PROF. KAHN, MAT 241 MIDTERM You have 2 hours to complete this exam
DIRECTIONS: Show all work. Put a box around your answers. Number your problems. Put your
name on every sheet. No internet. Honor system. Karma.
SECTION 1:
section).
10 POINT PROBLEMS (There are 6 problems for a total of 60 points in this
1) Resolve vector v into its components
2) Describe this shape (assume 3 dimensions): 𝑧 = 𝑥 2
3) Change the function, f ( x, y) = − x − y , so that it now has a vertex at (-3, 2, 4):
2
2
2
2
4) For the function, f ( x, y) = x + y , sketch two contours of your choice.
5) Write out the vector formed by starting at the P = (2, 7, -6) and ending at Q = (6,-2, 7)
6) Find a plane with a slope of 4 in the x-direction, and 9 in the y-direction, that passes
through the point (-10, 5, 6)
SECTION 2:
section).
20 POINT PROBLEMS (There are 7 problems for a total of 140 points in this
7) Find the equation of the linear function, f(x, y), whose contour diagram is below.
8) A cube is located such that its bottom four corners have the coordinates:
(-3, 4, 6), (-3, 8, 6), (1, 4, 6), (1, 8, 6)
Give the coordinates of the center of the cube.
9) A boat is heading due east at 50 km/hr (relative to the water).
The current is moving toward the southwest at 40 km/hr.
To someone standing on the shore, what is the boat’s direction?
(For the answer, I’m looking for the number of degrees that your angle makes above
the x-axis. For example, if it looked like the boat was heading north, you would put
90 degrees. If it looked like the boat were heading west, you would put 180 degrees)
10)
What is the angle between 5i + 4 j − k and 2i + 3 j + 8k ?
11)
Find the projection of vector ⇀
𝑅 that is parallel to vector ⇀
𝑆
⇀
𝑅 = 3⇀
𝑖 + 3⇀
𝑗 +4 𝑘
⇀
⇀
𝑆 = 7⇀
𝑖 − 3⇀
𝑗 +1 𝑘
12)
Write the equation for the table below in terms of X, Y, and Z, where Z is
represented by the number in bold face.
13)
Find the volume of the parallelepiped with the edges:
⇀
⇀
𝑄 = 6⇀
𝑖 − 4⇀
𝑗 +2 𝑘
⇀
⇀
𝑅 = −1 ⇀
𝑖 + 4⇀
𝑗 +3 𝑘
⇀
⇀
𝑆 = 2⇀
𝑖 + 5⇀
𝑗 −7 𝑘