Math 114 Exam 1 Fall 2010
With Helpful Hints for finding your mistakes
Rimmer
1. Match the equation with the graph of the surface it defines.
2
2
_____1. y + z = x
A.
B.
C.
D.
E.
F.
G.
H.
2
_____ 2. x = − y 2 − z 2
_____ 3. 9y 2 + z 2 = 16
_____ 4. z 2 + x 2 − y 2 = 1
_____ 5. x = y 2 − z 2
_____ 6. x 2 + 2 z 2 = 8
_____ 7. x = z 2 − y 2
_____ 8. z 2 + 4 y 2 − 4 x 2 = 4
_____ 9. x 2 + 4 z 2 = y 2
_____10. x 2 + y 2 + 4 z 2 = 10
Use the last slide of the 13.6 notes to
help you distinguish the different
types of graphs.
I.
J.
Math 114 Exam 1 Fall 2010
With Helpful Hints for finding your mistakes
Rimmer
2. If b = 1, 0,1 , c = 0, −1,1 , and a = 2, −3, k , find the value of the constant k so that
a, b, and c are coplanar.
a) -1
b) 0
c) 1
d) 2
e) 3
f) 4
g) 5
h) 6
Helpful Hint:
a, b, and c are coplanar if a ⋅ ( b × c ) = 0 (The parallelepiped determined by a, b, and c has no volume
since it has no height)
Find a ⋅ ( b × c ) by first finding b × c and then doing the dot product with a or by using the determinant
of the matrix that has a, b, and c as rows.
3. Find the point on the plane 3 x − 2 y + z = 17 closest to the point (1, 0, 0 ) . What is the
x − coordinate of this point.
a) −2
b) −1
c) 0
d) 1
e) 2
f) 3
g) 4
h) 5
Helpful Hint:
In order to find the point we have to see that the shortest distance is achieved by dropping a
perpendicular to the plane.
We first find the parametric equations of the line that contains our given point and is perpendicular to
the given plane. We use the normal vector for the direction of this line.
After finding the parametric equations for the line, we need to find the time t when our line intersects
the plane. Finally we plug this time into the parametric equations for the line to find the point that
represents the closest point on the plane to (1,0,0).
4. Find the arc length of the curve
r ( t ) = t 2 , cos t + t sin t ,sin t − t cos t
a)
1
5
b)
5
c)
1
5
for 0 ≤ t ≤ 2
d) 3 5
e)
5
5
f) 5 5
g) 1
h)
3 5
2
Helpful Hint:
b
Use the formula:
∫ r′ ( t ) dt .
a
Be careful to simplify r′ ( t ) first. You should get that r ′ ( t ) = 5t .
Math 114 Exam 1 Fall 2010
With Helpful Hints for finding your mistakes
Rimmer
5. Find the curvature of
r ( t ) = t 2 , cos t + t sin t ,sin t − t cos t
a)
1
5
b)
5
Use the formula: κ =
c)
1
5
d) 3 5
r′ (1) × r′′ (1)
r′ (1)
when t = 1.
3
e)
5
5
f) 5 5
g) 1
h)
3 5
2
. That is plug in t = 1 immediately after finding r′ ( t ) and r′′ ( t )
before doing the cross product.
6. Let L be the line given by
x = 2−t
y = 1+ t
z = 1 + 2t
L intersects the plane 2 x + y − z = 1 at a point P. Find the parametric equations for the line through P
which lies in the plane and is perpendicular to L.
Answer
First find the point P by substituting the x, y, and z from the
parametic equation for the line into the equation for the
plane. You should find the point to be (1,2,3).
x =
y =
The line in the plane that is perpendicular to the given
z
line is also perpendicular to the normal vector of the plane.
Take the cross product between the direction vector for the line and
normal vector for the plane to get the direction vector for the
desired line.
Use this direction vector along with P to get the desired result.
=
3 + 3t
Math 114 Exam 1 Fall 2010
With Helpful Hints for finding your mistakes
Rimmer
7. Show that the vector b − proja b is orthogonal to a
a) using the dot product
 a ⋅b 
b − projab = b −  2  a. To show that two vectors are
 a 


orthogonal we show that the dot product is 0.

 a ⋅b  
 a  ⋅ a.
2
 
a

 
( b − projab ) ⋅ a = b − 



a⋅b
Distributing we get:b ⋅ a −  2  a ⋅ a = b ⋅ a − a ⋅ b.
 a 


This is 0 since b ⋅ a = a ⋅ b.
b) geometrically
b − projab
− projab
b
projab
a