Section 2.8 Linear Approximation and Differentials                V63.0121.002.2010Su, Calculus I                         ...
Announcements           Quiz 2 Thursday on           Sections 1.5–2.5           No class Monday, May 31           Assignme...
Objectives           Use tangent lines to make           linear approximations to a           function.                   ...
Outline The linear approximation of a function near a point   Examples   Questions Differentials     Using differentials t...
The Big Idea Question Let f be differentiable at a. What linear function best approximates f near a?                      ...
The Big Idea Question Let f be differentiable at a. What linear function best approximates f near a? Answer The tangent li...
The Big Idea Question Let f be differentiable at a. What linear function best approximates f near a? Answer The tangent li...
The Big Idea Question Let f be differentiable at a. What linear function best approximates f near a? Answer The tangent li...
The tangent line is a linear approximation                                                              y                 ...
The tangent line is a linear approximation                                                              y                 ...
Example . Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0     (ii) about a = 60◦...
Example . Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0     (ii) about a = 60◦...
Example . Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0     (ii) about a = 60◦...
Example . Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0     (ii) about a = 60◦...
Example . Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0     (ii) about a = 60◦...
Example . Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0     (ii) about a = 60◦...
Example . Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0     (ii) about a = 60◦...
Example . Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0     (ii) about a = 60◦...
Example . Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0     (ii) about a = 60◦...
Example . Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0     (ii) about a = 60◦...
Example . Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0     (ii) about a = 60◦...
Illustration          y          .                                                                     y                  ...
Illustration          y          .                                                                  y                     ...
Illustration          y          .                                                                  y                     ...
Illustration          y          .                                                                  y                     ...
Illustration          y          .                                                                  y                     ...
Another Example Example                 √ Estimate         10 using the fact that 10 = 9 + 1.                             ...
Another Example Example                 √ Estimate         10 using the fact that 10 = 9 + 1. Solution                    ...
Another Example Example                 √ Estimate         10 using the fact that 10 = 9 + 1. Solution                    ...
Another Example Example                 √ Estimate         10 using the fact that 10 = 9 + 1. Solution                    ...
Another Example Example                 √ Estimate         10 using the fact that 10 = 9 + 1. Solution                    ...
Dividing without dividing? Example Suppose I have an irrational fear of division and need to estimate 577 ÷ 408. I write  ...
Dividing without dividing? Example Suppose I have an irrational fear of division and need to estimate 577 ÷ 408. I write  ...
Questions Example Suppose we are traveling in a car and at noon our speed is 50 mi/hr. How far will we have traveled by 2:...
Answers Example Suppose we are traveling in a car and at noon our speed is 50 mi/hr. How far will we have traveled by 2:00...
Answers Example Suppose we are traveling in a car and at noon our speed is 50 mi/hr. How far will we have traveled by 2:00...
Questions Example Suppose we are traveling in a car and at noon our speed is 50 mi/hr. How far will we have traveled by 2:...
Answers Example Suppose our factory makes MP3 players and the marginal cost is currently $50/lot. How much will it cost to...
Questions Example Suppose we are traveling in a car and at noon our speed is 50 mi/hr. How far will we have traveled by 2:...
Answers Example Suppose a line goes through the point (x0 , y0 ) and has slope m. If the point is moved horizontally by dx...
Answers Example Suppose a line goes through the point (x0 , y0 ) and has slope m. If the point is moved horizontally by dx...
Outline The linear approximation of a function near a point   Examples   Questions Differentials     Using differentials t...
Differentials are another way to express derivatives       f(x + ∆x) − f(x) ≈ f′ (x) ∆x                            y      ...
Differentials are another way to express derivatives       f(x + ∆x) − f(x) ≈ f′ (x) ∆x                            y      ...
Using differentials to estimate error                                                              y                      ...
Example A sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cutting machine will cut a rectangle whose width is e...
Example A sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cutting machine will cut a rectangle whose width is e...
