Mid-term Examination, Multivariable Calculus
Math 13, Winter 1998

Date: Thursday, 11 February; 19:00 - 20:20

Instructions:

• Complete all 4 questions. You may answer the questions in any order.
• Explain all your answers. Hand in all your working notes.
• You have 80 minutes to complete the exam.

1.

(a)

(20 points) Let $a>0$ be a positive number. Find the maximum of $f(r,z)=r^2 z$ subject to the constraint that $r^2 + z^2 = a^2$.

(b)

(5 points) What is the maximum possible volume of a cylinder contained inside a sphere of radius $a$?

2.

(a)

(5 points) Draw the region $R = \{ (x,y) : y^2 \leq x^2 \leq 1 - y^2,$    and $x\geq 0 \}$ in the $xy$-plane.

(b)

(20 points) Use polar coordinates to find the volume of the solid region lying above $R$ and below the surface      $z = (x^2 - y^2)/\sqrt{x^2 + y^2}$. (20 points)

3.

An ant is walking on a thin metal plate that occupies the region

$$\{ (x,y) : -3 \leq x\leq 3,\ \ -3 \leq y \leq 3 \}.$$

The temperature of the plate at the point $(x,y)$ is $T(x,y) = (8 - x^2 - y^2) e^x$.

(a)

(10 points) Suppose the ant is at the origin. In which direction should the ant move, so that its temperature decreases as quickly as possible?

(b)

(15 points) Where on the plate is the temperature the highest? What is the maximum temperature of any point on the plate? Justify your answer.

4.

A particle is travelling so that its position in $3$-space at time $t$ is given by

$$x(t) = \cos t, \quad y(t) = \sin t, \quad z(t) = 100 - \frac{1}{2} t^2$$

(a)

(10 points) Find the velocity, acceleration, and speed of the particle at time $t$.

(b)

(10 points) Find the tangential component of acceleration of the particle at time $t$.

(c)

(5 points) Find the length of the acceleration vector at time $t$. Use this to show that the curvature of the particle's trajectory at time $t$ is

$$\kappa = \frac{\sqrt{2+t^2}}{(1+t^2)^{\frac{3}{2}}}$$