0

I was preparing for Analysis exam and came across this exercise in one of the past exams.

Suppose $f: [a,b]\times[a,b] \to \Bbb R $ and $\frac{\partial f}{\partial t}: [a,b]\times[a,b] \to \Bbb R$ are continuous.

Prove that $F: [a,b] \to \Bbb R$ defined by $$F(t)=\int_a^t f(x,t)\ dx,\; t \in [a,b]$$ is differentiable and find $F'(t)$.

I think that I can use Leibniz integral rule to find the derivative of $F(t)$ $$F'(t)=\int_a^t \frac{\partial f}{\partial t}(x,t)\ dx + f(t,t)$$ but I don't understand the proof that is on Wikipedia. Specifically, why we need to use delta increments to apply MVT? Is there perhaps a version that is easier to understand?

Thanks

zyt
  • 31
  • Please be more specific. What part of the proof didn't you understand? – polfosol Apr 10 '18 at 15:40
  • The part about delta increments in the proof. Why do we need to use them to apply MVT? Also, where the non-integral part of the result comes from? I don't clearly see why taking the limit leads to there? – zyt Apr 10 '18 at 16:07
  • Does this one help? (It uses Fubini's theorem) – polfosol Apr 10 '18 at 20:19
  • Yes, in a sense that is understandable, but it doesn't help me to solve the question, since it doesn't deal with functions of t in the limits of integration. – zyt Apr 10 '18 at 22:49

0 Answers0