# 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?