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Starting problems

Starting problems. Write a tangent line approximation for each: g(x) = 3x 2 + 2x + 1 at x = 0 f(x) = cos x at x = 0 Write the formula for finding a tangent line approximation for function f at x=0. A closer look .

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Starting problems

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  1. Starting problems • Write a tangent line approximation for each: • g(x) = 3x2 + 2x + 1 at x = 0 • f(x) = cos x at x = 0 • Write the formula for finding a tangent line approximation for function f at x=0.

  2. A closer look • Investigate the tangent line approximation of f (x) = cos x near x = 0 by graphing. Notice: The farther a point x is away from 0, the worse the approximation!

  3. What next? • More accurate approximation to f (x) = cos x for x near 0? • Instead of using a line, what about a quadratic function that bends in the same way as the original function • At x = 0, the graphs of the original and the quadratic functions should have the • same value • same slope • same second derivative

  4. Quadratic approximation of the function • Find quadratic approximation to f (x) = cos x for x near 0 • Find a, b, and c: • Want at x = 0, two functions with first & second derivatives same value

  5. Quadratic approximations of the function • . • At x = 0 a = 1 b = 0 c =-.5

  6. Quadratic approximation of the function • So the quadratic approximation is • Graph this on the calculator in y3 to see how the graphs compare this time.

  7. Polynomial approximations of the functions • Do you think fitting a polynomial of higher degree would make a better fit? • Repeat the process for the quartic

  8. 2!c = -1 c = - 4!e =1 e = Polynomial approximations of the function 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a = 1 b = 0 3!d = 0 d = 0

  9. Polynomial approximations of the functions • So the quartic approximation is • Graph this on the calculator in y3 to see how the graphs compare this time.

  10. Taylor Polynomial of Degree n Approximating f (x) near x = 0 • Recapping … • Generalizing ...

  11. You try • Construct a Taylor polynomial of degree 7approximating the functionnear • Graph f and the Taylor polynomials of degree 2,3,4. • What do you notice about each successive approximation? • Write the Taylor polynomial for degree n. Express the expansion in sigma notation. Identify the type of series. ■

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