1. ## Derivative of integral

I got stuck on this problem. Hope anyone could give some help.
Let $\displaystyle f:R \rightarrow R$ be continuous and $\displaystyle \epsilon >0$. Define $\displaystyle g(t)=\int_{t-\epsilon}^{t+\epsilon}f(x)dx$ for all $\displaystyle t \in R$. Show that g is differentiable and find g'

2. Originally Posted by jackie
I got stuck on this problem. Hope anyone could give some help.
Let $\displaystyle f:R \rightarrow R$ be continuous and $\displaystyle \epsilon >0$. Define $\displaystyle g(t)=\int_{t-\epsilon}^{t+\epsilon}f(x)dx$ for all $\displaystyle t \in R$. Show that g is differentiable and find g'
Use any fixed real number $\displaystyle t_0$ to get
$\displaystyle g(t)=\int_{t_0}^{t+\epsilon}f(x)dx+\int_{t-\epsilon}^{t_0}f(x)dx$
Now do a linear substitution that eliminates $\displaystyle \epsilon$ from the upper and lower limit of the integral, respectively,
$\displaystyle =\int_{t_0-\epsilon}^tf(z+\epsilon)dz+\int_{t}^{t_0+\epsilon} f(z-\epsilon)dz=\int_{t_0-\epsilon}^tf(z+\epsilon)dz-\int_{t_0+\epsilon}^t f(z-\epsilon)dz$
Finally: apply fundamental theorem of calculus twice to find that
$\displaystyle g'(t)=f(t+\epsilon)-f(t-\epsilon)$