To approximate more complicated functions we may use rational functions (i.e., q

Question

To approximate more complicated functions we may use rational functions (i.e., quotients of polynomials) instead of polynomials (like in Taylor's approximation). Heres how in the case where g(x) = e^x (1) Let    f(x)=(1+ax)/(1+bx)    where a and b are constants. Write down the first three terms of the Taylor series for f about x = 0. (2) By equating the first three terms of the Taylor series about x = 0 for f and for g, find a and b so that f approximates g as closely as possible near x = 0. Both Questions must be answered and Show work for ratings and points. Calculus

Explanation / Answer

1)

taylor series of f is 1+x*(a-b) + bx^2 (b-a)

2)

taylor series of g is 1 + x + x^2/2

so (a-b) = 1
a = 1+ b

and
b (b-a) = 1/2
b*-1 = 1/2
b = -1/2
so a = 1/2

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