Applications of the derivative
Derivatives: monotonicity and linearization feed directly into curve sketching and optimization.
Learning goals
- Sketch curves using $f$, $f'$, $f''$ (monotonicity, concavity, inflection).
- Solve optimization on closed intervals: critical points + endpoints.
- Set up and solve related rates with a clear “constraint equation.”
- State and use the Mean Value Theorem as existence of a tangent slope.
Concept sheet
Critical points: interior points where $f'(c)=0$ or $f'(c)$ undefined. Candidates for extrema; on a closed interval compare values at critical points and endpoints.
First derivative test: sign changes of $f'$ indicate local max/min (if $f$ is continuous at the point).
Second derivative: $f'' \gt 0$ ⇒ concave up; $f'' \lt 0$ ⇒ concave down. Inflection where concavity changes (often $f''=0$ or undefined, verify sign change).
- Draw and label variables that change in time.
- Write one or more equations relating variables (geometry/physics).
- Differentiate with respect to $t$ using the chain rule.
- Substitute known instantaneous values; solve for the unknown rate.
Worked patterns
Pattern A — closest distance / minimize square distance
Minimize distance to a curve by minimizing squared distance (same minimizer, smoother algebra): $D^2 = (x-a)^2 + (f(x)-b)^2$.
Pattern B — related rates ladder (template)
Right triangle: $x^2+y^2=L^2$. Then $2x\dot x + 2y\dot y = 0$, so $\dot y = -\dfrac{x}{y}\dot x$ when $y\neq 0$.
Pattern C — MVT existence
If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, there exists $c\in(a,b)$ with $f'(c) = \dfrac{f(b)-f(a)}{b-a}$. Use when you need “some slope equals average slope.”
Practice
Original drills — use Check when available, then Show solution. A quick celebration plays on correct auto-checks (respects reduced motion).
Homework / practice
Syllabus §3.9–§4.4 plus §3.10 linearization. §4.7 optimization has no PDF in
materials/ yet; Appendix K (vector derivatives) is on the schedule but not in this folder.
| Section | Link | Focus |
|---|---|---|
| 3.9 — Related rates (export A) | Open PDF | Two files with similar names — likely duplicates; keep both links until you confirm |
| 3.9 — Related rates (export B, typo) | Open PDF | “Retes” in filename |
| Review 2 (pack) | Open PDF | Exam II: Appendix K & §3.1–§3.9 (per syllabus) |
| 3.10 — Linear approximations & differentials | Open PDF | Tangent line approximation, $dy$ vs $\Delta y$ |
| 4.1 — Maximum & minimum values | Open PDF | Extreme value theorem, critical numbers |
| 4.2 — Mean Value Theorem | Open PDF | MVT, consequences for monotonicity |
| 4.3 — Shape of a graph | Open PDF | $f'$, $f''$, concavity |
| 4.4 — L'Hôpital's rule | Open PDF | Indeterminate forms |
Mistakes & checks
- Optimization: forgetting endpoints on closed intervals.
- Related rates: substituting numeric values before differentiating (usually wrong).
- Inflection: $f''=0$ is not sufficient — need a sign change of $f''$.
Slide appendix (optional)
Optional condensed slide PDFs — see materials/README.md.
Quiz mode
Homework PDF quiz: each card links to the same WebAssign export filenames as the semester map (materials/). A session shows 20 shuffled cards from 45 PDF-linked slots; exact problem wording stays in the PDF unless you merge transcripts per quiz/transcripts/README.md. Regenerate: python3 scripts/gen_hw_quiz.py.