Example A sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cutting machine will cut a rectangle whose width is e...
Example A sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cutting machine will cut a rectangle whose width is e...
Why? Why use linear approximations dy when the actual difference ∆y is known?         Linear approximation is quick and re...
Outline The linear approximation of a function near a point   Examples   Questions Differentials     Using differentials t...
GravitationPencils down! Example         Drop a 1 kg ball off the roof of the Silver Center (50m high). We         usually...
GravitationPencils down! Example         Drop a 1 kg ball off the roof of the Silver Center (50m high). We         usually...
Gravitation Solution Solution We wonder if ∆F = F(re + ∆r) − F(re ) is small.         Using a linear approximation,       ...
Systematic linear approximation         √                    √          2 is irrational, but 9/4 is rational and 9/4 is cl...
Systematic linear approximation         √                    √          2 is irrational, but 9/4 is rational and 9/4 is cl...
Systematic linear approximation         √                    √          2 is irrational, but 9/4 is rational and 9/4 is cl...
Systematic linear approximation         √                    √          2 is irrational, but 9/4 is rational and 9/4 is cl...
Illustration of the previous example                                   .                                                  ...
Illustration of the previous example                                   .                                                  ...
Illustration of the previous example                                   .                                                  ...
Illustration of the previous example                                                               .                      ...
Illustration of the previous example                                                               .                      ...
Illustration of the previous example                                               . 2, 17 )                              ...
Illustration of the previous example                                               . 2, 17 )                              ...
Illustration of the previous example                                                                       .              ...
Illustration of the previous example                                                                       .              ...
Illustration of the previous example                                                                 .                    ...
Summary         Linear approximation: If f is differentiable at a, the best linear         approximation to f near a is gi...
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Lesson 12: Linear Approximation

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Lesson 12: Linear Approximation

  1. 1. Section 2.8 Linear Approximation and Differentials V63.0121.002.2010Su, Calculus I New York University May 26, 2010Announcements Quiz 2 Thursday on Sections 1.5–2.5 No class Monday, May 31 Assignment 2 due Tuesday, June 1 . . . . . .
  2. 2. Announcements Quiz 2 Thursday on Sections 1.5–2.5 No class Monday, May 31 Assignment 2 due Tuesday, June 1 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 2 / 27
  3. 3. 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. . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 3 / 27
  4. 4. Outline The linear approximation of a function near a point Examples Questions Differentials Using differentials to estimate error Advanced Examples . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 4 / 27
  5. 5. The Big Idea Question Let f be differentiable at a. What linear function best approximates f near a? . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 5 / 27
  6. 6. The Big Idea Question Let f be differentiable at a. What linear function best approximates f near a? Answer The tangent line, of course! . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 5 / 27
  7. 7. The Big Idea Question Let f be differentiable at a. What linear function best approximates f near a? Answer The tangent line, of course! Question What is the equation for the line tangent to y = f(x) at (a, f(a))? . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 5 / 27
  8. 8. The Big Idea Question Let f be differentiable at a. What linear function best approximates f near a? Answer The tangent line, of course! Question What is the equation for the line tangent to y = f(x) at (a, f(a))? Answer L(x) = f(a) + f′ (a)(x − a) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 5 / 27
  9. 9. 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 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 6 / 27
  10. 10. 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 to a, the better the approxmation f .(a) . . x−a L(x) is to f(x) . x . a . x . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 6 / 27
  11. 11. Example . Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0 (ii) about a = 60◦ = π/3. . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
  12. 12. Example . Example Estimate 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 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
  13. 13. Example . Example Estimate 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 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
  14. 14. Example . Example Estimate 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 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
