Questions: How can you compute the net change of a function f between a and b using an integral? Question: How many antiderivatives does a given continuous function have? Question: What do all antiderivatives of a given continuous function all have in common?

The Mean Value Theorem for Integrals

Basic Real Analysis 1

2. PAST PAPER QUESTION (2018): Let f be a function that satisfies the following properties: f(x + y) = f(x)f(y) for all x, y ER, and ƒ'(0) = 1. a. Explain why f is continuous at x = 0. b. Show the following two facts: that f(0) = 1, and that f(x) #0 for all x E R. (You may prove them in either order, and use whichever you prove first to prove the second if you wish.) c. Show that f is continuous for all x € R. d. Show that f is differentiable for all x ER, and compute f'(x) in terms of x and f(x). e. In fact, f is uniquely defined by the above properties, and is a well-known function. Which is it?