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# CALCIII Test III

Linearization: L(x, y) = fx(x0 , y0)*(x – x0) + fy(x0 , y0)*(y – y0) + z0
Total Differential: df fx(x0 , y0)dx + fy(x 0 , y 0)dy
Tangent Line to a Level Curve f x (x 0 , y 0) (x – x 0) + f y (x 0 , y 0) (y – y 0) = 0
Tangent Plane (explicit) f x (x 0 , y 0) (x – x 0) + f y (x 0 , y 0) (y – y 0) – (z – z 0) = 0
Tangent Plane (implicit) f x (P 0) (x – x 0) + f y (P 0) (y – y 0) + f z (P 0) (z – z 0) = 0
Normal Line x = f x (x 0 , y 0) t + x 0 , y = f y (x 0 , y 0) t + y 0 , z = - t + z 0
f has a local max at (a, b) if f xx < 0 and f xx f yy - f xy 2 > 0 at (a, b)
f has a local min at (a, b) if f xx > 0 and f xx f yy - f xy 2 > 0 at (a, b)
f has a saddle at (a,b) if f xx f yy - f xy 2 < 0 at (a, b)
inconclusive if f xx f yy - f xy 2 = 0 at (a, b)
(r, θ, z) → (x, y, z) x = r cos θ, y = r sin θ, z = z
(x, y, z) → (r, θ, z) r = sqrt(x^2 + y^2), tan θ = y/x, z = z
(ρ, θ, Φ) → (r, θ, z) r = ρ sin Φ, θ = θ, z = ρ cos Φ
(r, θ, z) → (ρ, θ, Φ) r = sqrt(x^2 + y^2), tan θ = y/x, z = z
Created by: timmay23