a
Name
Dylan Zwick
Summer, 2012
Math22lO Midterm 1
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—
Date
7-)Z--oJ
Instructions: Please show all of your work as partial credit will be given where
appropriate, and there may be no credit given for problems where there is no
work shown. All answers should be completely simplified, unless otherwise
stated.
1. For
—2 and
x=2
t
1
y=3’/4—t such that 2t4
,
do the following:
(a) (10 pts) Eliminate the parameter to obtain the corresponding Cartesian
equation.
()
yD
L
—
7
Answer
(b) (10 pts) Graph the curve.
1
(c)
1
(5 pts) Indicate if the curve is simple and/or closed.
Simple:
or
Closed: T
or
F
(circle one)
(circle one)
+
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7. (10 pts each) For a=3i+4j+5k and
following:
(a) Direction cosines for a
b=2i+j+3k ,find each of the
cos c
=
=__________
cosy
=
—
(Just write a simplified expression.
(b) The angle 0 between a and b
If you don’t have a calculator just write the numerical formula for the angle.)
(iii
Oc°y
-‘
I
Cc) Find the projection of
0 =
b onto a
roj()
LI
/3
Projection of
6
b onto
a
=
,
52
i
(Z5)
)
8. (10 pts each) For the planes given by
3x 2y + 5z = 16
and
4x + 2y + z = 13,
answer the following questions.
-
(a) Find the line of intersection between the planes and write that line in
parametric equations.
yt/6
Ly+
17, 1
(o
/ i-z,
(Th
Line:
y(l7t
—±/
(b) Find the equation of the plane that is perpendicular to the line of
intersection and goes through the point (1, 3, 2).
i( i)
17 (3) ±
Equation of plane:
7
91 +L 7
—I
+
(7 \/
I
7
a
9. (a) (10 pts) Convert 2
+
x
=
2y-i-2x
y from a Cartesian coordinate equation
into an equation in cylindrical coordinates.
yy r
xr(ox
+lrco
r
Z(+(9)
Z (c
Answer:
(b) (10 pts) Convert r=2sine from a cylindrical coordinate equation into
an equation in Cartesian coordinates.
()
‘-k
—
1
y
Y+yly
(y-I)
(j)t
Answer:
8
Y
/