Lesson 12: Linear Approximations and Differentials (handout)

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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 …

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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  • 1. . V63.0121.001: Calculus I . Sec on 2.8: Linear Approxima ons . March 2, 2011 Notes Sec on 2.8 Linear Approxima ons and Differen als V63.0121.001: Calculus I Professor Ma hew Leingang . New York University March 2, 2011 . . Notes Announcements Quiz in recita on this week on 1.5, 1.6, 2.1, 2.2 Midterm March 7 in class on 1.1–2.5 No quiz in recita on next week . . Notes Objectives Use tangent lines to make linear approxima ons to a func on. Given a func on and a point in the domain, compute the lineariza on of the func on at that point. Use lineariza on to approximate values of func ons Given a func on, compute the differen al of that func on Use the differen al nota on to es mate error in linear approxima ons. . . . 1.
  • 2. . V63.0121.001: Calculus I . Sec on 2.8: Linear Approxima ons . March 2, 2011 Notes Outline The linear approxima on of a func on near a point Examples Ques ons Differen als Using differen als to es mate error Advanced Examples . . Notes The Big Idea Ques on What linear func on best approximates f near a? Answer The tangent line, of course! Ques on What is the equa on for the line tangent to y = f(x) at (a, f(a))? Answer L(x) = f(a) + f′ (a)(x − a) . . Notes tangent line = linear approximation y The func on L(x) L(x) = f(a) + f′ (a)(x − a) f(x) is a decent approxima on to f near a. f(a) x−a How decent? The closer x is to a, the be er the approxima on L(x) is to f(x) . x a x . . . 2.
  • 3. . V63.0121.001: Calculus I . Sec on 2.8: Linear Approxima ons . March 2, 2011 Notes Example Example Es mate sin(61◦ ) = sin(61π/180) by using a linear approxima on (i) about a = 0 (ii) about a = 60◦ = π/3. Solu on (i) Solu on (ii) ( ) √ 3 We have f π = ( ) 1 3 2 and If f(x) = sin x, then f(0) = 0 ′ π and f′ (0) = 1. f 3 = 2. So the linear approxima on So the linear approxima on is √ 3 1( π) near 0 is L(x) = 0 + 1 · x = x. L(x) = + x− ( ) 2 2 3 61π 61π ( ) sin ≈ ≈ 1.06465 61π 180 180 sin ≈ 0.87475 180 . Calculator check: sin(61◦ ) ≈ 0.87462. . Notes Illustration y y = L1 (x) = x √ ( ) y = L2 (x) = 2 3 + 1 2 x− π 3 big difference! y = sin x very li le difference! . x 0 π/3 61◦ . . Notes Another Example Example √ Es mate 10 using the fact that 10 = 9 + 1. Solu on . . . 3.
  • 4. . V63.0121.001: Calculus I . Sec on 2.8: Linear Approxima ons . March 2, 2011 Notes Dividing without dividing? Example A student has an irra onal fear of long division and needs to es mate 577 ÷ 408. He writes 577 1 1 1 = 1 + 169 = 1 + 169 × × . 408 408 4 102 1 Help the student es mate . 102 . . Notes Dividing without dividing? Solu on . . Notes 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? Answer . . . 4.
  • 5. . V63.0121.001: Calculus I . Sec on 2.8: Linear Approxima ons . March 2, 2011 Notes Questions 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 . . Notes Questions 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 ver cal movement? Answer . . Notes Outline The linear approxima on of a func on near a point Examples Ques ons Differen als Using differen als to es mate error Advanced Examples . . . 5.
