cabt v3

how to find critical points

take partial derivatves solve for zero, crit points satisfy both equations

max min saddle

max:f_xx<0

min;f_xx>0

saddle:D<0

D=f_xx*f_yy-(f_xy)²

linearization of f(x,y) from (x,y) to(x₁,y₁)

f(x,y)+f_x(x,y)(x₁-x)+f_y(x,y)(y₁-y)

freshen up on lambda

please bc you know its going to be worth like 20%

tangent plane

take partial derivateves, plug in the point(abc) for slope of each section

f_x(x-a)+f_y(y-b)+f_z(z-c)=0

level set

2d splices of a 3d surface, check points and what theyre supost to equal, prob eaisest if you set one too zero

drawing rotational symetry

draw the plane with a loopy arrow around the axis of symmetry

make parametric

f(x,y,z)=r(u,v)

isolate a varible and set the others to u and v then plug in the restulting equation the that varibles portion of the vector equation

D_uhatf(x,y)

dot product of the gradient of f(x,y) and uhat

uhat dot the gradient will be greatest when uhat points

in the same direction as the gradient

chainrule

same,

d_xof the outside with the inside plugged in * d_x of the inside