Homework Assignment 10

# Differentiation

1. Show that $f(x)=x^3$ is everywhere differentiable (and find its derivative). Use the limit definition of the derivative directly.

2. Complete Problem 1 in Chapter 3 (page 186).

# Mean Value Theorem

3. Check the Mean Value Theorem for the function $f(x)=x^3$ on the interval $[0,1]$. (That is, find a $c\in(0,1)$ satisfying the requirements of the theorem.)

4. Suppose that $f'(x)>0$ for all $x\in[a,d]$ and f is differentiable on the interval $(a,d)$. Use the Mean Value Theorem to show that if $a<b<c<d$ then $f(b)<f(c)$. This proves that f is strictly monotone increasing (see problem 3 in Chapter 3). [Hint: Use a proof by contradiction.]

5. Suppose that $f:\mathbb{R}\to\mathbb{R}$ and that $f(0)=0$ and $|f'(x)|\leq M$ for all x. Prove that $|f(x)|\leq M |x|$. Apply this to the function $f(x)=\sin(x)$.

6. What does the Mean Value Theorem say about your walk to Thayer Hall in the morning?