1. Related Rates

Strategy: (1) Draw and label. (2) Write equation connecting variables. (3) Differentiate w.r.t. t. (4) Substitute known values and solve.

Do NOT substitute numbers until AFTER differentiating.

Sliding Ladder: x² + y² = L² ⇒ 2x(dx/dt) + 2y(dy/dt) = 0.

2. Linearization and Differentials

L(x) = f(a) + f'(a)(x − a)

dy = f'(x) dx (approximates Δy)

Relative error = dy/y Percentage error = 100 · dy/y

3. Extreme Values

Extreme Value Theorem: f continuous on [a,b] ⇒ f has an absolute max and an absolute min on [a,b].

Critical Point: c where f'(c) = 0 or f'(c) DNE. Local extrema only occur at critical points.

Closed Interval Method

Find all critical points in (a, b).
Evaluate f at critical points and at endpoints.
Largest value = absolute max; smallest = absolute min.

4. Mean Value Theorem

f continuous on [a,b], differentiable on (a,b) ⇒ ∃ c ∈ (a,b) with f'(c) = [f(b) − f(a)]/(b − a)

Rolle's Theorem: If also f(a) = f(b), then ∃ c with f'(c) = 0.

Key Consequences

f' = 0 everywhere ⇒ f is constant.
f' > 0 ⇒ f is increasing. f' < 0 ⇒ f is decreasing.
f' = g' everywhere ⇒ f = g + C.