The maximum value of f(x, y) = x y subject to the constraint x + y = 8 is:
By AM-GM inequality or Lagrange multipliers: x = y = 4. Maximum f(4,4) = 16.
If z = f(x, y) where x = r $\cos \theta$ and y = r $\sin \theta, find \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2$ when $\frac{\partial z}{\partial r} = 3$ and $\frac{\partial z}{\partial \theta} = 4$ at r = 2.
In polar coordinates: $(z_x)^2 + (z_y)^2$ = $(z_r)^2$ + $\frac{1}{r^2} (z_\theta)^2 = 3^2 + \frac{1}{4} (4^2) = 9 + 4 = 13$.
Which of the following statements about saddle points of f(x, y) are TRUE?
A saddle point is a stationary point (grad f = 0) where the Hessian determinant is negative, meaning the surface curves up in one direction and down in another.
Find the shortest distance from the origin to the plane 2x + 3y + 6z = 14.
Distance from origin to ax + by + cz = d is |d| / $\sqrt{a^2 + b^2 + c^2} = 14 / \sqrt{4 + 9 + 36} = 14 / \sqrt{49} = 14 / 7 = 2$.
The radius of curvature of the curve $y = e^x$ at x = 0 is:
Radius of curvature R = (1 + (y'$)^2$)^{3/2} / |y''|. At x=0: y = 1, y' = 1, y'' = 1. R = (1 + $1)^{3/2$} / $1 = 2^{3/2$} = 2$\sqrt{2}$.
