PRACTICE EXAM 1 - SECTIONS 11.1-12.5 (version
1)
General Exam Information
You will not be allowed anything to use on the exam besides something to write with and
a scientific calculator. The real exam will be much shorter than this, I have included many
problems for extra practice.
This exam will cover sections 11.1-12.5 in the textbook. The following is a list of things
we did NOT talk about in class and will NOT show up on the exam. Everything else from
these sections is considered fair for the exam.
- Section 11.1 - Midpoint formula
- Section 11.3 - ”Direction Angles and Cosines”
- Section 11.3 - Formula for Work Done (I may, however, give you a problem which states
the work formula for you and asks you to use it).
- Section 11.4 - Torque
- Section 11.5 - ”Kepler’s Laws of Planetary Motion”
- Section 11.7 - Components of Acceleration, Binormals
- Section 11.9 - ”Spherical Coodinates in Geography”
- Section 12.1 - ”Computer Graphs”
- Section 12.1 - ”Applications of Contour Plots” (you should know about Level Curves
though)
Practice Problems
1. A steel beam in a bridge bears two loads, one with a force vector of < −4, −3 > and
another with a force vector of < 3, −7 >. What is the vector of the force that must
hold the steel beam in place in order for it to stay in place?
2. Find the angle between the vectors < −1, 3 > and < 4, 7 >. Write your answer in
degrees. Draw the vectors in the same coordinate plane.
3. Find the projection of the vector < 1, 4, 3 > onto the vector < 6, −1, 5 >. Are the
vectors perpendicular? How do you know?
4. Find the area of the triangle with vertices (1, 1, 0), (2, 3, −2), (0, 4, 5). (Use the cross
product)
5. Find the equation of the plane that passes through the points (0, 0, 4), (1, 0, −3), (−2, 3, 5)
1
6. Find the tangent line to the vector-valued function F(t) =< 3t, t2 − 5, sin(t) > at the
point (0, −5, 0). Write your answer in both symmetric and parametric form. Also write
a unit vector in the direction of the line.
7. Find the parametric form of the line with symmetric equations
x−4
−2
=
y−1
6
=
z+3
.
1
8. Find the speed, acceleration, and curvature of a particle moving in 3 dimensions with
parametric equations x = et , y = e−t , z = 2t when t = ln(2)
9. Find the curvature of the equation y = sin(x) + x when x = π.
2
2
10. Sketch the cross sections for z = y4 − x9 corresponding to y = 0, and corresponding to
z = 2. What type of quadric surface is this?
11. Sketch level curves for the function F (x, y) =
x2 +2x
y
at z = −2, 0, and 4.
12. Change the rectangular coordinates (4, 1, 6) into cylindrical coordinates and spherical
coordinates. List all angle measures in degrees.
13. Rewrite the equation x2 +y 2 = z in cylindrical coordinates and in spherical coordinates.
14. Rewrite the equation 6 = sin2 (φ) sin(θ) cos(θ) in rectangular coordinates.
15. Find the following limit or give a reason that it does not exist:
x2 + 2y
x,y→(2,2) x2 − 2y
lim
16. For the function F (x, y) = x2 + 2xy + 1, what direction represents the fastest rate of
increase from the point (1, −1)? What is the directional derivative in the direction of
2i + j? Name a unit vector for which the directional derivative in that direction is 0.
17. Find the tangent plane to the surface F (x, y) = (x − 2y)2 at the point (2, 2). Use the
tangent plane to approximate the value of the function at the point (2.1, 2.1).
2