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Drill

Drill. Find dy / dx. Linearization and Newton’s Method. Section 4.5 Day 1: page 242 2-18, even. Objectives. Students will be able to find linearizations and use Newton’s method to approximate the zeros of a function. estimate the change in a function using differentials.

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Drill

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  1. Drill • Find dy/dx

  2. Linearization and Newton’s Method Section 4.5 Day 1: page 2422-18, even

  3. Objectives • Students will be able to • find linearizations and use Newton’s method to approximate the zeros of a function. • estimate the change in a function using differentials.

  4. Linearization • 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 of f(x) = L(x) is the standard linear approximation of f at a. The point x = a is the center of the approximation.

  5. Example: Finding a Linearization • Find the linearization of f(x) = (1+x)1/2 at x = 0, and use it to approximate (1.02)1/2 without a calculator. • Since f(0) = 1, the point of tangency is (0,1) • f’(0) = ½

  6. Example: Finding a Linearization • L(x) = f(a) + f’(a)(x –a) • L(x) = f(0) + f’(0)(x-0) • L(x) = 1 + ½ (x – 0) • L(x) = 1 + ½ x • To approximate (1.02)1/2 , we use x = .02, because the original equation was (1 + x)1/2 • f(.02) = L(.02) = 1 + ½ (.02) = 1.01

  7. Example: Finding a LinearizationL(x) = f(a) + f’(a)(x –a) Find the linearization of and use it to approximate sin 1.25 without a calculator. Then use a calculator to determine the accuracy of the approximation.

  8. Example 1 Finding a Linearization Find the linearization of and use it to approximate sin 1.25 without a calculator. Then use a calculator to determine the accuracy of the approximation.

  9. Example 1 Finding a Linearization Find the linearization of and use it to approximate sin 1.25 without a calculator. Then use a calculator to determine the accuracy of the approximation. The calculator gives sin 1.25 = 0.9489846194, so the approximation error is

  10. Example Approximating Binomial Powers Use the formula to find polynomials that will approximate the following functions for values of x close to zero.

  11. Example: Approximating Roots Use linearizations to approximate each. L(x) = f(a) + f’(a)(x –a) The closest perfect square to 123 is 121 L(123) = f(121) + f’(121)(123 –121) L(123) = 11+ 1/22(2) = 11.09

  12. Example: Approximating Roots The closest perfect cube to 123 is 125 L(123) = f(125) + f’(125)(123 –125) L(123) = 5+ 1/75(-2) = 4.973

  13. Newton’s Method • Guess a first approximation to a solution of the equation f(x) = 0. • Use the first approximation to get a second, the second to get a third, and so on using the formula

  14. Example 4 Using Newton’s Method Use Newton’s Method to solve x3 + 7x + 3 = 0. Look at the graph to determine an approximate solution. We can use ANS on calculator once we have stored the equation to generate multiple xn

  15. Example 4 Using Newton’s Method Use Newton’s Method to solve x4 – 2x – 8 = 0 over the interval [–2, 0].

  16. Homework • Page 242: 2-4, 8, 10, 12, 16

  17. Drill: Find the linearization of L(x) of f(x) at aL(x) = f(a) + f’(a)(x –a) • F(x) = x3 – 2x + 3, a = 2 • F(2) = 7 • F’(x) = 3x2 – 2 F’(2) = 10 • L(x) = 7+ 10(x –2) = 10x - 13 • F(x) = x + 1/x, a = 1 • F(1) = 2 • F’(x) = 1 – 1/x2 F’(1) = 0 • L(x) = 2+ 0(x –1) = 2

  18. Differentials • Let y = f(x) be a differentiable function. • The differential dx is an independent variable. • The differential dy is dy = f’(x)dx

  19. Example 5 Finding the Differential dy Find the differential dy and evaluate dy for the given values of x and dx.

  20. Example 5 Finding the Differential dy Find the differential dy and evaluate dy for the given values of x and dx.

  21. Example 5 Finding the Differential dy Find the differential dy and evaluate dy for the given values of x and dx. To find y, sub x = 4 into original equation: 4 – 2y = 3(4)y 4 – 2y = 12y 4 = 14y 2/7 = y

  22. Finding differentials of functions If dx ≠ 0, then the quotient of the differential dy by the differential dx is equation to the derivative f’(x) because We sometimes write df = f’(x)dx in place of dy = f’(x)dx, calling df the differential of f. For example, if f(x) = 3x2 – 6, df = 6x dx

  23. Finding differentials of functions • d(tan2x)= 2sec22x dx • d(arctan4x)=

  24. 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 + dxis df = f’(a)dx

  25. Example 6 Estimating Change with Differentials The length of a side of a cube, s, increases from s = 5 inches to s = 5.01 inches. Use dV to estimate the increase in the cube’s volume, V. Compare this estimate with the true change ΔV and find the approximation error. Estimate Approximation Error Exact

  26. Homework • Page 242: 20, 21, 26, 28, 30, 31-36

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