polynomial approximations
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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
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.
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 . . . . .
Polynomial Approximations
To approximate near x = 0:
same y – intercept:
xy e
1y at 0xy e x
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
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
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
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)
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
Example::Find the Taylor (Maclaurin) Polynomial of degree n approximating the function f at x = 0.
( ) 1, 2f x x n
Example::Find the Taylor (Maclaurin) Polynomial of degree n approximating the function f at x = 0.
(2 )( ) , 3xf x e n
Example:Find the Taylor (Maclaurin) Polynomial of degree n approximating the function f at x = a.
3( ) 2 8 f x x n a
Example:Find the Taylor (Maclaurin) Polynomial of degree n approximating the function f at x = a.
( ) , 3, 4f x x n a
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.
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
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
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
Last update:4/10/2012
Assignment:
Wksht: DW 6053
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