  • 6. . V63.0121.001: Calculus I . Sec on 2.8: Linear Approxima ons . March 2, 2011 Notes Differentials are derivatives The fact that the the tangent line is an approxima on means that y f(x + ∆x) − f(x) ≈ f′ (x) ∆x ∆y dy Rename ∆x = dx, so we can write this as dy ∆y ≈ dy = f′ (x)dx. ∆y dx = ∆x Note this looks a lot like the Leibniz-Newton iden ty dy . x = f′ (x) x x + ∆x dx . . Notes Using differentials to estimate error Es ma ng error with differen als y If y = f(x), x0 and ∆x is known, and an es mate of ∆y is desired: Approximate: ∆y ≈ dy ∆y dy Differen ate: dy = f′ (x) dx dx = ∆x Evaluate at x = x0 and . x dx = ∆x. x x + ∆x . . Notes Using differentials to estimate error Example A regular sheet of plywood measures 8 ft × 4 ft. Suppose a defec ve plywood-cu ng 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? . . . 6.
  • 7. . V63.0121.001: Calculus I . Sec on 2.8: Linear Approxima ons . March 2, 2011 Notes Solution Solu on 1 Write A(ℓ) = ℓ2 . We want to know ∆A when ℓ = 8 ft and 2 ∆ℓ = 1 in. ( ) 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 es mate for ∆ℓ. dℓ When ℓ = 8 and dℓ = 12 , we have dA = 12 = 2 ≈ 0.667. 1 8 3 So we get es mates close to the hundredth of a square foot. . . Notes Why should we care? Why use linear approxima ons dy when the actual difference ∆y is known? Linear approxima on is quick and reliable. Finding ∆y exactly depends on the func on. With more complicated func ons, linear approxima on much simpler. See the “Advanced Examples” later. In real life, some mes only f(a) and f′ (a) are known, and not the general f(x). . . Notes Outline The linear approxima on of a func on near a point Examples Ques ons Differen als Using differen als to es mate error Advanced Examples . . . 7.
  • 8. . V63.0121.001: Calculus I . Sec on 2.8: Linear Approxima ons . March 2, 2011 Notes Gravitation 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. GMm In fact, the force felt is F(r) = − 2 , where M is the mass of the earth and r r is the distance from the center of the earth to the object. G is a constant. GMm GM At r = re the force really is F(re ) = , and g is defined to be 2 r2 e re 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 rela ve error? The percentage error? . . Notes Gravitation Solution Solu on We wonder if ∆F = F(re + ∆r) − F(re ) is small. Using a linear approxima on, dF GMm ∆F ≈ dF = dr = 2 3 dr dr re re ( ) GMm dr ∆r = = 2mg r2e re re ∆F ∆r The rela ve error is ≈ −2 F re . . Notes Solution continued re = 6378.1 km. If ∆r = 50 m, ∆F ∆r 50 ≈ −2 = −2 = −1.56 × 10−5 = −0.00156% F re 6378100 . . . 8.
  • 9. . V63.0121.001: Calculus I . Sec on 2.8: Linear Approxima ons . March 2, 2011 Notes Systematic linear approximation √ √ 2 is irra onal, but 9/4 is ra onal and 9/4 is close to 2. So √ √ √ 1 17 2 = 9/4 − 1/4 ≈ 9/4 + 3 (−1/4) = 2( /2) 12 This is a be er approxima on since (17/12)2 = 289/144 Do it again! √ √ √ 1 2 = 289/144 − 1/144 ≈ 289/144 + 17 (−1/144) = 577/408 2( /12) ( )2 577 332, 929 1 Now = which is away from 2. 408 166, 464 166, 464 . . Illustration of the previous Notes example (2, 17 ) 12 (2, 17/12) 9 3 ( 577 ) ( 289 17()4 , 2 ) 2, 408 144 , 12 . 2 . . . Notes Summary 2 Linear approxima on: If f is differen able at a, the best linear approxima on to f near a is given by Lf,a (x) = f(a) + f′ (a)(x − a) Differen als: If f is differen able at x, a good approxima on to ∆y = f(x + ∆x) − f(x) is dy dy ∆y ≈ dy = · dx = · ∆x dx dx Don’t buy plywood from me. . . . 9.