Assignment 9

Group 1 Question

The set of all points \displaystyle S = \left\{ (x,y,z) : (x-1)^2 + y^2 + \frac{z^2}{4} = 1\right\} forms a sphere-like object called an ellipsoid centered at the point (1, 0, 0) (a picture of this surface is shown below).


Find the points on this ellipsoid that are closest to and farthest away from the point the origin.

Group 2 Question

Find the equation for the tangent plane to the surface determined by

\displaystyle y^z + x^z + x^y = 3

at the point (1, 1, 1).

Group 3 Question

Find the maximum and minimum values of f(x,y,z) = xyz over all points in the region of space W = \left\{(x,y,z) : 2x^2+2y^2+z^2 \leq 2 \right\}.

Group 4 Question

Find the linear approximation of the function

\displaystyle F(u, v) = \left(\,e^u, \arctan\left(\frac{v^2}{1+u^2}\right), e^v\right)

at the point $latex (0, 1)4.

Group 5 Question

Let f(x,y,z) be a real-valued function that satisfies

\displaystyle \nabla f\,(1, -1, \sqrt{2}) = \langle 1, 2, -2 \rangle

(and note that there are many functions that satisfy this).  Given this information, compute the partial derivative \partial f/\partial \theta at the point with spherical coordinates (\rho, \theta, \varphi) = (2, -\pi/4, \pi/4).

Group 6 Question

For this question let A be a symmetric, 2\times 2 matrix of real numbers

\displaystyle A = \left[ \begin{array}{cc} a& b \\ b & c \end{array}\right]

and define a function f:\mathbb{R}^2\to\mathbb{R} by

\displaystyle f(x,y) = f(\vec{x}) = \vec{x}\cdot\,A\vec{x}.

(a) Show that the gradient of f satisfies \nabla f = 2A\vec{x}.

(b) Compute the Hessian matrix of f.

(c) Show that f has a critical point \vec{x}_0 = (x_0, y_0) \textit{on the unit circle} precisely when

\displaystyle A\vec{x}_0 = \lambda\,\vec{x}_0

for some \lambda \in \mathbb{R}.

(d) (Bonus) What is a vector \vec{x}_0 that satisfies an equation like that in part (c) called?



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