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(a). Let f(x) = where a and b are constants. Write down the first three 1 + b terms of the Taylor series for f(x) about x = 0. (b) By equating the first three terms of the Taylor series in part (a) with the Taylor series for e* about x = 0, find a and b so that f(x) approximates e as closely as possible near x = 0 (e) (c) Use the Padé approximant to e' to approximate e. Does the Padé approximant overstimate or underestimate the value of e? (d) Use MATLAB to plot the graphs of e* and the Padé approximant to e' on the same axes. Submit your code and graphs. Use your graph to explain why the Pade approximant overstimates or underestimates the value of e. Indicate the error on the graph

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Answer:

(a). Let f(x) = where a and b are constants. Write down the first three 1 + b terms of the Taylor series for f(x) about x = 0.

To find the Taylor series for f(x), we first need to find its derivatives:

f(x) = (1 + ax)/(1 + bx) f'(x) = a(1 + bx) - ab(1 + ax)/(1 + bx)^2 f''(x) = ab(1 - 2ax + b + 2a^2x)/(1+bx)^3 f'''(x) = ab(2a^3 - 6a^2bx + 3ab^2x^2 - 2abx + b^3)/(1+bx)^4

Using these derivatives , we can write the Taylor series for f(x) about x=0:

f(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + ... = 1 + ax - abx^2 + 2a^2bx^3/3 + ...

Thus , the first three terms of the Taylor series for f(x) about x=0 are:

1 + ax - abx^2

(b) By equating the first three terms of the Taylor series in part (a) with the Taylor series for e* about x = 0 , find a and b so that f(x) approximates e as closely as possible near x = 0 (e)

We have the Taylor series for e* about x=0:

e* = 1 + x + x^2/2! + x^3/3! + ...

Comparing this to the first three terms of the Taylor series for f(x) from part (a), we can equate coefficients to get:

1 = 1 a = 1 -ab/2 = 1/2

Solving for a and b, we get:

a = 1 b = -1

Thus , the function f(x) = (1 + x)/(1 - x) approximates e as closely as possible near x=0.

(c) Use the Padé approximant to e' to approximate e. Does the Padé approximant overestimate or underestimate the value of e?

The Padé approximant to e' is:

e'(x) ≈ (

Step-by-step explanation:

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