Some questions about Calculus III

MATH 162, FALL 2017: ASSIGNMENT 2

Due: Wednesday, October 11, in class

Please answer the following questions, explaining all of the steps that you took to find solutions to the following problems.

1. Find the arc length of the following curves; (a) ~r(t) = 〈t, 4 cos(t), 4 sin(t)〉 with −3 ≤ t ≤ 3. (b) ~r(t) = 12ti + 8t3/2j + 3t2k with 0 ≤ t ≤ 3.

2. Find the vectors T, N and B associated to the following curves at the given points. (a) ~r(t) = 〈t2, 2

3 t3, t〉 at (1, 2

3 , 1).

(b) ~r(t) = 〈sin(t), ln(cos(t)), cos(t)〉 at (0, 0, 1). 3. Let ~r(s) be a space curve.

(a) Show that dB/ds is orthogonal to B. (b) Show that dB/ds is orthogonal to T. (c) Use this to conclude that there is some number τ(s) so that

dB/ds = −τ(s)N. This number is called the torsion of ~r(s). (d) Show that if ~r(s) = 〈f(s),g(s), 0〉 for some functions f(s) and

g(s) then τ(s) = 0, or equivalently, that dB/ds = 0. 4. Find the following limits or show that they don’t exist.

(a)

lim (x,y)→(0,0)

xy√ x2 + y2

(b)

lim (x,y)→(1,3)

ex 2y sin(x−y)

(c)

lim (x,y)→(0,0)

x4 −y4

x2 −y2

(d)

lim (x,y,z)→(0,0,0)

yz

x2 + 4y2 + 9z2

5. Verify the conclusion of Clairaut’s theorem in the following cases, that is, show uxy = uyx. (a)

u = y − 5x2y3 + x (b)

u = exy sin(y −x)

MATH 162, FALL 2017: ASSIGNMENT 2

6. Verify that the function u = 1/ √ x2 + y2 + z2 is a solution to the

three-dimensional Laplace equation

uxx + uyy + uzz = 0.