Mathematics 241, section 1 -- Multivariable Calculus

Review Sheet for Exam 1

September 20, 2013

General Information

1. Coordinates in R³, subsets defined by coordinate equations, slices, etc.
2. Vectors, the dot product, lengths and angles
3. The cross product in R³
4. Equations of lines and planes
5. Parametric curves and motion (know how to parametrize any segment of the line through a given point with a given direction vector, any arc of the circle in the plane with a given center and radius, and any arc of an ellipse with given center, major and minor axes),
6. Tangent vectors and tangent lines to parametric curves

There will be 4 or 5 problems, each with several parts. Some may ask for a graph or the result of a calculation; others may ask for a description or explanation of some phenomenon (similar to some questions from the lab).

I will happy to try to schedule a review session before the exam, but there are some constraints. Wednesday evening is not good for me because of the Chamber Orchestra concert. Thursday morning at 8:00am is a possibility(!)

Suggested Review Problems

• Section 1.1/5,6,11
• Section 1.2/4,5,6,7,9
• Section 1.3/1bde, 7bd, 8,10
• Section 1.4/1,3,6,9,questions like 11 and permutations thereof
• End of Chapter Exercises Chapter 1/15, 16 (we did not discuss this but you should be able to figure out the idea if you think about it)
• Section 2.1/1,2,5bd, 7bd, 9, 10
• Section 2.2/1c, 2bd, 3, 8

Sample Exam Questions

Disclaimer: The following questions indicate the range of topics that may be covered and the ``style'' of the questions I might ask. The actual exam questions may be posed differently and may combine the topics we have discussed in different ways. It will also be shorter.

I. Consider the set Q = {(x,y,z) in R³ : x² - y²/4 - z² = 1}.

• A) Identify the slices of Q in planes parallel to the three coordinate planes. Take x,y,z = -2,-1,0,1,2 (at least).
• B) Use that information to generate a rough sketch of Q or a verbal description. (This is called a hyperboloid of two sheets. Can you see why?)
• C) Show that the image of the parametric curve α(t) = ((e^t + e^-t)/2, 0, (e^t - e^-t)/2) for t in R lies entirely in Q.

II.

• A) Find the equation of the plane in R³ containing the point P = (1,1,1) and the line through Q = (0,1,-1) with direction vector v = (-4,1,2).
• B) The points P,Q,Q + v from part A are three corners of a parallelogram in the plane you found. Find the fourth corner and the area of that parallelogram.
• C) Two lines with direction vectors v = (0,1,2) and w = (4,3,2) meet at (0,0,0). Find the acute angle between the two lines.
• D) Show that if u = (u₁,u₂,u₃), v = (v₁,v₂,v₃), and w = (w₁,w₂,w₃) are general vectors in space, then u x (v + w) = u x v + u x w.

III.

• A) Sketch the parametric curve α(t) = (4 cos(π t), 3 sin(π t)) for 0 ≤ t ≤ 1. Show the starting and final points and indicate the direction the curve is traced with an arrow.
• B) Give a parametrization of the circle with center (1,3) and radius r = 4 so that one full circuit of the circle is traced out clockwise for 0 ≤ t ≤ 4 π.

IV. All parts of this problem refer to the parametric curve

• A) Show that for all t, the distance from the origin to the point α(t) is given by 1 + cos(t).
• B) At how many different times t in [0,2 π] is the y-coordinate of α(t) equal to zero? Find them.
• C) Find a parametrization of the tangent line to the cardioid at the point α(π/4).
• D) Give a parametrization for any one circle whose interior region completely contains the cardioid. Explain how you determined your center and radius.