Exam 2 Solution Spring 2015 on Calculus III - MAC 2313 Exam...

MAC 2313 Mar 5, 2015
Exam II Prof. S. Hudson
1) [10 pts] Evaluate the integral:
R
2
0
h e 3 t , sin( πt/ 2) i dt .
2) [10 pts] Find the equation of the osculating plane for the curve cos( t ) i + sin( t ) j + k at
t = π/ 4. Explain your method briefly and circle your answer.
3) [10 pts] A particle moves along a curve with speed || v || =
√
t 2 + e - 3 t . Find the scalar
tangential component of acceleration when t = 0.
4) [10 pts] Evaluate the limit (if it exists) as ( x, y ) → (0 , 0) by converting to polar coordi-
nates, lim
p
x 2 + y 2 ln( x
2
+ y
2
). Remember to show all your work.
5) [15 pts] Let f ( x, y, z ) = 2 xy
2
z
3
, P (1 , - 1 , 2) and Q (0 . 99 , - 1 . 02 , 2 . 02). Use a total differ-
ential to approximate the change in the value of f from P to Q . Simplify completely.
6) [10 pts] Compute D
u
f at P , given f ( x, y ) = e
2 xy
, P (4 , 0), u =
3
5
i +
4
5
j .
7) [15 pts] Answer T or F; you do not have to justify your answers (but this sometimes
helps, if there is some minor misunderstanding):
If f
x
( a, b ) and f
y
( a, b ) exist then f is continuous at ( a, b ).
Gradients are normal vectors to level curves.
If f is differentiable at P and || u || = 1 then | D
u
f ( P ) | ≤ ||∇ f ( P ) || .
If D is a closed set in R
2
then every point in D is a boundary point.
If f
xy
( a, b ) and f
yx
( a, b ) exist then they are equal.
8) [10 pts] State the definition of differentiable for a function f ( x, y ) at a point ( x
0
, y
0
). If
you use df