AP® Calculus BC · Unit 5 · Topic 5.1
Using the Mean Value Theorem
The Mean Value Theorem says that if a function is continuous on [a,b] and differentiable on (a,b), somewhere inside that interval its instantaneous slope must equal its average slope over the whole interval.
Picture a car that drives 60 miles in one hour. Its average speed is 60 mph, but its speedometer reading changes the whole trip. It cannot jump straight from 0 to 60 and back without passing through every speed in between, so at some instant it must read exactly 60. That instant is guaranteed by the Mean Value Theorem, which just restates this with slopes: if f is continuous on [a,b] and differentiable on (a,b), there is some c in (a,b) where f'(c) equals the average rate of change, (f(b)-f(a))/(b-a).
Geometrically, the average rate of change is the slope of the secant line connecting (a, f(a)) and (b, f(b)). The theorem promises that somewhere between a and b, the tangent line to the curve is parallel to that secant. To actually find c, set f'(x) equal to the secant slope and solve. For f(x) = x2 on [1,4], the secant slope is (16-1)/(4-1) = 5, and f'(x) = 2x, so 2c = 5 and c = 2.5, which sits safely inside (1,4).
Both hypotheses matter: continuity on the closed interval and differentiability on the open interval. Skip either and the guarantee can fail. This is where Unit 2's lesson on differentiability and continuity comes back: a corner, cusp, or vertical tangent breaks differentiability even where the function stays continuous, so the theorem cannot promise anything there. This same continuity-and-differentiability checklist resurfaces constantly in Unit 5, since finding extrema and reading concavity both depend on knowing exactly where a function fails to be smooth.
Key terms
- Mean Value Theorem
- If f is continuous on [a,b] and differentiable on (a,b), some c in (a,b) has f'(c) equal to the average rate of change over [a,b].
- Secant line
- The straight line connecting two points on a curve, whose slope equals the average rate of change between them.
- Tangent line
- The line touching a curve at one point whose slope equals the function's instantaneous rate of change there.
- Differentiable
- Having a defined derivative at every point in question, which requires the graph to have no corners, cusps, or breaks.
Watch it explained
- How it worksHow does the mean value theorem promise a tangent line parallel to the secant?
- Worked exampleFind the value of c guaranteed by the mean value theorem for f(x) = x2 on [1, 4].
- Common mistakeWhy doesn't the mean value theorem apply to the absolute value of x on [-1, 1]?
- Ties togetherHow does a corner breaking differentiability connect back to Unit 2's differentiability and continuity topic?
Check yourself
End of topic.