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).

screen-shot-2016-10-26-at-9-31-23-pm

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?

 

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s