Module 4 Quick Reference

Applications of Derivatives I – Calculus I

Related Rates Strategy

1. Draw & label variables

2. Write equation relating variables

3. Differentiate w.r.t. t (chain rule)

4. Substitute known values, solve

Do NOT plug in numbers before differentiating!

Common Related Rates Equations

Pythagorean

x² + y² = L²

Circle Area

A = πr²

Sphere Volume

V = (4/3)πr³

Cone Volume

V = (π/3)r²h

Linearization & Differentials

L(x) = f(a) + f'(a)(x − a)
dy = f'(x) dx ≈ Δy
Relative error = |dy/y|    % error = 100|dy/y|

Extreme Values

EVT: continuous on [a,b] ⇒ absolute max and min exist

Critical points: f'(c) = 0 or f'(c) DNE

Closed Interval Method: evaluate f at critical points + endpoints; compare

Mean Value Theorem

f'(c) = [f(b) − f(a)] / (b − a) for some c in (a,b)

Rolle's

f(a) = f(b) ⇒ f'(c) = 0

Consequences

f' > 0 ⇒ f increasing
f' = 0 ⇒ f constant