MAT 240 Calculus III
Fall 2014 Quiz#5
NAME:_________________
1. If r(t) = cos(3t ) i + t 2 j + sin( 3t ) k describes the path of an object, find the following:
A) The velocity of the object.
B) The speed of the object.
1
3
2. A sketch of the position vector r(t) =  t 3 
2
 i + 2t  j is shown below. Do the following:
3
7
6
5
4
3
2
1
A) Find the position vector, the velocity vector, and the
acceleration vector when t  2 .
-7 -6 -5 -4 -3 -2 -1
B) On the graph provided, sketch the velocity vector
and the acceleration vector at t  2 such that the
initial points of these two vectors are at the terminal
point of the position vector.
-1
-2
-3
-4
-5
-6
-7
3. Find the position vector given the following: v(t) = e  t i + sin( t ) j, and r(0) = i +2j
4. Find a set of parametric equations for the tangent line to the curve given by
r(t) =
1 2
1
t , t ,  t 3 at t = 4.
16
4
y
x
1
2
3
4
5
6
7
5. A ball rolls off (launch angle 0 ) a 64 foot cliff at a constant speed of 8 ft/s, do the following:
A) Find a vector valued function for the path of the ball. The gravitational constant is 32 ft/s2
B) How long does it take for the ball to hit the floor after it leaves the table?
C) How far horizontally did the ball travel when it hits the floor?
6. If r(t) = 2t 2 i + 3t j
A) Find T(1)
7. If r(t) = 2 sin( t ) i +
B) Find aT at t = 1
5 t j + 2 cos(t ) k
A) Find T(t)
B) Find N(t)
C) Find aT
D) Find aN
C) Find aN at t = 1
8. An object has a T(2) =
1 1 1
,
,
3 3 3
and N(2) =
1
2
3
,
,
. Furthermore, at the
14 14 14
time value of t = 2, aT = 5 3 and aN = 4 14 . Use this information to help you find the
acceleration of the object at the time value of t = 2.
9. Find the arclength of the curve given by r(t) =
2 32
t i + t j + 3t k over the interval 0,6
3
10. If f ( x, y)  3xy  2 y 2 , do the three items below (simplify your answers):
A) f  2,3
B) f ( x,4 x)
C) f  x  x, y   f  x, y 
11. Sketch the domain of the following function on the xy-plane provided. Be clear on whether
or not the boundary of the domain is included or excluded from the domain.
f ( x, y) 
y 1
7
6
5
4
3
2
1
x 1
-7 -6 -5 -4 -3 -2 -1
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-3
-4
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-7
y
x
1
2
3
4
5
6
7
12. Sketch the domain of the following function on the xy-plane provided. Be clear on whether
or not the boundary of the domain is included or excluded from the domain.
7
6
5
4
3
2
1
1
f ( x, y ) 
4 x 2  9 y 2  36
-7 -6 -5 -4 -3 -2 -1
13. If f ( x, y )  y 2  4 x 2 then do the following:
-7 -6 -5 -4 -3 -2 -1
B) Sketch the level curve for z = -16
x
1
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-7
7
6
5
4
3
2
1
A) Sketch the level curve for z = 16
y
y
6
5
Z=20
Z=10
4
Z=5
3
Z=0
2
1
x
-5 -4 -3 -2 -1
-1
-2
-3
1
2
3
4
5 6
7
8
9 10 11
3
4
5
6
7
y
-1
-2
-3
-4
-5
-6
-7
13. A contour map of a function f ( x, y ) is shown below. Use it to sketch the 3D surface
represented by f ( x, y )
7
2
x
1
2
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