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Patrick's question at Yahoo! Answers (First fundamental theorem of Calculus)

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
Hello Patrick,

The problem is:

Find $\dfrac{f''(2)}{\pi}$ if $f(x)=\displaystyle\int_1^x\sin (\pi t^2)\;dt$. Enter your answer as an integer.

Solution. Using the First fundamental theorem of Calculus, $f'(x)=\sin (\pi x^2)$. Deriving again, $f''(x)=2\pi x\cos (\pi x^2)$ so, $$\dfrac{f''(2)}{\pi}=\frac{4\pi\cos(4\pi)}{\pi}=4\cos(4\pi)=4$$