Exam 2 formulas Flashcards
tangent plane to a curve of 3 variables w=f(x,y,z)
fx(p0)(x-x0) + fy(p0)(y-y0) +fz(p0)(z-z0)=0
normal line of a curve of three variables w=f(x,y,z)
x=x0 + fx(p0)t
y=y0 + fy(p0)t
z=z0 + fz(p0)t
tangent plane to a surface of 2 variables z=f(x,y)
fx(x-x0) + fy(y-y0) - (z-z0)
parametric equation for line tangent to curve of intersection of surfaces (any amount of variables)
gradient f(x,y,z) X gradient g(x,y,z)
parametrize normally
linearization
f(x0,y0)+fx(x0,y0)(x-x0)+fy(x0,y0)(y-y0)+(z)…
contour curve
when f(x,y,z) = c
when c = f(x0,y0,z0)
all second partial derivatives of a two variable function
fxx, fyy, fxy, fyx
chain rule of dw/dt when w=f(x,y) and when x=x(t) and y=y(t)
(∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt) (just add variables when necessary)
limit definition of a plane directional derivative
lim (f(x0+su1, y0 +su2) - f(x0,y0)) /s
s –> 0
Implicit differentiation 3 variables
(dy/dx) = -(Fx/Fy)
(dx/dy) = -(Fy/Fx)
(dz/dx) = -(Fx/Fz)
(dx/dz) = -(Fz/Fx)
(dz/dy) = -(Fy/Fz)
(dy/dz) = -(Fz/Fy)
derivative in direction of most rapid increase, most rapid decrease, 0 change
∇f ⋅ (∇f/(|∇f|)),
∇f ⋅ -(∇f/(|∇f|),
normal to ∇f ⋅ (∇f/(|∇f|)
gradient vector ∇f
∇f = (∂f/∂x)i + (∂f/∂y)j (+ (∂f/∂z)k) (for three dimensions)
derivative in a direction
∇f|p0 ⋅ U
(U must be a unit vector)
differentials
estimation of change
df = fx(x0,y0)dx + fy(x0,y0)dy
Is fxx positive or negative for local minimums
positive (hessian must be > 0)
