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Linearization and Newton’s Method 4.5 QUICK REVIEW Find dy / dx. 1. sin x 2 2 x cos x x 1 Solve the equation graphically. 2. 3. xe 1 0 -x 4. x 3 x 1 0 3 5. Let f ( x) xe 1. Write an equation for the line tangent to f at x 0. -x Slide 42 QUICK REVIEW SOLUTIONS Find dy / dx. 1. sin x 2. 2 2 x cos x 2 x cos x x 1 2 2 x sin x sin x cos x x 1 2 Solve the equation graphically. 3. xe 1 0 x 0.567 -x 4. x 3 x 1 0 x 0.322 3 5. Let f ( x) xe 1. Write an equation for the line tangent to f at x 0. -x y x 1 Slide 4- 3 WHAT YOU’LL LEARN ABOUT Linear Approximation Differentials Estimating Change with Differentials Absolute, Relative, and Percent Change …and why Engineering and science depend on approximation in most practical applications; it is important to understand how approximation techniques work. Slide 44 LINEARIZATION TANGENT LINE 𝒚 − 𝒚 = 𝒎(𝒙 − 𝒙) If f is differentiable at x a, then the equation of the tangent line, L( x) f (a) f '(a)( x - a), defines the linearization of f at a. The approximation f ( x) L( x) is the standard linear approximation of f at a. The point x a is the center of the approximation. http://www.calculusapplets.com/linearapprox.html Slide 45 EXAMPLE FINDING A LINEARIZATION Find the linearization of f ( x) cos x at x / 2 and use it to approximate cos 1.75 without a calculator. Since f ( / 2) cos( / 2) 0, the point of tangency is ( / 2, 0). The slope of the tangent line is f '( / 2) sin( / 2) 1. Thus L( x) 0 ( 1) x x . 2 2 To approximate cos 1.75 f (1.75) L(1.75) 1.75 . 2 Let’s talk to our bowtie friend! Slide 46 DIFFERENTIALS Let y f ( x) be a differentiable function. The differential dx is an independent variable. The differential dy is dy f '( x)dx. 𝑑𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 ′ 𝑠 ′ = = 𝑓 𝑥 ′ 𝑑𝑥 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥 𝑠 𝑑𝑦 = 𝑓 ′ 𝑥 𝑑𝑥 (Just multiplied dx to other side) Slide 47 EXAMPLE FINDING THE DIFFERENTIAL DY Find the differential dy and evaluate dy for the given value of x and dx. y x 5 2 x, x 1, dx 0.01 dy 5 x 4 2 dx dy 5 2 0.01 0.07 Slide 48 DIFFERENTIAL ESTIMATE OF CHANGE Let f ( x) be differentiable at x a. The approximate change in the value of f when x changes from a to a dx is df f '(a)dx. Slide 49 ESTIMATING CHANGE WITH DIFFERENTIALS Slide 410 EXAMPLE ESTIMATING CHANGE WITH DIFFERENTIALS The radius of a circle increases from a 5 m to 5.1 m. Use dA to estimate the increase in the circle's area A. Since A r , the estimated increase is 2 dA 2 rdr 2 5 0.1 m 2 Slide 411