NORTHEASTERN UNIVERSITY
Final Examination
MTH1223 Fall '97
1) (10 pts) A weather ballon filled with helium
has volume V, pressure P and temperature T related by
V=(8.2) T/P
with volume in liters, pressure in atmospheres and temperature in degrees
K. If the temperature is decreasing at .3 degrees per minute, and the pressure
is decreasing at .002 atmospheres per minute, how fast is the volume changing
when the temp.= 250 degrees, and pressure=.8 atmospheres?
What kind of a problem is this?
Solution
2) A ball is rolling down a ramp which is 3 feet high and 10 feet long.
The total force on the ball due to gravity is 50 lbs (straight down), but
only the component of the force along the ramp affects the motion of the
ball.
a) Sketch the crossection of the ramp, the force due to the load, and the
components of the force due to the load along the ramp and perpendicular
to the ramp. (2 pts.)
Solution
b) Find a vector in the direction of the ramp (Write
it in
| terms of |
 |
and |
 |
) (2 pts.). |
Solution
c) Find the vector which gives the force affecting
the motion of the
ball. What angle does this vector make with respect to the x axis? (5 points)
What kind of a problem is this?
Solution
d) Find the magnitude of the force affecting the motion
of the ball. (2 points)
Solution
For another problem like this see Quiz
2, Problem 1
3) (12 points) The following table gives the
temperature readings T at various positions on a plate in degrees Celsius,
x and y measured in centimeters.
|
y=3 |
y=3.5 |
y=4 |
y=4.5 |
| x=.5 |
35 |
34.5 |
33 |
31 |
| x=1 |
36.5 |
35 |
33.5 |
33 |
| x=1.5 |
36 |
35.5 |
34 |
33.5 |
a) Use the table to find approximate values for Tx
(1,4), Ty (1,4).
Solution
b) Using the table and your answers to a) give the
local linearization for T at (1,4).
Solution
c) Use the local linearization to approximate the change
in T if x increases by .1cm and y decreases by .2cm .
Solution
For another problem like this see Quiz 4, problem
2.
4) (15 points) The temperature T at the point (x,y) (in cm) on a metal plate
is given by
a) Compute the gradient of T(x,y).
Solution
For more practice calculating partial derivatives see Quiz
3.
b) Starting from the point (3,-4) what is the rate
of change of the temperature (with respect to position) in the direction
Solution
c) From the point (3,-4), in what direction (unit vector)
should one move so that the temperature decreases as quickly as possible?
Solution
d) The iso-thermal curves are the curves of
constant temperature.
Find a normal vector to the iso-thermal curve through the point (3,-4).
Solution
e) Find a vector tangent to the isothermal curve at
the point (3,-4).
Solution
For other problems like 4b), 4c), 4d) and 4e), see Quiz
5.
5) (12 points) Find and classify the critical points
of
f(x,y)=2x3-15x2+36x+y3-3y.
For another problem like this see Quiz 6, problem
1
6.) (10 points) Let R be the trapezoid with vertices at
(-1,2), (-1,-2), (1,4), and (1,-4).
a) Find the integral of x over this region.
For another problem like this see Quiz 8, problem
3.
b) What is the average value of x on the
trapezoid?
Solution
7) (10 points) An engineer is designing a pipeline which
is supposed to connect two points P and S. The engineer decides to do it
in three sections. The first section runs from point P to point Q, and costs
$3 a mile to lay, the second section runs from point Q to point R and costs
$2 mile , the third runs from point R to point S and costs $1 a mile. Looking
at the diagram below, you see that if you know the lengths marked x and
y, then you know the positions of Q and R. Find the values of x and y which
minimize the cost of the pipeline.
What kind of a problem is this?
How
do you solve these things anyway? (with Lagrange multipliers)
Solution (with Lagrange multipliers)
8) (10 points) Use Lagrange multipliers
to find the maximum and minimim values of f(x,y)= 2x+y on the ellipse with
equation 2x2+2y2-xy=1
Solution
Another Lagrange Multiplier problem
9.) (10 points) Let S be the solid which is
bounded above by the plane with equation z=50-y/4, bounded below by the
plane z=10 and on the sides by the xz plane and the surface with equation
y=100-x2. Suppose you know that the density of the solid S(x,y,z)
is given by S(x,y,z)=x+y+z. Sketch the solid. Set up the triple integral
for the mass of the solid in rectangular coordinates; The first variable
of integration should be z. Do not calculate the integral.
Solution
For more problems on triple integrals see Quiz 9
| 10.)(10 points). Compute |
 |
and |
 |
for the half-washer R with inside radius 1 and outside radius 2. (The half
washer is sketched below.)
What kind of a problem is this anyway?
Solution
For another problem like this see Quiz 9 problem
3
11a.) (8 pts.) A level curve and gradient plot appears
below.
Circle the critical points and indicate whether they are maxes, mins, or
saddles.
Solution
11b) (2 pts.) Draw the path that a particle
will take if it starts at (0.2,-0.8) and moves tangent to the gradient field.
Solution
