TEXAS A&M UNIVERSITY
DEPARTMENT OF MATHEMATICS
MATH 251-510
Exam 2 version A, 11 Oct 2007
Name:
Points:
/60
In all questions no work =⇒ no points!
1.
True or false? Give explanations!
(a) There is a function f such that fx = y sin(x) and fy = cos(x).
2
2
(b) If f (x, y) = ex then ∇f = 2xex .
2
2
(c) If f (x, y) = ex then df = 2xex dx.
(d) A drop of water placed in point p on a surface will go down in the direction −∇f (p).
(e) Range of f (x, y) = ln(x2 + y 2) is {(x, y) : (x, y) 6= (0, 0)} (all points apart from the
origin).
(10 points)
2.
Below, “derivative” refers to the directional derivative. Give your answers with respect to
the gradient ∇f (in (a)-(c) just giving an answer is enough):
(a) In which direction is the derivative of f maximal?
(b) What is the maximal value of the derivative?
(c) In which direction is the derivative of f minimal?
(d) In which direction is the derivative zero? Prove your answer.
(e) In which direction is the derivative half of its maximal value? Prove your answer.
(10 points)
3.
If u = f (x, y) where x = es cos t and y = es sin t, then
(a) Find ∂u/∂s and ∂u/∂t
(b) Show that
∂u
∂x
2
+
∂u
∂y
2
= e−2s
"
∂u
∂s
2
+
∂u
∂t
2 #
(10 points)
4.
The volume of a circular cylinder is V (h, r) = πr 2 h. If h = 30m, r = 50m, the height
decreases at the rate 0.5m/hr and the radius increases at the rate 1m/hr, then does the
volume increase or decrease? At what rate?
(10 points)
5.
A package in the shape of a rectangular box can be mailed parcel post if the sum of its
length and girth (the perimeter of a cross-section perpendicular to the length) is at most
78in. Find the dimensions of the package with largest volume that can me mailed parcel
post.
(10 points)
6.
Find the maximal and minimal values and their locations for
f (x, y) = x2 + xy + y 2
over the domain {(x, y) : x2 + y 2 ≤ 4}.
(10 points)
7.
(Bonus question +5%) Prove that
x2 y 3
= 0.
(x,y)→(0,0) x2 + y 2
lim