13. Advanced Topics or Review

Differentiation/integration Interchange Issues — Quiz

Test your understanding of differentiation/integration interchange issues with 5 practice questions.

Practice Questions

Question 1

Which theorem is commonly used to justify $\lim_{n\to\infty}\int_a^b f_n(x)\,dx = \int_a^b \lim_{n\to\infty} f_n(x)\,dx$ when the functions are dominated by an integrable function?

Question 2

Suppose $f_n\to f$ uniformly on $[a,b]$. What can you conclude about the integrals?

Question 3

Which rule describes the formula $\frac{d}{dx}\int_a^b f(x,t)\,dt = \int_a^b \frac{\partial}{\partial x}f(x,t)\,dt$ under suitable hypotheses?

Question 4

Let $F(x)=\int_0^1 x t\,dt$. What is $F'(x)$?

Question 5

Let $G(x)=\int_0^x t^2\,dt$. What is $G'(x)$?