Taylor Polynomials & Approximation (9.7)

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Presentation transcript:

Taylor Polynomials & Approximation (9.7) March 7th, 2017

Taylor & Maclaurin Polynomials Definitions of nth Taylor Polynomial and nth Maclaurin Polynomial: If f has n derivatives at c, then the polynomial is called the nth Taylor Polynomial for f at c. If c=0, then is also called the nth Maclaurin Polynomial for f.

*The Taylor Polynomial can be used to compare an elementary function f(x) to a infinite polynomial expansion P(n), or series, by using two functions that have that pass through the same point (c,f(c)) and have the same slope (c, f(c)).

*Ex. 1: Find the nth-Maclaurin Polynomial for .

*Ex. 2: Find the nth Maclaurin Polynomial for .

*Ex. 3: Find the nth Maclaurin Polynomial for .

*Ex. 4: Find the nth Maclaurin Polynomial for .

Ex. 5: Find the third Taylor polynomial for expanded about . Graph each to compare.

Remainder of a Taylor Polynomial *The Taylor polynomial allows us to approximate a particular value of f(x) within a certain amount of error given by the remainder, . Exact value=Approximate value + Remainder

*Thm. 9.19: Taylor’s Theorem: If a function f is differentiable through order n+1 in an interval I containing c, then, for each x in I, there exists z between x and c such that where

Ex. 6: Use Taylor’s Theorem to obtain an upper bound for the error of the approximation given by Then calculate the exact value of the error.

Ex. 7: Determine the degree of the Maclaurin polynomial required for the error in approximation of the function at the indicated value of x to be less than 0.001.