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Published byJuliana Ray Modified over 9 years ago
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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).
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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. Go over “Before the Lesson” problems
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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?
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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.
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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.
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Replace ?? with last term in
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
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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.
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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.
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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?
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Taylor Series for sin x To get the Taylor series for sin x take the derivative of both sides.
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Taylor expansions About x = 0 About x = 1 ■
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