Calculus and Vectors

Limits and rates of change

A limit describes the value a function approaches; a derivative captures instantaneous change.

f^{'} (a) = h \to 0 lim \frac{f ( a + h ) - f ( a )}{h}

Derivative definition

A small set of rules differentiates broad families of composite functions.

(f g)^{'} = f^{'} g + f g^{'}

Product rule

\frac{d}{d x} f (g (x)) = f^{'} (g (x)) g^{'} (x)

Chain rule

Derivative signs reveal increasing, decreasing, extreme, and concavity behaviour.

f^{'} (x) = 0 or DNE

Critical-number candidates

Vectors encode magnitude and direction; dot and cross products reveal geometric relationships.

∥ v ∥ = v_{1}^{2} + v_{2}^{2} + v_{3}^{2}

Vector magnitude

a \cdot b = ∥ a ∥∥ b ∥ cos θ

Dot product