Mathematics 241, section 1 – Multivariable Calculus Exam 1 October 1, 2010

Directions

Do all work in the blue exam booklet. Do not place anything you want considered for credit on this sheet. All answers should be given in exact form (e.g. as radicals, not decimal approximations), and all angles should be given in radian measure. There are 100 possible points. If you have time, you can also try for the 10 Extra Credit points.

I. In this question, P = (0, 0, 1), Q = (1, 2, −4), and R = (1, −1, 0) in R^{3}. A) (15) Find the equation of the plane containing P, Q, R. B) (10) Give a parametrization of the line segment from R to Q (in that direction),

including the proper range of t-values. C) (10) Which t-value gives the midpoint of the line segment from part B (the point

equidistant from R and Q) in your parametrization? D) (10) Compute the angle between the vectors P⃗ Q and P⃗ R.

II. Let u, v, w be vectors in R^{3} with tails at the origin. A) (10) Show that

w × (u + v) = w × u + w × v.

B) (5) Show that if w is in the plane spanned by u and v, then (u × v) · w = 0.

III. All parts of this question refer to α(t) = (cos(3t) cos(t), cos(3t) sin(t)) called a 3-leaved rose curve:

A) (15) Starting from t = 0, what is the first t with α(t) = (0, 0)? At how many different

t is α(t) = (0, 0) in the range 0 ≤ t < π? B) (15) Find a parametrization of the tangent line to the rose curve at t = π/3. C) (10) Give a parametrization of a circle surrounding the “leaf” of α(t) in quadrants I

and IV, but not completely enclosing either of the other leaves. (Any circle that does that is OK.)

Extra Credit (10) Let β(t) be a parametric curve in R^{3}. Suppose that β(t) · β(t) = 1 (dot product) for all t. Show that β′(t) is orthogonal to β(t) for all t.