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Page 1: Polynomial Approximations

Polynomial Approximations

BC Calculus

Page 2: Polynomial Approximations

Intro:

REM: Logarithms were useful because highly involved problems like

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

4

3

271*(123)(317)

1log ( ) log 271 log123 3log3174y

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

Page 3: Polynomial Approximations

Polynomial ApproximationsTo approximate near x = 0:

a) the same y – intercept:

b) the same slope:

c) the same concavity:

d) the same rate of change of concavity:

xy eRequires a Polynomial with:

e) the same . . . . .

Page 4: Polynomial Approximations

Polynomial Approximations

To approximate near x = 0:

same y – intercept:

xy e

1y at 0xy e x

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xy ePolynomial Approximations

To approximate near x = 0:same y – intercept:

the same slope:

1y

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

y a bx

at 0xy e x

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xy ePolynomial ApproximationsTo approximate near x = 0:

same y – intercept:

the same slope:

the same concavity:

1y 1 1y x

2y ax bx c

at 0xy e x

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xy ePolynomial Approximations

To approximate near x = 0:same y – intercept:

the same slope:

the same concavity:

the same rate of change of concavity.

1y 1 1y x

211 1 2y x x

2 31 11 1 2 6y x x x

at 0xy e x 3 2y ax bx cx d

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2 3 4 51 1 1 11 1 ...2 6 24 120y x x x x x

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

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Method:(A)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

( )

1

( )( )!

nn

n

f a x an

0( ) y a P a1( ) y a P a bx

21( ) y a P a bx cx

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

( ) 1, 2f x x n

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

(2 )( ) , 3xf x e n

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

3( ) 2 8 f x x n a

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

( ) , 3, 4f x x n a

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Taylor and Maclaurin Polynomials

( )( )!

nnf a x a

n 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.

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

(0) 3, (0) 4, (0) 8 (0) 4f f f f

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

(0) 1, (0) 2, (0) 8 (0) 48f f f f

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Taylor’s on TI - 89

taylor ( f (x) , x , order , point)

F-3 Calc

#9 taylor (

sin 3 6y x n a

taylor ( sin (x) , x , 3 , )6

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Last update:4/10/2012

Assignment:

Wksht: DW 6053