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Polynomial Approximations. BC Calculus. Intro:. REM: Logarithms were useful because highly involved problems like Could be worked using only add, subtract, multiply, and divide. - PowerPoint PPT Presentation

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Polynomial Approximations

Polynomial ApproximationsBC Calculus

1Intro:REM: Logarithms were useful because highly involved problems like

Could be worked using only add, subtract, multiply, and divide

THE SAME APPLIES TO FUNCTIONS - The easiest to evaluate are polynomials since they only involve add, subtract, multiply and divide.2

Polynomial ApproximationsTo approximate near x = 0:a) the same y intercept:b) the same slope:c) the same concavity:the same rate of change of concavity:

Requires a Polynomial with:e) the same . . . . . 3

Polynomial ApproximationsTo approximate near x = 0:same y intercept:

4

Polynomial ApproximationsTo approximate near x = 0:same y intercept:the same slope:

We want the First Derivative of the Polynomial to be equal to the derivative of the function at x = a

5

Polynomial ApproximationsTo approximate near x = 0:same y intercept:the same slope:the same concavity:

6

Polynomial ApproximationsTo approximate near x = 0:same y intercept:the same slope:the same concavity:the same rate of change of concavity.

7

Called a Taylor Polynomial (or a Maclaurin Polynomial if centered at 0)Method:

Find the indicated number of derivatives ( for n = ).

Beginning pointSlope: Concavity:etc..

(B) Evaluate the derivatives at the indicated center. ( x = a )

(C) Fill in the polynomial using the Taylor Formula

Example::Find the Taylor (Maclaurin) Polynomial of degree n approximating the function f at x = 0.

10Example::Find the Taylor (Maclaurin) Polynomial of degree n approximating the function f at x = 0.

11Example:Find the Taylor (Maclaurin) Polynomial of degree n approximating the function f at x = a.

12Example:Find the Taylor (Maclaurin) Polynomial of degree n approximating the function f at x = a.

13Taylor and Maclaurin Polynomials

In General (for any a ) Taylor Polynomial Maclaurin if a = 0 Theorem: If a function has a polynomial (Series) representation that representation will be the TAYLOR POLYNOMIAL (Series)Theorem: the Polynomial (Series) representation of a function is unique.14Example::Find the Taylor (Maclaurin) Polynomial of degree 3 approximation of f at x = 0. Use it to approximate f (.2)

15Example::Find the Taylor (Maclaurin) Polynomial of degree 3 approximation of f at x = 0. Use it to approximate f (.2)

16Taylors on TI - 89taylor ( f (x) , x , order , point)F-3 Calc#9 taylor (

taylor ( sin (x) , x , 3 , )

17Last update:4/10/2012

Assignment:

Wksht:DW 6053