Lesson 12: Linear Approximation and Differentials (Section 21 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 another set of notation for the same problem.
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Lesson 12: Linear Approximation and Differentials (Section 21 handout)
- 1. V63.0121.021, CalculusSection 2.8 : Linear Approximation and Differentials I October 14, 2010 Notes Section 2.8 Linear Approximation and Differentials V63.0121.021, Calculus I New York University October 14, 2010 Announcements Quiz 2 in recitation this week on §§1.5, 1.6, 2.1, 2.2 Midterm on §§1.1–2.5 Announcements Notes 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 and Differentials October 14, 2010 2 / 27 Objectives Notes 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 and Differentials October 14, 2010 3 / 27 1
- 2. V63.0121.021, CalculusSection 2.8 : Linear Approximation and Differentials I October 14, 2010 Outline Notes The linear approximation of a function near a point Examples Questions Differentials Using differentials to estimate error Advanced Examples V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 2010 4 / 27 The Big Idea Notes 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) V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 2010 5 / 27 The tangent line is a linear approximation Notes 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 V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 2010 6 / 27 2
- 3. V63.0121.021, CalculusSection 2.8 : Linear Approximation and Differentials I October 14, 2010 Example Notes 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) √ If f (x) = sin x, then f (0) = 0 We have f π = 23 and 3 and f (0) = 1. f π = 1. 3 2 √ So the linear approximation near 3 1 π So L(x) = + x− 0 is L(x) = 0 + 1 · x = x. 2 2 3 Thus Thus 61π 61π 61π sin ≈ 0.87475 sin ≈ ≈ 1.06465 180 180 180 Calculator check: sin(61◦ ) ≈ 0.87462. V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 2010 7 / 27 Illustration Notes y y = L1 (x) = x √ 3 1 π y = L2 (x) = 2 + 2 x− 3 big difference! y = sin x very little difference! x 0 π/3 61◦ V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 2010 8 / 27 Another Example Notes Example √ Estimate 10 using the fact that 10 = 9 + 1. Solution V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 2010 9 / 27 3
- 4. V63.0121.021, CalculusSection 2.8 : Linear Approximation and Differentials I October 14, 2010 Dividing without dividing? Notes 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 V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 2010 10 / 27 Questions Notes 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? V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 2010 11 / 27 Answers Notes 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 V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 2010 12 / 27 4
- 5. V63.0121.021, CalculusSection 2.8 : Linear Approximation and Differentials I October 14, 2010 Answers Notes 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 V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 2010 14 / 27 Answers Notes 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 V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 2010 16 / 27 Outline Notes The linear approximation of a function near a point Examples Questions Differentials Using differentials to estimate error Advanced Examples V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 2010 17 / 27 5
- 6. V63.0121.021, CalculusSection 2.8 : Linear Approximation and Differentials I October 14, 2010 Differentials are another way to express derivatives Notes 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 . V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 2010 18 / 27 Using differentials to estimate error Notes y If y = f (x), x0 and ∆x is known, and an estimate of ∆y is desired: Approximate: ∆y ≈ dy dy Differentiate: dy = f (x) dx ∆y Evaluate at x = x0 and dx = ∆x dx = ∆x. x x x + ∆x V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 2010 19 / 27 Example Notes 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( ) = . We want to know ∆A when = 8 ft and ∆ = 1 in. 2 97 9409 9409 (I) A( + ∆ ) = A = So ∆A = − 32 ≈ 0.6701. 12 288 288 dA (II) = , so dA = d , which should be a good estimate for ∆ . d When = 8 and d = 12 , we have dA = 12 = 2 ≈ 0.667. So we get 1 8 3 estimates close to the hundredth of a square foot. V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 2010 20 / 27 6
- 7. V63.0121.021, CalculusSection 2.8 : Linear Approximation and Differentials I October 14, 2010 Why? Notes 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). V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 2010 21 / 27 Outline Notes The linear approximation of a function near a point Examples Questions Differentials Using differentials to estimate error Advanced Examples V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 2010 22 / 27 Gravitation Pencils down! Notes 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 ) = 2 = −mg . 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 relative error? The percentage error? V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 2010 23 / 27 7
- 8. V63.0121.021, CalculusSection 2.8 : Linear Approximation and Differentials I October 14, 2010 Gravitation Solution Notes 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 V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 2010 24 / 27 Systematic linear approximation Notes √ 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 and Differentials October 14, 2010 25 / 27 Illustration of the previous example Notes (2, 17 ) 12 (9, 2) 4 3 2 V63.0121.021, Calculus I (NYU) Section 2.8 17 12Approximation and Differentials (2, / ) Linear 9 3 October 14, 2010 26 / 27 (4, 2) 577 289 17 2, 408 144 , 12 8
- 9. V63.0121.021, CalculusSection 2.8 : Linear Approximation and Differentials I October 14, 2010 Summary Notes 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 and Differentials October 14, 2010 27 / 27 Notes Notes 9