Calculus II, Exam 3, Fall 1998

1. [30 pts total] A particle moves along a curve with position vector given by ${\bf {r}}(t)~=~<~\;t,\; t^2,\; 4/t\;>$ for $t \geq 1$, and P is the point (2, 4, 2).

(a) [10 pts] Find the velocity vector for the motion at P.

ANS.

(b) [10 pts] Find parametric equations for the tangent line to the curve at P.

ANS. x = y = z =

(c) [10 pts] Find a formula for the distance D traveled by the particle along the curve from P = (2, 4, 2) to the point (4, 16, 1). Simplify your answer as much as possible, but do NOT evaluate.

ANS. D =

2. [34 pts total] L is the line x = 2t - 2, y = 4t, and z = - t + 1, and P is the plane 3x - y + 2z = - 6.

(a) [10 pts] Find a unit vector ${\bf {u}}$ which is parallel to L.

ANS. ${\bf {u}}$ =

(b) [10 pts] Verify that the plane P and the line L are parallel, using vector methods.
HINT: What vectors are associated with P and L? How are they related?

(c) [14 pts] Q is the plane which contains L and is perpendicular to P (see sketch). Find an equation for Q.

ANS.

3. [36 pts total]
(a) [16 pts] Match the formulas for the functions z = f(x,y) with numbers for their graphs and letters for their contour maps.

$$z = \frac{1}{1+x^2+y^2}$$     -          -     

$$z = xe^{(-x^2 + y^2)}$$         -          -     

$$z = xye^{-(x^2 + y^2)}$$         -          -     

$$z = (x^2 + 3y^2)e^{-(x^2 + y^2)}$$     -          -     

1.		2.
3.		4.
A.		B.
C.		D.

(b) [10 pts] A surface S is given by the equations

$$\left\{ \begin{array}{rl} x(x,t) = & e^t\cos\,s,\\ y(s,t)... ...end{array}\right\} \;0 \leq s \leq 2\pi, \; 0 \leq t < \infty.$$

Show that S consists of (part of) a quadric surface, by finding its equation in terms of x, y, and z. Name the surface, and describe its cross sections in planes perpendicular to the y-axis.

(c) [10 pts] Sketch the region in space specified in spherical coordinates by $0~\leq~\rho~\leq~5,\newline 0 \leq \phi \leq \pi/6,\;\;0~\leq \theta \leq 2\pi$.

BONUS: What does this region remind you of?