# Midterm ```MAT 240 Calculus III
Fall 2014 Midterm
NAME:_________________
1. Find the first derivatives for the following equations. You do not need to simplify your
A) f ( x) 
x2  4
tan( x)
C) x 2  y 2  ln xy
B) h( x)  e  x cos(5x)
2. Use your integration techniques to find the following:
2x  1
A) 
B)  sin 2 ( x) cos 3 ( x) dx
dx
x
 x cos( x) dx
C)
D)

1
x 9
2
dx
3. SET UP ONLY - The integral(s) you would use to find the volume of the shape formed by
2
revolving the region bounded by the graphs of x  4   y  3 , and x  0 about the x-axis.
You can use any method, but help me by telling me which method you want to use. THX.
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
y
x
1
2
3
4
5
6
7
4. Find the equation of the tangent line at t  1 for the curve whose parametric equations are:
x  3t  1
y  sin(  t )
5. Given the vectors u =  2,  1, 3  and v =  1, 2,  1  , find the following:
A) PROJ v u
B) The area of the parallelogram made up by u and v.
C) The angle between the vectors u and v
D) Find a vector that has a magnitude of 10 and in the direction of v.
7. SET UP ONLY find the vector component forms of the force vector
and distance vector you would use to find the torque created on the bolt
by the wrench shown. There are many possible answers, so make sure
you clearly mark your x and y axes you used to orientate the picture.
You can leave your answer in terms of the angle(s) involved. (I know the
angles are not nice ones....)
8. Find the parametric equations of the line that passes through the points 1, 2, 3 and
 4, 5,1
9. Find an equation of the plane that contains the point  1, 4,  2 and is parallel to the plane
3x  2 y  z  6
10. Find a point that lies on the line of intersection of the planes 2 x  3 y  z  12 and
2 x  2 y  z  4 . There are many, just fine ONE!
11. Find the point of intersection between the line x  3t  1, y  t  2, z  2t and the plane
given by 2 x  y  3z  26
12. Find the distance between the point 4, 2, 3 and the line given by the parametric
equations x  3t  1, y  t  2, z  2t
13. Find the distance between the point 4, 2, 3 and the plane given by 2 x  y  3z  6
14. Sketch the 3D surface represented by the equation z  y 2  4
15. For the equation x 2  25 y 2  4 z 2  100 sketch the yz – trace.
16. Sketch the 3D solid that has the following spherical coordinate description:
2    4, 0   

2
, 0

2
  
17. Convert the cylindrical point  3, ,5  to rectangular coordinates.
 6 
 3  
18. Convert the spherical coordinates  4,
,  to rectangular coordinates.
 4 6
  
19. Convert the spherical coordinates 10, ,  to cylindrical coordinates.
4 3

20. Convert the rectangular equation 2 x  3 y  6 z  0 to a cylindrical equation. Simplify
your equation by solving for r .
21. Convert the rectangular equation z  x 2  y 2 to a spherical equation. Simplify your
equation by solving for  .
22. Find the domain of the vector valued function: r(t) =
t 2  25 i +
t 1
j
t2
23. Let r(t) =
2t 2  3t  1 cos( t )  1
,
2t
t 2 1
Find lim r (t )
t1


24. Do parts A, B, and C below for the vector valued function r(t) = 2  t  i + t 2  2t  2 j .
Then use your information to sketch the curve given by r(t). Make sure to show the
orientation.
y
5
A) Find r(-2) and sketch it.
4
3
2
B) Find r(0) and sketch it.
1
x
-5
-4
-3
C) Find r(2) and sketch it.
-2
-1
1
2
-1
-2
-3
-4
-5
25. Let r(t) = i +
1
d
j + ln( t ) k and u(t) = 2t i + t 2 j + k find the following:
[r(t)  u(t)]
dt
t
26. If r(t) = t 2 i + t j + k and u(t) = i + 2t j + 3t 2 k, find the following:
27. Let r(t) = t 3 i + 2t 2 j + 3t k Find
d
[r(t)  u(t)]
dt
d
r (t )
dt
EXTRA CREDIT - Find a vector valued function that describes a NON-LINEAR path way
from the point (3,0) to the point 0,3 . (There are many correct answers!!)
3
4
5
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