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Calculus Notes 12.10pdf

# Calculus Notes 12.10pdf - Calculus-Stewart Dr Berg Summer...

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Calculus- Stewart Dr. Berg Summer ‘09 Page 1 12.10 12.10 Taylor Series Taylor Polynomials Consider the problem of using a polynomial to approximate a function. Suppose we wish to use a third order polynomial to calculate values for the exponential function f ( x ) = e x . We cannot expect the polynomial to be accurate over the entire domain, so we make it as accurate as we can near some point, say zero. First: Let P ( x ) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 be the third order polynomial. Since we want accuracy near zero and f (0) = e 0 = 1 , we need P (0) = a 0 + a 1 0 + a 2 0 2 + a 3 0 3 = a 0 = e 0 = 1 so a 0 = 1 . Next: We want the polynomial to have the same slope at zero as f ( x ) = e x and f (0) = e 0 = 1 , so we need P (0) = a 1 + 2 a 2 x + 3 a 3 x 2 x = 0 = a 1 = D x e x x = 0 = 1 , so a 1 = 1 . Next: We also want the polynomial to have the same curvature at zero as the exponential function, so P (0) = 2 a 2 + 3 2 a 3 x x = 0 = 2 a 2 = D xx e x x = 0 = 1 , so a 2 = 1/2 . Next: We also want the polynomial to have the same change in curvature at zero as the exponential function, so P (0) = 3 2 a 3 = D xxx e x x = 0 = 1 , so a 3 = 1/6 . Hence, the appropriate polynomial would be P ( x ) = f (0) + f (0) x + f (0) x 2 + f (0) x 3 = 1 + x + x 2 2 + x 3 6 . Definition The n th Taylor polynomial (centered at a ) is P n ( x ) = f ( a ) + f ( a )( x a ) + f ( a ) ( x a ) 2 2!

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Calculus Notes 12.10pdf - Calculus-Stewart Dr Berg Summer...

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