Calculus III Test 1 Jan. 30, 2003 NAME_________________________________
No calculators, books, or notes allowed. Justify your answers by giving
appropriate arguments and steps. Circle answers. All problems will be of equal
value. Be sure to work the given problem; otherwise you will not receive credit.
1. Let the curve Cbe given by !
r(t) = t2;1t3;1 + t3. Find parametric
equations for the tangent line to the curve at the point !
r(2) = h4;7;9i:
2. Find the length of the curve !
r(t) = (cos t)i+(sin t)j+2t3=2kfor 0t7.
3. Consider an ellipse x2=4 + y2=9 = 1. Find its curvature at the vertices
(2;0) and (0;3):(Hint: Try letting x= 2 cos t; y = 3 sin t:)
4. Find the velocity, speed, and acceleration at time tfor a particle with
position given by !
r(t) = (cos 2t)i(sin 2t)j+tk:
5. There is a battle on the planet Xix. A projectile is …red from ground level
with an initial speed of 200m=s at a 30angle of elevation above the horizontal.
Ignoring friction, …nd (horizontal) the range of the projectile (Important: On
Xix downward acceleration due to gravity is 5m=s2).
6. Let f(x; y) = xsin(3x+ 5y):Find the …rst partial derivatives of f.
7. Let f(x; y) = y=(x2+ 1). Sketch and label the level curves given by
f(x; y) = kfor k=2;1;0;1;2:
8. (10) Suppose z=g(y
x):Assume ghas continuous …rst and second deriv-
atives. Find @ z=@x and @ z=@y:
9.Determine which of the following functions solve the partial di¤erential
equation uxx +uyy = 0:
(A) u= tan1(y
x)
(B) u= ln(x2+y2)
10. Let f(x; y) = xy: (A) Sketch and label the level curves for f(x; y)=1
and f(x; y) = 4:(B) Let !
V(x; y) = D@f
@x ;@f
@y Eand sketch representations of
!
V(x; y)with tail at the point (x; y)on each of the two level curves with xvalues
given by x=1;2:(C) What can you conclude about the relation between
the vectors !
V(x; y)and the level curves?
11. Do one of the following: (Circle the letter of the problem you attempt).
(A) Find the following limit, or show it does not exist: lim
(x;y)!(0;0)
x2y2
x4+y4
(B) Find the unit tangent !
Tand unit normal !
Nat each point on !
r(t) =
t; t2; t3.
Extra Credit: The gas law for a …xed mass mof an ideal gas at absolute
temperature T, pressure P, and volume Vis P V =mRT , where Ris the gas
constant. Show that @P
rV
@V
@T
@T
@P =1:
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