  15. 15. Example . Example Estimate 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 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
  16. 16. Example . Example Estimate 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 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
  17. 17. Example . Example Estimate 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 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
  18. 18. Example . Example Estimate 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 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
  19. 19. Example . Example Estimate 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 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
  20. 20. Example . Example Estimate 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 Calculator check: sin(61◦ ) ≈ . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
  21. 21. Example . Example Estimate 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 Calculator check: sin(61◦ ) ≈ 0.87462. . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 7 / 27
  22. 22. Illustration y . y . = sin x . x . . 1◦ 6 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 8 / 27
  23. 23. Illustration y . y . = L1 (x) = x y . = sin x . x . 0 . . 1◦ 6 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 8 / 27
  24. 24. Illustration y . y . = L1 (x) = x b . ig difference! y . = sin x . x . 0 . . 1◦ 6 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 8 / 27
  25. 25. 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 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 8 / 27
  26. 26. 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 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 8 / 27
  27. 27. Another Example Example √ Estimate 10 using the fact that 10 = 9 + 1. . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 9 / 27
  28. 28. Another Example Example √ 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 = 9 to estimate f(10) = 10. . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 9 / 27
  29. 29. Another Example Example √ 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 = 9 to estimate f(10) = 10. √ √ d√ 10 ≈ 9 + x (1) dx x=9 1 19 =3+ (1) = ≈ 3.167 2·3 6 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 9 / 27
  30. 30. Another Example Example √ 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 = 9 to estimate f(10) = 10. √ √ d√ 10 ≈ 9 + x (1) dx x=9 1 19 =3+ (1) = ≈ 3.167 2·3 6 ( )2 19 Check: = 6 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 9 / 27
  31. 31. Another Example Example √ 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 = 9 to estimate f(10) = 10. √ √ d√ 10 ≈ 9 + x (1) dx x=9 1 19 =3+ (1) = ≈ 3.167 2·3 6 ( )2 19 361 Check: = . 6 36 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 9 / 27
  32. 32. Dividing without dividing? Example Suppose I have an irrational fear of division and need to estimate 577 ÷ 408. I write 577 1 1 1 = 1 + 169 = 1 + 169 × × . 408 408 4 102 1 But still I have to find . 102 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 10 / 27
  33. 33. Dividing without dividing? Example Suppose I have an irrational fear of division and need to estimate 577 ÷ 408. I write 577 1 1 1 = 1 + 169 = 1 + 169 × × . 408 408 4 102 1 But still I have to find . 102 Solution 1 Let 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 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 10 / 27
  34. 34. Questions Example Suppose 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? . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 11 / 27
  35. 35. Answers Example Suppose 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? . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 12 / 27
  36. 36. Answers Example Suppose 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?) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 12 / 27
  37. 37. Questions Example Suppose 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? Example Suppose our factory makes MP3 players and the marginal cost is currently $50/lot. How much will it cost to make 2 more lots? 3 more lots? 12 more lots? . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 13 / 27
  38. 38. Answers Example Suppose our factory makes MP3 players and the marginal cost is currently $50/lot. How much will it cost to make 2 more lots? 3 more lots? 12 more lots? Answer $100 $150 $600 (?) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 14 / 27
  39. 39. Questions Example Suppose 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? Example Suppose our factory makes MP3 players and the marginal cost is currently $50/lot. How much will it cost to make 2 more lots? 3 more lots? 12 more lots? Example Suppose a line goes through the point (x0 , y0 ) and has slope m. If the point is moved horizontally by dx, while staying on the line, what is the corresponding vertical movement? . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 15 / 27
  40. 40. Answers Example Suppose a line goes through the point (x0 , y0 ) and has slope m. If the point is moved horizontally by dx, while staying on the line, what is the corresponding vertical movement? . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 16 / 27
  41. 41. Answers Example Suppose a line goes through the point (x0 , y0 ) and has slope m. If the point is moved horizontally by dx, while staying on the line, what is the corresponding vertical movement? Answer The slope of the line is rise m= run We are given a “run” of dx, so the corresponding “rise” is m dx. . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 16 / 27
