Mt201 Last Min Formula

1

D_u f = ∇f · u

Directional derivative (almost always asked)

2

Tangent plane: z − z₀ = fₓ(x₀,y₀)(x−x₀) + fᵧ(y−y₀)

Graph z=f(x,y)

3

Normal = ⟨fₓ, fᵧ, −1⟩ or ∇F

Tangent plane / normal line / surface

4

Chain rule (one parameter): dz/dt = fₓ dx/dt + fᵧ dy/dt

Composite with curve

5

zₓ = −Fₓ/Fz , zᵧ = −Fᵧ/Fz

Implicit surfaces (very common!)

6

Hessian determinant D = fxx fyy − fxy²

Classify critical points

7

∇f = direction of steepest ascent, |∇f| = max rate

Conceptual + computation

8

Linear approximation: Δz ≈ fₓ Δx + fᵧ Δy

Sometimes appears

9

fxx fyy − fxy² > 0 and fxx > 0 → min

Quick check for nature of crit. point

10

Directional derivative max = |∇f|

Almost always asked at least once