Mathematics 241, section 1 – Multivariable Calculus Exam 1 – September 27, 2013

Directions

Do all work in the blue exam booklet. Do not place anything you want considered for credit on this sheet. There are 100 possible points. If you have time, you can also try for the 10 Extra Credit points (see back).

I. In this question, P = (1, 0, 1), Q = (1, 2, −2), and R = (1, −1, 1) in ℝ^{3}.

A) (15) Find the equation of the plane containing P, Q, R.

B) (15) Give a parametrization of the line segment from R to Q (in that direction), including the proper range of t-values. Which t-value gives the midpoint of the line segment from part B (the point equidistant from R and Q) in your parametrization?

C) (10) Compute the angle between the vectors v = Q − P and w = R − P.

II. Let α(t) = (u(t), v(t)) and β(t) = (w(t), z(t)) be parametric curves in ℝ^{2}.

A) (10) Show that if α(t) and β(t) are differentiable, then

\frac{d}{dt}(α(t) · β(t)) = α′(t) · β(t) + α(t) · β′(t),

(where · denotes the dot product).

B) (10) Let β(t) be a parametric curve in ℝ^{2}. Suppose that β(t) · β(t) = 1 (dot product) for all t. Show that the vector β′(t) is orthogonal to the vector β(t) for all t.

III. All parts of this question refer to α(t) = (cos(2t) cos(t), cos(2t) sin(t)) called a 4-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/4.

C) (10) Give a parametrization of a circle that will fit completely inside the “leaf” of α(t) in quadrants I and IV. (Any circle that does that is OK.)

Extra Credit (10) Identify and sketch the slices of the surface

{(x, y, z) ∈ ℝ^{3} | x^{2} − y^{2} + z^{2} = 1}

in the planes x = 1 and x = 2.