
64. 64. AnswersExampleSuppose a line goes through the point (x0 , y0 ) and has slope m. If thepoint is moved horizontally by dx, while staying on the line, what is thecorresponding vertical movement?AnswerThe slope of the line is rise m= runWe are given a “run” of dx, so the corresponding “rise” is m dx. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 25 / 36
65. 65. OutlineThe linear approximation of a function near a point Examples QuestionsDifferentials Using differentials to estimate errorAdvanced Examples . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 26 / 36
66. 66. Differentials are another way to express derivativesThe fact that the the tangentline is an approximation meansthat y . f(x + ∆x) − f(x) ≈ f′ (x) ∆x ∆y dyRename ∆x = dx, so we canwrite this as . . dy . ∆y ′ ∆y ≈ dy = f (x)dx. . . dx = ∆xNote this looks a lot like theLeibniz-Newton identity . x . dy x x . . + ∆x = f′ (x) dx . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 27 / 36
67. 67. Using differentials to estimate error y .Estimating error withdifferentialsIf y = f(x), x0 and ∆x is known,and an estimate of ∆y isdesired: . . Approximate: ∆y ≈ dy dy . ∆y Differentiate: dy = f′ (x) dx . . dx = ∆x Evaluate at x = x0 and dx = ∆x. . x . x x . . + ∆x . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 28 / 36
68. 68. ExampleA sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cuttingmachine will cut a rectangle whose width is exactly half its length, butthe length is prone to errors. If the length is off by 1 in, how bad can thearea of the sheet be off by? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 29 / 36
69. 69. ExampleA sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cuttingmachine will cut a rectangle whose width is exactly half its length, butthe length is prone to errors. If the length is off by 1 in, how bad can thearea of the sheet be off by?Solution 1 2Write A(ℓ) = ℓ . We want to know ∆A when ℓ = 8 ft and ∆ℓ = 1 in. 2 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 29 / 36
70. 70. ExampleA sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cuttingmachine will cut a rectangle whose width is exactly half its length, butthe length is prone to errors. If the length is off by 1 in, how bad can thearea of the sheet be off by?Solution 1 2Write A(ℓ) = ℓ . We want to know ∆A when ℓ = 8 ft and ∆ℓ = 1 in. 2 ( ) 97 9409 9409 (I) A(ℓ + ∆ℓ) = A = So ∆A = − 32 ≈ 0.6701. 12 288 288 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 29 / 36
71. 71. ExampleA sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cuttingmachine will cut a rectangle whose width is exactly half its length, butthe length is prone to errors. If the length is off by 1 in, how bad can thearea of the sheet be off by?Solution 1 2Write A(ℓ) = ℓ . We want to know ∆A when ℓ = 8 ft and ∆ℓ = 1 in. 2 ( ) 97 9409 9409 (I) A(ℓ + ∆ℓ) = A = So ∆A = − 32 ≈ 0.6701. 12 288 288 dA (II) = ℓ, so dA = ℓ dℓ, which should be a good estimate for ∆ℓ. dℓ When ℓ = 8 and dℓ = 12 , we have dA = 12 = 2 ≈ 0.667. So we 1 8 3 get estimates close to the hundredth of a square foot. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 29 / 36
72. 72. Why?Why use linear approximations dy when the actual difference ∆y isknown? Linear approximation is quick and reliable. Finding ∆y exactly depends on the function. These examples are overly simple. See the “Advanced Examples” later. In real life, sometimes only f(a) and f′ (a) are known, and not the general f(x). . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 30 / 36
73. 73. OutlineThe linear approximation of a function near a point Examples QuestionsDifferentials Using differentials to estimate errorAdvanced Examples . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 31 / 36
74. 74. GravitationPencils down!Example Drop a 1 kg ball off the roof of the Silver Center (50m high). We usually say that a falling object feels a force F = −mg from gravity. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 32 / 36
75. 75. GravitationPencils down!Example Drop a 1 kg ball off the roof of the Silver Center (50m high). We usually say that a falling object feels a force F = −mg from gravity. In fact, the force felt is GMm F(r) = − , r2 where M is the mass of the earth and r is the distance from the center of the earth to the object. G is a constant. GMm At r = re the force really is F(re ) = = −mg. r2 e What is the maximum error in replacing the actual force felt at the top of the building F(re + ∆r) by the force felt at ground level F(re )? The relative error? The percentage error? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 32 / 36
76. 76. Gravitation SolutionSolutionWe wonder if ∆F = F(re + ∆r) − F(re ) is small. Using a linear approximation, dF GMm ∆F ≈ dF = dr = 2 3 dr dr re re ( ) GMm dr ∆r = 2 = 2mg re re re ∆F ∆r The relative error is ≈ −2 F re re = 6378.1 km. If ∆r = 50 m, ∆F ∆r 50 ≈ −2 = −2 = −1.56 × 10−5 = −0.00156% F re 6378100 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 33 / 36
77. 77. Systematic linear approximation √ √ 2 is irrational, but 9/4 is rational and 9/4 is close to 2. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 34 / 36
78. 78. Systematic linear approximation √ √ 2 is irrational, but 9/4 is rational and 9/4 is close to 2. So √ √ √ 1 17 2 = 9/4 − 1/4 ≈ 9/4 + (−1/4) = 2(3/2) 12 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 34 / 36
79. 79. Systematic linear approximation √ √ 2 is irrational, but 9/4 is rational and 9/4 is close to 2. So √ √ √ 1 17 2 = 9/4 − 1/4 ≈ 9/4 + (−1/4) = 2(3/2) 12 This is a better approximation since (17/12)2 = 289/144 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 34 / 36
80. 80. Systematic linear approximation √ √ 2 is irrational, but 9/4 is rational and 9/4 is close to 2. So √ √ √ 1 17 2 = 9/4 − 1/4 ≈ 9/4 + (−1/4) = 2(3/2) 12 This is a better approximation since (17/12)2 = 289/144 Do it again! √ √ √ 1 2 = 289/144 − 1/144 ≈ 289/144 + (−1/144) = 577/408 2(17/12) ( )2 577 332, 929 1 Now = which is away from 2. 408 166, 464 166, 464 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 34 / 36
81. 81. Illustration of the previous example . . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
82. 82. Illustration of the previous example . . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
83. 83. Illustration of the previous example . 2 . . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
84. 84. Illustration of the previous example . . 2 . . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
85. 85. Illustration of the previous example . . 2 . . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
86. 86. Illustration of the previous example . 2, 17 ) ( 12 . . . 2 . . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
87. 87. Illustration of the previous example . 2, 17 ) ( 12 . . . 2 . . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
88. 88. Illustration of the previous example . . 2, 17/12) ( . . 4, 3) (9 2 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
89. 89. Illustration of the previous example . . 2, 17/12) ( .. ( . 9, 3) ( )4 2 289 17 . 144 , 12 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
90. 90. Illustration of the previous example . . 2, 17/12) ( .. ( . 9, 3) ( ( 577 ) )4 2 . 2, 408 289 17 . 144 , 12 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
91. 91. Summary Linear approximation: If f is differentiable at a, the best linear approximation to f near a is given by Lf,a (x) = f(a) + f′ (a)(x − a) Differentials: If f is differentiable at x, a good approximation to ∆y = f(x + ∆x) − f(x) is dy dy ∆y ≈ dy = · dx = · ∆x dx dx Don’t buy plywood from me. . . . . . .V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 36 / 36