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Math103 -- Test 1 -- September 22, 1999

Instructions

There are six questions plus an extra credit question.
If your work won't fit in the space provided, clearly indicate where it is continued. Use the blank page at the end for scratch.
Organize your work in an unambiguous order. Show all necessary steps.
You may use a calculator where appropriate, but be sure to show the work leading up to your calculations. Do not use your calculator for vector operations.
Good luck!

1.

(10%) Find two unit vectors that are perpendicular to both of the following vectors.

$\begin{displaymath} \vec{v}=-\vec{\imath}+8\vec{\jmath}\end{displaymath}$

$\begin{displaymath} \vec{w}=10\vec{\imath}-2\vec{\jmath}+3\vec{k}\end{displaymath}$

2.

(15%) Determine whether the following two lines are parallel, intersecting, or skew.

L₁: x = 4t + 5, y = -t - 2, z = 6t

L₂: x = s + 7, y = 5s, z = -3s + 1

3.

(15%)

(a)

Find the angle between the planes given by the following two equations. You may leave your answer in terms of the arccosine function.

3x+4y+5z=6

-(x-5)+4(y-1)+7(z+2)=0

(b)

Find a point on either of the two planes above.

4.

(20%) Suppose a submarine starts at time t=0 with a position vector of $\vec{r}_0=-100 \vec{k}$ and a velocity of $\vec{v}_0=500\vec{\jmath}$ , and proceeds to accelerate at a rate at time t of

$\begin{displaymath} \vec{a}(t)=e^t \vec{\imath} - e^{t/2} \vec{\jmath} - 10 \vec{k}.\end{displaymath}$

What will the position vector $\vec{r}(t)$ be at time t?

5.

(20%)

(a)

Find the arc-length parametrization of the following curve; that is, find the x, y, and z coordinates in terms of the arc length s measured from the point (0, 1, 0), where t=0. (Use the table of integrals on the last page, and don't forget to put things over a common denominator when integrating!)

$\begin{displaymath} x=\frac{t}{2}, y= \sqrt{1-t^2}, z=\frac{\sqrt{3}}{2} t.\end{displaymath}$

(b)

Use any method to find the unit tangent vector and the unit normal vector for the above curve at the point (0, 1, 0).

6.

(20%)

(a)

Put the following equation into a form from which you can tell what type of quadric surface the graph is without plotting it. What type of surface is it?

7x²+4y²-10z=0

(b)

Now find and graph the traces of the above equation in the planes x=1, y=1, and z=1. (I.e., find and graph the intersection of the surface with those planes.) Be sure to mark the centers and the intercepts of the curves you graph and label them with their exact coordinates. (Other than that your graph does not have to be very accurate.)

7.

THIS IS AN EXTRA CREDIT PROBLEM WORTH AN ADDITIONAL 5%.
Re-do problem 5(b) using a different method!

A Very Short Table of Indefinite Integrals

For $a\ne 0$ ,

$\begin{displaymath} \int \frac{1}{\sqrt{a^2-x^2}}\,dx = \sin^{-1}\frac{x}{a}+C\end{displaymath}$

$\begin{displaymath} \int \frac{1}{\sqrt{x^2\pm a^2}}\,dx = \ln \left\vert x + \sqrt{x^2\pm a^2} \right\vert + C\end{displaymath}$

For $a\ne 0$ ,

$\begin{displaymath} \int\frac{1}{x^2+a^2}\,dx = \frac{1}{a}\tan^{-1} \frac{x}{a}+C\end{displaymath}$

$\begin{displaymath} \int\frac{1}{x^2-a^2}\,dx = \frac{1}{2a} \ln \left\vert \frac{x+a}{x-a}\right\vert + C\end{displaymath}$

For $a\ne b$ ,

$\begin{displaymath} \int\frac{1}{(x-a)(x-b)}\,dx=\frac{1}{a-b}\left(\ln\vert x-a\vert-\ln\vert x-b\vert\right)+C\end{displaymath}$

$\begin{displaymath} \int\frac{cx+d}{(x-a)(x-b)}\,dx=\frac{1}{a-b}\left[(ac+d)\ln\vert x-a\vert- (bc+d)\ln\vert x-b\vert\right]+C\end{displaymath}$

About this document ...

This document was generated using the LaTeX2HTML translator Version 97.1 (release) (July 13th, 1997)

The command line arguments were:
latex2html -split 1 -link 1 test1.tex.

The translation was initiated by Joshua Holden on 9/27/1999

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Joshua Holden
9/27/1999