Module 5: Applications of Derivatives II
Calculus I – Study Guide
Learn Without Walls
1. First Derivative Test
f' changes + to − at c ⇒ local max. f' changes − to + at c ⇒ local min. No sign change ⇒ no extremum.
Build a sign chart: find critical points, test f' in each interval.
2. Second Derivative and Concavity
| f''(x) | Concavity |
|---|---|
| > 0 | Concave up (cup) |
| < 0 | Concave down (frown) |
Inflection point: where f'' changes sign.
Second Derivative Test
f'(c) = 0 and f''(c) > 0 ⇒ local min
f'(c) = 0 and f''(c) < 0 ⇒ local max
f'(c) = 0 and f''(c) = 0 ⇒ inconclusive
f'(c) = 0 and f''(c) < 0 ⇒ local max
f'(c) = 0 and f''(c) = 0 ⇒ inconclusive
3. Curve Sketching Checklist
- Domain
- Intercepts (x and y)
- Symmetry (even/odd/periodic)
- Asymptotes (VA, HA)
- f': increasing/decreasing, local extrema
- f'': concavity, inflection points
- Plot key points and sketch
4. Optimization
Procedure: (1) Draw picture, assign variables. (2) Write objective function. (3) Use constraint to reduce to one variable. (4) Find critical points. (5) Verify max/min.
Fencing example: 2x + y = P, maximize A = xy = x(P − 2x). Optimal x = P/4.