Derivative of integral

**jackie** · April 14th 2010, 12:24 AM

I got stuck on this problem. Hope anyone could give some help.

Let $f:R \rightarrow R$ be continuous and $\epsilon >0$ . Define $g(t)=\int_{t-\epsilon}^{t+\epsilon}f(x)dx$ for all $t \in R$ . Show that g is differentiable and find g'

**Failure** · April 14th 2010, 03:32 AM

Originally Posted by jackie

I got stuck on this problem. Hope anyone could give some help.

Let $f:R \rightarrow R$ be continuous and $\epsilon >0$ . Define $g(t)=\int_{t-\epsilon}^{t+\epsilon}f(x)dx$ for all $t \in R$ . Show that g is differentiable and find g'

Use any fixed real number $t_0$ to get
$g(t)=\int_{t_0}^{t+\epsilon}f(x)dx+\int_{t-\epsilon}^{t_0}f(x)dx$
Now do a linear substitution that eliminates $\epsilon$ from the upper and lower limit of the integral, respectively,
$=\int_{t_0-\epsilon}^tf(z+\epsilon)dz+\int_{t}^{t_0+\epsilon} f(z-\epsilon)dz=\int_{t_0-\epsilon}^tf(z+\epsilon)dz-\int_{t_0+\epsilon}^t f(z-\epsilon)dz$
Finally: apply fundamental theorem of calculus twice to find that
$g'(t)=f(t+\epsilon)-f(t-\epsilon)$

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