- #1
azatkgz
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Homework Statement
Question 5(10 marks)
Assume that a function f(x) is continuous on the interval [0,1].Express the following derivatives as formulae in terms of
[tex]x,f(x),f'(x) and \int_{a}^{x}f(t)dt[/tex]
[tex]a)\frac{d}{dx}\int_{x}^{0}tf(t)dt[/tex] [tex]b)\frac{d}{dx}\int_{0}^{x}xf(t)dt[/tex]
[tex]c)\frac{d}{dx}\int_{0}^{x}xf(x)dt[/tex]
The Attempt at a Solution
I got 6 marks from 10.Can you help me to find my mistakes?
a)[tex]xf(x)[/tex]
b)[tex]\frac{d}{dx}\int_{0}^{x}xf(t)dt=\int_{0}^{x}f(t)dt+x\frac{d}{dx}\int_{0}^{x}f
(t)dt=\int_{0}^{x}f(t)dt+xf(x)[/tex]
c)[tex]\frac{d}{dx}\int_{0}^{x}xf(x)dt=\int_{0}^{x}f(x)dt+f'(x)x\int_{0}^{x}dt+xf
(x)\frac{d}{dx}\int_{0}^{x}dt=f(x)x+f'(x)x^2+xf(x)=2xf(x)+x^2f'(x)[/tex]