Find f(4) if

Find f(4) if

Postby Guest » Thu Nov 08, 2018 8:39 pm

can somebody help please?

find f(4) if
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Guest
 

Re: Find f(4) if

Postby Guest » Tue Sep 10, 2019 7:59 am

I would start by differentiating both sides of the equation with respect to x:
[tex]2xf(x^2)= cos(\pi x)- \pi x sin(\pi x)[/tex].

Now set x= 2 and solve for f(4).

(Differentiate the integral using the "Fundamental Theorem of Calculus", [tex]\frac{d}{du}\int_0^u f(t)dt= f(u)[/tex] together with the substitution [tex]u= x^2[/tex] and the chain rule.)
Guest
 

Re: Find f(4) if

Postby Guest » Wed Nov 13, 2019 8:59 am

"Leibniz's rule" is a generalization of the "Fundamental Theorem of Calculus":

[tex]\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} f(x, t)dt= \frac{d\beta(x)}{dx}f(x,\beta(x)- \frac{d\alpha(x)}{dx}f(x, \alpha(x))+ \int_{\alpha(x)}^{\beta(x)} \frac{\partial f(x,t)}{\partial x} dt[/tex].

In this problem, [tex]\alpha(x)= 0[/tex], [tex]/beta(x)= x^2[/tex], f is a function of t only so that reduces to [tex]\frac{d}{dx}\int_0^{x^2} f(t)dt= 2x f(x^2)= x cos(\pi x)[/tex].
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