Applications of Derivatives: every key term you need

Critical Numbers

Where f'(x) = 0 or undefined. Candidates for local extrema. Find by solving f'(x)=0.

First Derivative Test

f' changes from + to - → local max. f' changes from - to + → local min. f' doesn't change → not extremum.

Second Derivative Test

f''(c) < 0 → local max. f''(c) > 0 → local min. f''(c) = 0 → inconclusive. Faster than first derivative test.

Concavity

f'' > 0 → concave up (∪). f'' < 0 → concave down (∩). f''(x) = 0 → inflection point (changes concavity).

Inflection Point

Point where concavity changes. Second derivative changes sign. Graph looks like S-shaped curve.

Mean Value Theorem

If f continuous on [a,b] and differentiable on (a,b), then ∃c where f'(c) = [f(b)-f(a)] / (b-a).

Optimization (Max/Min)

Find critical numbers in domain, evaluate at endpoints and critical points. Compare to find global max/min.

Related Rates

Two quantities related; differentiate with respect to time. Example: water draining from cone.

Linear Approximation

Near point (a, f(a)), use tangent line: L(x) = f(a) + f'(a)(x-a). Approximates f(x) for x near a.

Curve Sketching

Find domain, intercepts, critical points, inflection points, asymptotes, end behavior. Sketch using all info.

Unit 3 Summary

Derivatives determine extrema, concavity, inflection points. Optimization uses critical points. MVT guarantees derivative value.

More AP Calculus BC guides

• Limits & Continuity
• Parametric & Polar
• Infinite Series
• Differential Equations