Unit 5: Analytical Applications of Differentiation
1. The Mean Value Theorem (free preview)
If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then somewhere in between, the instantaneous rate equals the average rate:
$f'(c) = \dfrac{f(b) - f(a)}{b - a} \quad \text{for some } c \text{ in } (a,b)$
Intuition: if you averaged 60 mph over a trip, at some instant your speedometer read exactly 60. On the FRQ, stating the two hypotheses (continuous, differentiable) is usually worth a point by itself.
Worked example
Find the $c$ guaranteed by MVT for $f(x) = x^2$ on $[0, 4]$.
- Average rate: $\dfrac{f(4) - f(0)}{4 - 0} = \dfrac{16}{4} = 4$.
- Set $f'(c) = 2c = 4$.
- $c = 2$, which lies in $(0, 4)$. Done.
Common mistake: applying MVT where the hypotheses fail. $|x|$ on $[-1, 1]$ has average rate $0$ but no point with $f'(c) = 0$: the corner at $0$ breaks differentiability, and the theorem simply does not apply.
🔒
The rest of Unit 5 is in the full course
The exam's single biggest unit, worth 15–18% of your score:
- First and second derivative tests, and when each one fails
- The candidates test for absolute extrema
- Concavity, inflection points, and the $x^4$ trap
- Optimization: boxes, fences, distances, profit
- Connecting the graphs of $f$, $f'$, and $f''$
- 45 exam-style questions with verified step-by-step solutions
All 8 units, 300+ questions, 2 full-length exams. 7-day refund.