Worksheet for Section 3.10

Secton 3.10 is about several theorems involving derivatives, most importantly the Mean Value Theorem and its applications.

• Mean Value Theorem: Suppose $f(x)$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$. Then there is a number $c$ in the interval $(a,b)$ such that:
  $$f'(c)=\frac{f(b)-f(a)}{b-a}$$

This says that there must be a tangent line which is parallel to the secant line through the points $(a,f(a))$ and $(b,f(b))$. The Theorem can be proven using techniques which will be covered in Chapter 4. There is an important picture which accompanies the Mean Value Theorem, and an applet available on the course web page provides an interactive version of this picture. To demonstrate the Theorem, consider the function $f(x)=\sqrt{x-2}$ on the interval $[2,6]$. Find all of the values $c$ in the interval which satisfy the Mean Value Theorem (the Theorem guarantees at least one) -- that is, find $c$ in $[2,6]$ such that:

$$f'(c)=\frac{f(6)-f(2)}{6-2}$$

Here are a few more examples -- in each case, determine first whether the Mean Value Theorem applies (so check that the function is differentiable), and if so, find the value of $c$ that the Theorem guarantees.

• $$f(x)=\frac{x+1}{x}$$, on interval $[-1,1]$
• $g(x)=\sin x$, on interval $[-\pi,\pi]$
• $h(x)=x^2$, on interval $[-1,2]$

The other three theorems in this section are all direct consequences of the Mean Value Theorem:

• The Increasing Function Theorem: Suppose $f(x)$ is continuous on the interval $[a,b]$ and differentiable on $(a,b)$. If $f'(x)>0$ for $x$ in $(a,b)$, then $f(x)$ is increasing on $[a,b]$. If $f'(x)\geq 0$ on $(a,b)$, then $f(x)$ is nondecreasing on $[a,b]$.
• The Constant Function Theorem: Suppose $f(x)$ is continuous on the interval $[a,b]$ and differentiable on $(a,b)$. If $f'(x)=0$ for $x$ in $(a,b)$, then $f(x)$ is constant on $[a,b]$.
• The Racetrack Principle: Suppose $g(x)$ and $h(x)$ are both continuous on the interval $[a,b]$ and differentiable on $(a,b)$, and that $g'(x)\leq h'(x)$ for $x$ in $a,b)$. If $g(a)=h(a)$, then $g(x)\leq h(x)$ for all $x$ in $[a,b]$. If $g(b)=h(b)$, then $g(x)\geq h(x)$ for all $x$ in $[a,b]$.

I will show you in class how each of these follows from the Mean Value Theorem. As an example, consider the inequality $\ln(1+x)\leq x$ (for $x>-1$). Graph the functions $g(x)=\ln(1+x)$ and $h(x)=x$ together to graphically see what the inequality means. Then prove the inequality using the Racetrack Principle.