Warm up Construct the Taylor polynomial of degree

Taylor Polynomials p The polynomial Pn(x) which agrees at x = 0 with f

Polynomials not centered at x = 0 Suppose we want to approximate f(x) =

Polynomials not centered at x = 0 p We modify the definition of a

Taylor Polynomial of degree n approximating f(x) near x = a p p Construct

How does the graph look? Graph y 1 = ln x p Graph Taylor

Conclusions Taylor polynomials centered at x = a give good approximations to f(x) for

Taylor Polynomials to Taylor Series p Recall the Taylor polynomials centered at x =

Taylor Series or Taylor expansion p For an infinite number of terms we can

Taylor Series for sin x p To get the Taylor series for sin x

Taylor expansions p About x = 0 p About x = 1 ■

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

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

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. p How can we write a polynomial to approximate a function about a point other than x = 0? p

Polynomials not centered at x = 0 p We modify the definition of a Taylor approximation of f in two ways. n n p 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.

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

How does the graph look? Graph y 1 = ln x p Graph Taylor polynomial of degree 4 approximating ln x for x near 1: p p Graph each of the following one at a time to see what is happening around x = 1. n n n y 5 = y 4(x) + ? ? y 6 = y 5(x) + ? ? Y 7 = y 6(x) + ? ? Replace ? ? with last term in the Taylor polynomial of next degree

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. p The higher the degree of the Taylor polynomial, the larger the interval over which it fits the function closely. p

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

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

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

Taylor expansions p About x = 0 p About x = 1 ■