Section 2.8
 Linear Approximation and Differentials

                      V63.0121.041, Calculus I

                     ...
Announcements




         Quiz 2 in recitation this
         week on §§1.5, 1.6, 2.1,
         2.2
         Midterm on §§...
Objectives
         Use tangent lines to make
         linear approximations to a
         function.
                Given...
Outline



The linear approximation of a function near a point
  Examples
  Questions


Differentials
    Using differenti...
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 ...
The Big Idea

Question
Let f be differentiable at a. What linear function best approximates f
near a?

Answer
The tangent ...
The Big Idea

Question
Let f be differentiable at a. What linear function best approximates f
near a?

Answer
The tangent ...
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
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Illustration

       y
       .
                                                                   y
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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:...
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:...
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 ...
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 ...
Answers


Example
Suppose a line goes through the point (x0 , y0 ) and has slope m. If the
point is moved horizontally by ...
Outline



The linear approximation of a function near a point
  Examples
  Questions


Differentials
    Using differenti...
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 r...
Outline



The linear approximation of a function near a point
  Examples
  Questions


Differentials
    Using differenti...
Gravitation
Pencils down!

Example

       Drop a 1 kg ball off the roof of the Silver Center (50m high). We
       usuall...
Gravitation
Pencils down!

Example

       Drop a 1 kg ball off the roof of the Silver Center (50m high). We
       usuall...
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 close...
Systematic linear approximation

      √                    √
       2 is irrational, but 9/4 is rational and 9/4 is close...
Systematic linear approximation

      √                    √
       2 is irrational, but 9/4 is rational and 9/4 is close...
Systematic linear approximation

      √                    √
       2 is irrational, but 9/4 is rational and 9/4 is close...
Illustration of the previous example




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Illustration of the previous example




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Illustration of the previous example




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Illustration of the previous example




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Illustration of the previous example




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Illustration of the previous example




                                                   . 2, 17 )
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Illustration of the previous example




                                                   . 2, 17 )
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Illustration of the previous example




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Illustration of the previous example




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Illustration of the previous example




                                                                .
               ...
Summary


     Linear approximation: If f is differentiable at a, the best linear
     approximation to f near a is given ...
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Lesson 12: Linear Approximation and Differentials (Section 41 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 41 slides)

  1. 1. Section 2.8 Linear Approximation and Differentials V63.0121.041, Calculus I New York University October 13, 2010 Announcements 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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. . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 4 / 27
  5. 5. The Big Idea Question Let f be differentiable at a. What linear function best approximates f near a? . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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! . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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))? . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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) . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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. . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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◦ ) ≈ . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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. . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 7 / 27
  22. 22. Illustration y . y . = sin x . x . . 1◦ 6 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 8 / 27
  23. 23. Illustration y . y . = L1 (x) = x y . = sin x . x . 0 . . 1◦ 6 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 8 / 27
  24. 24. Illustration y . y . = L1 (x) = x b . ig difference! y . = sin x . x . 0 . . 1◦ 6 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 8 / 27
  27. 27. Another Example Example √ Estimate 10 using the fact that 10 = 9 + 1. . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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. . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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? . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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? . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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?) . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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? . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 (?) . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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? . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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? . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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. . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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? . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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. . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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). . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 22 / 27
  52. 52. Gravitation Pencils 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. . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 23 / 27
  53. 53. Gravitation Pencils 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? . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 24 / 27
  55. 55. Systematic linear approximation √ √ 2 is irrational, but 9/4 is rational and 9/4 is close to 2. . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 25 / 27
  59. 59. Illustration of the previous example . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 26 / 27
  60. 60. Illustration of the previous example . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 26 / 27
  61. 61. Illustration of the previous example . 2 . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 26 / 27
  62. 62. Illustration of the previous example . . 2 . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 26 / 27
  63. 63. Illustration of the previous example . . 2 . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 26 / 27
  64. 64. Illustration of the previous example . 2, 17 ) ( 12 . . . 2 . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 26 / 27
  65. 65. Illustration of the previous example . 2, 17 ) ( 12 . . . 2 . . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 26 / 27
  66. 66. Illustration of the previous example . . 2, 17/12) ( . . 4, 3) (9 2 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 26 / 27
  67. 67. Illustration of the previous example . . 2, 17/12) ( .. ( . 9, 3) ( )4 2 289 17 . 144 , 12 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 26 / 27
  68. 68. Illustration of the previous example . . 2, 17/12) ( .. ( . 9, 3) ( ( 577 ) )4 2 . 2, 408 289 17 . 144 , 12 . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 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. . . . . . . V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 27 / 27

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