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Warm up

Warm up. Construct the Taylor polynomial of degree 5 about x = 0 for the function f(x)=e x . Graph f and your approximation function for a graphical comparison. To check for accuracy, find f(1) and P 5 (1). Taylor Polynomials.

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Warm up

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  1. Warm up • Construct the Taylor polynomial of degree 5 about x = 0 for the function f(x)=ex. • Graph f and your approximation function for a graphical comparison. • To check for accuracy, find f(1) and P5(1).

  2. Taylor Polynomials • The polynomial Pn(x) which agrees at x = 0 with f and its n derivatives is called a Taylor Polynomial at x = 0. • Taylor polynomials at x = 0 are called Maclaurin polynomials.

  3. Polynomials not centered at x = 0 • Suppose we want to approximate f(x) = ln x by a Taylor polynomial. The function is not defined for x < 0. • How can we write a polynomial to approximate a function about a point other than x = 0?

  4. Polynomials not centered at x = 0 • We modify the definition of a Taylor approximation of f in two ways. • The graph of P must be shifted horizontally. This is accomplished by replacing x with x – a. • The function value and the derivative values must be evaluated at x = a rather than at x = 0.

  5. Taylor Polynomial of degree n approximating f(x) near x = a • Construct the Taylor polynomial of degree 4 approximating the function f(x) = ln x for x near 1.

  6. How does the graph look? • Graph y1 = ln x • Graph Taylor polynomial of degree 4 approximating ln x for x near 1: • Graph each of the following one at a time to see what is happening around x = 1. • y5 = y4(x) + ?? • y6 = y5(x) + ?? • Y7 = y6(x) + ?? Replace ?? with last term in the Taylor polynomial of next degree

  7. Conclusions • Taylor polynomials centered at x = a give good approximations to f(x) for x near a. Farther away, they may or may not be good. • The higher the degree of the Taylor polynomial, the larger the interval over which it fits the function closely.

  8. Taylor Polynomials to Taylor Series • Recall the Taylor polynomials centered at x = 0 for cos x: • The more terms we added the better the approximation.

  9. Taylor Series or Taylor expansion • For an infinite number of terms we can represent the whole sequence by writing a Taylor series for cos x: • How would represent the series for ex?

  10. Taylor Series for sin x • To get the Taylor series for sin x take the derivative of both sides.

  11. Taylor expansions • About x = 0 • About x = 1 ■

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