Question

compute Taylor expansion for secx at x=π/4 upto minimum 4 points
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Answer 1

Given :

sec x at a=π/4

To find :

Compute Taylor expansion for secx centered at a=π/4 upto minimum 4 points.

Solution :

f(x) = secx

f(π/4) = sec(π/4) = √2

f'(x) = secx·tanx

f'(π/4) = √2 · 1 = √2

f"(x) = (secx·tanx) tanx + secx·sec²x = secx ( tan²x + sec²x ) = secx

f"(π/4) = √2

f'''(x) = secx·tanx

f'''(π/4) = √2

$f(x) = \sum\frac{ f^{n} (a)}{n!} (x-a)^{n} = f(x) + f'(x)(x-a) + \frac{f''(a)}{2!}(x-a)^{2!} + \frac{f'''(a)}{3!}(x-a)^{3} + .............$

0≤n<∞

$secx = f(\pi /4) + f'(\pi /4)(x-\pi /4)+\frac{f''(\pi /4)(x-\pi /4)^{2}}{2!} +\frac{f'''(\pi /4)(x-\pi /4)^{3}}{4!}+.....$

$secx = \sqrt{2}+ \sqrt{2}(x-\pi /4) +\frac{ \sqrt{2}(x-\pi /4)^{2} }{2} +\frac{ \sqrt{2}(x-\pi /4)^{3} }{6}+.....$