Math 224 Fall 00
Final Exam Dec
12
Good
luck! Please show all work since this will be graded with partial credit.
Problem 
Points 
Score 
Prob 
Pts 
Score 


1 
30 

9 
15 

Test 1 

2 
10 

10 
10 

Test 2 

3 
10 

11 
10 

Test 3 

4 
10 

12 
15 

Test 4 

5 
15 

13 
15 

HW/Q 

6 
10 

14 
10 

Final 

7 
10 

15 
10 

Total 

8 
10 

16 
10 

% 




Total 
200 

Grade 

Note: Homework scores
were only used if they helped the final grade.
They were not used if they lowered the grade.
_{} _{}
Spherical Coords’s
_{}
_{}
_{} _{} _{} _{}
_{} _{} _{} _{}
1.(3pts
each) Circle T for True, F for False.
a) T F Two vectors are orthogonal if their dot
product equals zero.
b) The direction of a vector is defined to be a _______ vector
that
points in the same direction as the vector.
c) T F _{}
d) T F A region is open if it consists of all
interior points.
e) A point (x, y) is ________________ point of a region R if it is the
center
of
a disk that lies entirely in R.
f) The gradient of a differentiable function of two variables is always
___________
to the function’s level
curves.
g) T F The absolute maximum of a bounded function
may not occur at a boundary point.
h) T F If the density of an object is constant, the
center of mass is called the centroid
of
the object.
i) T F The area of a closed and bounded region R in
the polar coordinate plane is
_{}
j) A function has a saddle point when the ________________ is evaluated and
found
to be negative.
2.(10pts)
Find the parametric equations for the line that passes thru the point (0, 1, 3)
and
is
parallel to the vector v = 2i – 3j + 2k.
3.
(10pts) For the vectors A = 2i – 3j + 2k, and B =
i – 1j + 2k find their cross product.
4.(10pts) Sketch the figure: _{}
5.(15pts) Show that the dot product of the position function r = sin(t)i + cos(t)j + 8k and the corresponding acceleration has a constant value, and find this value._{}
6.(10pts)
Find and describe the level curve for the function _{} at the point (2, 1).
7.(10pts)
Given _{}write the chain rule formula for
finding
_{}
8.(10pts)
Solve the IVP, _{}with initial conditions, when t = 0, _{}
9.
(15pts) To land on the green you must first clear a pine tree with your
approach shot. If you
hit
the golf ball at a speed of 90 ft/sec and the ball leaves the ground at a 60
degree angle, will
it
clear the top of the 60 foot tree that is 40 yards away?
10.(10pts)
Find the limit: _{}
11.(10pts)
The three dimensional Laplace equation is _{} appears in applied
math.
Show
that _{} satisfies a Laplace
equation.
12.
(15pts) The dimensions L, H, W of a box
are changing with respect to time. At
an instant in time
L =
1 m, H = 2 m, and W = 3 m, dL/dt = 1 m/s, dH/dt = 1 m/s, and dW/dt =
3m/s. At what rate is the
volume
of the box changing? (V = LHW).
13.
(15pts) Find the direction in which
the function _{} is increasing most
rapidly
at the point (1, 1).
14.(10pts)
Find all the local max, mins, and saddle points of the function, f(x, y), given
that
_{}
15.(10pts)
Integrate the function _{} over the disk _{}
16.(10pts)
Set up the double integral to find the volume of the region that lies under the
paraboloid
_{}, and above the triangle enclosed by the lines y = x, x = 0,
and x + y = 2, in the xyplane.