MATH 1920 SUMMER 2009 PRELIM 1
Please write your name on all of the exam booklets you use. Show all your work
and put all your work in the exam booklet. Circle your final answers and be sure that
you have explained them in detail. No calculators are permitted. Good luck!
1. (7 pts) The position vector of a moving particle in space is given by
r(t) = (t−sin t)i+ (1 −cos t)j,0≤t≤2π.
Find the time or times in the given time interval when the velocity and acceleration
vectors are orthogonal.
2. (7 pts) The temperature at a point (x, y) on a flat metal plate is given by
T(x, y) = 60
1 + x2+y2,
where Tis measured in ◦Cand x, y in meters. Find the rate of change of temperature
with respect to distance at the point (2,1) in
(a) the i-direction.
(b) the j-direction.
3. (10 pts) Find the following limits, if they exist.
(a) lim
(x,y)→(0,0)
xy
x2+y2.
(b) lim
(x,y)→(0,0)
x2y2
x2+y2.
4. Consider the function f(x, y) = 1
p9−x2−y2.
(a) (6 pts) Find the domain Dand the range Rof f.
(b) (2 pts) Determine if the domain of fis an open region, a closed region, or
neither.
(c) (3pts) Describe and graph the function’s level curves {(x, y)∈D:f(x, y) = c}
for c= 1/3, c= 1/2 and c= 1.
5. (10 pts) Let z=f(x, y) be a differentiable function with equal partial derivatives
everywhere, that is, fx(a, b) = fy(a, b) for all (a, b) in R2. Consider the function
g(s, t) = f(s2−t2, t2−s2)
and find ∂g
∂s and ∂g
∂t . What can you say about the graph of g? Explain.
OVER PLEASE —
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