Lesson 15: Linear Approximation and Differentials

Slide 1: Section 3.8 Linear Approximation and Differentials Math 1a March 10, 2008 Announcements ◮ Midterm is graded. Come to office hours if you don’t have it back yet. ◮ Problem Sessions Sunday, Thursday, 7pm, SC 310 ◮ Office hours Tues, Weds, 2–4pm SC 323 . Image: Flickr user cobalt123 . . . . . .

Slide 2: Announcements ◮ Midterm is graded. Come to office hours if you don’t have it back yet. ◮ Problem Sessions Sunday, Thursday, 7pm, SC 310 ◮ Office hours Tues, Weds, 2–4pm SC 323 . . . . . .

Slide 3: Outline The linear approximation of a function near a point Examples . . . . . .

Slide 4: The Big Idea Question Let f be differentiable at a. What linear function best approximates f near a? . . . . . .

Slide 5: The Big Idea Question Let f be differentiable at a. What linear function best approximates f near a? Answer The tangent line, of course! . . . . . .

Slide 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! Question What is the equation for the line tangent to y = f(x) at (a, f(a))? . . . . . .

Slide 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))? Answer L(x) = f(a) + f′ (a)(x − a) . . . . . .

Slide 8: Outline The linear approximation of a function near a point Examples . . . . . .

Slide 9: Example Example Estimate ln(1.02). . . . . . .

Slide 10: Example Example Estimate ln(1.02). Solution We know ln(1) = 0, so ln(1.02) should not be too far from 0. . . . . . .

Slide 11: Example Example Estimate ln(1.02). Solution We know ln(1) = 0, so ln(1.02) should not be too far from 0. In fact d ln(1.02) ≈ ln(1) + ln x (0.02) dx x=0 = 0 + 1(0.02) = 0.02. . . . . . .

Slide 12: Example Example Estimate ln(1.02). Solution We know ln(1) = 0, so ln(1.02) should not be too far from 0. In fact d ln(1.02) ≈ ln(1) + ln x (0.02) dx x=0 = 0 + 1(0.02) = 0.02. Calculator check: . . . ln(1.02) ≈ . . . . . .

Slide 13: Example Example Estimate ln(1.02). Solution We know ln(1) = 0, so ln(1.02) should not be too far from 0. In fact d ln(1.02) ≈ ln(1) + ln x (0.02) dx x=0 = 0 + 1(0.02) = 0.02. Calculator check: . . . ln(1.02) ≈ 0.0198. . . . . . .

Slide 14: Another Example Example √ Estimate 10 using the fact that 10 = 9 + 1. . . . . . .

Slide 15: Another Example Example √ Estimate 10 using the fact that 10 = 9 + 1. Solution √ √d√ 10 ≈ 9+x (1) dx x=9 1 19 =3+ (1 ) = ≈ 3.167 2·3 6 . . . . . .

Slide 16: Another Example Example √ Estimate 10 using the fact that 10 = 9 + 1. Solution √ √ d√ 10 ≈ 9+ x (1) dx x=9 1 19 =3+ (1 ) = ≈ 3.167 2·3 6 2 19 Check: = 6 . . . . . .

Slide 17: Another Example Example √ Estimate 10 using the fact that 10 = 9 + 1. Solution √ √ d√ 10 ≈ 9+ x (1) dx x=9 1 19 =3+ (1 ) = ≈ 3.167 2·3 6 2 19 361 Check: = . 6 36 . . . . . .

Slide 18: Differentials are another way to express derivatives If y = f(x), . . . . . .

Slide 19: Differentials are another way to express derivatives dy If y = f(x), then = f′ (x), dx . . . . . .

Slide 20: Differentials are another way to express derivatives dy If y = f(x), then = f′ (x), and dy = f′ (x)dx. dx y . . ∆y . dy . d x = ∆x . x . x x . . + ∆x . . . . . .

Slide 21: Differentials are another way to express derivatives dy If y = f(x), then = f′ (x), and dy = f′ (x)dx. dx y . . ∆y . dy . d x = ∆x . x . x x . . + ∆x Then ∆y ≈ dy = f′ (x0 ) dx near x0 . . . . . . .

Slide 22: Another example Example Drop a 1kg ball off the roof of the Science Center (30m high). We usually say that a falling object feels a force F = mg from gravity. . . . . . .

Slide 23: Another example Example Drop a 1kg ball off the roof of the Science Center (30m 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 = mg. What is the maximum r2 e error in making this approximation? The relative error? The percentage error? . . . . . .

Slide 24: Another example Example Drop a 1kg ball off the roof of the Science Center (30m 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 = mg. What is the maximum r2 e error in making this approximation? The relative error? The percentage error? Note: re = 6378.1 km, M = 5.9724 × 1024 kg, and G = 6.6742 × 10−11 N · m2 · kg−2 . . . . . .

Slide 25: Systematic linear approximation √ ◮ 2 is irrational, but 9/4 is rational and 9/4 is close to 2. . . . . . .

Slide 26: 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 . . . . . .

Slide 27: 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 . . . . . .

Slide 28: 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 . . . . . .

Slide 29: Illustration of the previous example . . . . . . .

Slide 30: Illustration of the previous example . . . . . . .

Slide 31: Illustration of the previous example . 2 . . . . . . .

Slide 32: Illustration of the previous example . • . 9/4, 3/2) ( . . 2 . . . . . .

Slide 33: Illustration of the previous example . • . 9/4, 3/2) ( . . 2 . . . . . .

Slide 34: Illustration of the previous example . 2, 17/12) ( . . • • . 9/4, 3/2) ( . 2 . . . . . . .

Slide 35: Illustration of the previous example . 2, 17/12) ( . . • • . 9/4, 3/2) ( . 2 . . . . . . .

Slide 36: Illustration of the previous example . • . 2, 17/12) ( . 9/4, 3/2) ( . • . . . . . .