Taylor Expansion Formula for f(x) and Example with e^(x^3-1)

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I still need help with my first Taylors expansion thread, but this is a separate problem

Homework Statement



a) Give the formula for the Taylor expansion of the function f(x) about the point x=a. You should include the formula for the coeffiecients of the expansion in terms of the derivatives of f(x)
b) Find the first three non-zero terms of the Taylor expansion about x=1 of the function
f(x)=exp(x^3-1)

2. The attempt at a solution

a)
f(x + a) = f(a) + xf'(a) + x^2f''(a)/2! + ... + x^nf^n(a)/n! ... where f^n(a) means the nth derivative of f(a)

I think I might have missed the point of this question.

b)
f(x+1) = 1 + 3x^3.e^(3x-1) + [9x^6.e^(3x-1)+6x^3.e^(3x-1)]/2

I'm assuming here that exp(x^3-1) means e^(x^3-1), but this isn't notation that's used anywhere else on the paper.
 
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Taylor expansion about x = a by definition:

f(x) = f(a) + (x-a)*f'(a) + (x-a)^2/2!*f''(a) + ...
 
Use the formula:

NumberedEquation1.gif
 

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Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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