  42. 42. Outline The linear approximation of a function near a point Examples Questions Differentials Using differentials to estimate error Advanced Examples . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 17 / 27
  43. 43. Differentials are another way to express derivatives f(x + ∆x) − f(x) ≈ f′ (x) ∆x y . ∆y dy Rename ∆x = dx, so we can write this as . ∆y ≈ dy = f′ (x)dx. . dy . ∆y And this looks a lot like the . . dx = ∆x Leibniz-Newton identity dy . = f′ (x) x . dx x x . . + ∆x . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 18 / 27
  44. 44. Differentials are another way to express derivatives f(x + ∆x) − f(x) ≈ f′ (x) ∆x y . ∆y dy Rename ∆x = dx, so we can write this as . ∆y ≈ dy = f′ (x)dx. . dy . ∆y And this looks a lot like the . . dx = ∆x Leibniz-Newton identity dy . = f′ (x) x . dx x x . . + ∆x Linear approximation means ∆y ≈ dy = f′ (x0 ) dx near x0 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 18 / 27
  45. 45. Using differentials to estimate error y . If y = f(x), x0 and ∆x is known, and an estimate of ∆y is desired: Approximate: ∆y ≈ dy . Differentiate: dy = f′ (x) dx . ∆y . dy Evaluate at x = x0 and . . dx = ∆x dx = ∆x. . x . x x . . + ∆x . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 19 / 27
  46. 46. Example A sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cutting machine will cut a rectangle whose width is exactly half its length, but the length is prone to errors. If the length is off by 1 in, how bad can the area of the sheet be off by? . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 20 / 27
  47. 47. Example A sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cutting machine will cut a rectangle whose width is exactly half its length, but the length is prone to errors. If the length is off by 1 in, how bad can the area of the sheet be off by? Solution 1 2 Write A(ℓ) = ℓ . We want to know ∆A when ℓ = 8 ft and ∆ℓ = 1 in. 2 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 20 / 27
  48. 48. Example A sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cutting machine will cut a rectangle whose width is exactly half its length, but the length is prone to errors. If the length is off by 1 in, how bad can the area of the sheet be off by? Solution 1 2 Write 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 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 20 / 27
  49. 49. Example A sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cutting machine will cut a rectangle whose width is exactly half its length, but the length is prone to errors. If the length is off by 1 in, how bad can the area of the sheet be off by? Solution 1 2 Write 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. . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 20 / 27
  50. 50. Why? Why use linear approximations dy when the actual difference ∆y is known? 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). . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 21 / 27
  51. 51. Outline The linear approximation of a function near a point Examples Questions Differentials Using differentials to estimate error Advanced Examples . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 22 / 27
  52. 52. 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. . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 23 / 27
  53. 53. 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? . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 23 / 27
  54. 54. Gravitation Solution Solution We 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 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 24 / 27
  55. 55. Systematic linear approximation √ √ 2 is irrational, but 9/4 is rational and 9/4 is close to 2. . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 25 / 27
  56. 56. 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 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 25 / 27
  57. 57. 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 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 25 / 27
  58. 58. 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 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 25 / 27
  59. 59. Illustration of the previous example . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
  60. 60. Illustration of the previous example . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
  61. 61. Illustration of the previous example . 2 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
  62. 62. Illustration of the previous example . . 2 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
  63. 63. Illustration of the previous example . . 2 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
  64. 64. Illustration of the previous example . 2, 17 ) ( 12 . . . 2 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
  65. 65. Illustration of the previous example . 2, 17 ) ( 12 . . . 2 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
  66. 66. Illustration of the previous example . . 2, 17/12) ( . . 4, 3) (9 2 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
  67. 67. Illustration of the previous example . . 2, 17/12) ( .. ( . 9, 3) ( )4 2 289 17 . 144 , 12 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
  68. 68. Illustration of the previous example . . 2, 17/12) ( .. ( . 9, 3) ( ( 577 ) )4 2 . 2, 408 289 17 . 144 , 12 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 26 / 27
  69. 69. 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. . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 27 / 27

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