∫Calculus I Unit 4 Review

The Mean Value Theorem (MVT) connects a function's average rate of change over an interval to its instantaneous rate of change at some point within that interval. It builds on Rolle's Theorem and serves as the foundation for several major results in calculus, including why a zero derivative means a function is constant and why functions with equal derivatives differ by a constant.

Rolle's Theorem

Rolle's Theorem is the starting point for understanding the MVT. It applies when three conditions are met:

The function $f$ is continuous on the closed interval $[a, b]$
The function $f$ is differentiable on the open interval $(a, b)$
The endpoint values are equal: $f(a) = f(b)$

When all three hold, there's at least one point $c$ in $(a, b)$ where $f'(c) = 0$ .

The geometric picture is straightforward: if the function starts and ends at the same height, it has to turn around somewhere. At that turning point (a local max or min), the tangent line is horizontal, meaning the slope is zero.

Rolle's Theorem is actually a special case of the Mean Value Theorem. The MVT relaxes the requirement that $f(a) = f(b)$ , making it applicable to a much wider range of situations.

Rolle's theorem interpretation, The Mean Value Theorem · Calculus

Mean Value Theorem Statement and Application

The MVT requires only two conditions (no equal-endpoint requirement):

$f$ is continuous on $[a, b]$
$f$ is differentiable on $(a, b)$

Under these conditions, there exists at least one point $c$ in $(a, b)$ such that:

$f'(c) = \frac{f(b) - f(a)}{b - a}$

The right side is the average rate of change (the slope of the secant line between $(a, f(a))$ and $(b, f(b))$ ). The left side is the instantaneous rate of change at $c$ . So the MVT says: somewhere between $a$ and $b$ , the tangent line is parallel to the secant line.

How to apply the MVT to find $c$ :

Check the hypotheses. Verify that $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ . If either fails, the MVT doesn't apply.
Compute the average rate of change. Calculate $\frac{f(b) - f(a)}{b - a}$ .
Find $f'(x)$ . Differentiate the function.
Set $f'(c)$ equal to the average rate of change and solve for $c$ . Only keep solutions where $c$ is in the open interval $(a, b)$ .

Example: Let $f(x) = x^2$ on $[1, 3]$ .

$f$ is a polynomial, so it's continuous and differentiable everywhere. Hypotheses are satisfied.
Average rate of change: $\frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = 4$
$f'(x) = 2x$
Set $2c = 4$ , so $c = 2$ . Since $2 \in (1, 3)$ , this confirms the MVT.

Rolle's theorem interpretation, Mean value theorem - Wikipedia

Consequences of the Mean Value Theorem

The MVT proves several results you'll use throughout calculus:

Zero derivative means constant function. If $f'(x) = 0$ for all $x$ in $(a, b)$ , then $f$ is constant on that interval. The proof uses the MVT: for any two points in the interval, the difference $f(x_2) - f(x_1)$ equals $f'(c)(x_2 - x_1) = 0$ .

Equal derivatives means functions differ by a constant. If $f'(x) = g'(x)$ for all $x$ in $(a, b)$ , then $f(x) = g(x) + C$ for some constant $C$ . This follows directly from the previous result applied to $f - g$ .

Sign of the derivative determines monotonicity. If $f'(x) > 0$ for all $x$ in $(a, b)$ , then $f$ is strictly increasing on that interval. If $f'(x) < 0$ , then $f$ is strictly decreasing. This is the theoretical justification for the first derivative test you use when analyzing function behavior.

These consequences also support broader applications: connecting integration and differentiation, estimating errors in linear approximations, and analyzing rates of change in applied problems (velocity, cost, etc.).

Two other existence theorems often come up alongside the MVT:

The Intermediate Value Theorem says that a continuous function on $[a, b]$ takes on every value between $f(a)$ and $f(b)$ . This is about output values, not slopes.
The Extreme Value Theorem guarantees that a continuous function on a closed interval attains both an absolute maximum and an absolute minimum.

Both continuity and differentiability are essential to the MVT, but they play different roles. Continuity on the closed interval $[a, b]$ ensures no breaks or jumps (including at the endpoints). Differentiability on the open interval $(a, b)$ ensures a well-defined tangent line exists at every interior point. A function can be continuous but not differentiable (think of $f(x) = |x|$ at $x = 0$ ), so you need to check both conditions separately.

∫Calculus I Unit 4 Review