1.  (True or False) Please indicate whether the following statements are either TRUE or FALSE. You do not have to explain your answer.

(a)  (2 points) If a curve C ⊆ R2 has well-defined tangent vector at a point (x0 ,y0 ) ∈ C, and r(t) is a

differentiable parameterization of this curve with r(0) = ⟨x0 ,y0 ⟩, then r\ (0) is a non-zero tangent vector to C at (x0 ,y0 ).

TRUE or FALSE

(b)  (2 points) Let f(x,y) be a differentiable function of two variables. If (x0 ,y0 ) is such that ∇f(x0 ,y0 ) =

0, then (x0 ,y0 ) is either a local maximum or a local minimum of f

TRUE or FALSE

(c)  (2 points) There is no function f(x,y) so that, for all x,y ∈ R2 , fxy (x,y) = x2 − 3y

and

fyx (x,y) = y2 − 3x.

TRUE or FALSE

(d)  (2 points) Let f : R2 → R2 be differentiable at (x0 ,y0 ), and suppose that f(x0 ,y0 ) = (x0 ,y0 ). Then if h = f 。f 。f , Dh(x0 ) = (Df(x0 ,y0 ))3 .

TRUE or FALSE

(e)  (2 points) Let r1 (t), r2 (t) denote the position vectors of two particles moving on the unit sphere

with speed 1. If the angle between the position vectors r1 (t), r2 (t) is constant, then the sum of the angles between r(t) and r2 (t) and r1 (t) and r(t) is constant.

TRUE or FALSE

2. (7 points) Suppose f(x,y,z) is a function on R3 , and let d > 0 be an integer. We say f(x,y,z) is homogeneous of degree d if

f(tx,ty,td) = td f(x,y,z)

for every t ∈ R. Show that if f(x,y,z) is homogeneous of degree d, then

f(x,y,z) = (xf北 (x,y,z) + yfy (x,y,z) + zfz (x,y,z)).

3.   (a)  (3 points) Parameterize the ellipse E obtained by intersecting the cylinder C : x2 + y2 = 9 and the plane P : −4x + 3z = 0 in R3 .

(b)  (3 points) Find the lengths of the minor and major axes of the ellipse E .

(c)  (5 points) Find the curvature of the ellipse E\ parameterized by x(t) = 3cost, y(t) = 5sint, z(t) = 0 at the point (0, 5, 0).

(d)  (2 points) Explain, without computation, why the curvature of our original ellipse E at the point (3, 0, 4) agrees with the number you found in the previous part.

4. In this problem, we will maximize the value of the function f(x,y,z) = xyz over the unit sphere x2 + y2 + z2 = 1 in two different ways.

(a)  (10 points) First, observe that the following is a parameterization of the sphere1 , as φ runs over

[0,π] and θ runs over [−π,π):

x(φ,θ) = sinφcosθ

y(φ,θ) = sinφsinθ

z(φ,θ) = cosφ .

Use this parameterization to determine