MATH 2210 Final Exam (Fall 2005) Name ________________________________
Directions: Show your work. Correct answers without relevant supporting work will not
receive full credit.
1. Determine the limit (with justification) or show that the limit does not exist.
(a)
42
2
)0,0(),(
3
2
lim yx
xy
yx
(b)
xx
yx
ee
yx
2
123
lim
2
2
)0,0(),(
2. The speed of sound traveling through ocean water is a function of temperature, salinity, and
pressure. It has been modeled by the function
DSTTTTC 016.0)35)(01.034.1(00029.0055.06.42.1449
32
where C is the speed of sound (in meters per second), T is temperature (in degrees Celsius), S
is the salinity (concentration of salts in parts per thousand), and D is the depth below the
ocean surface (in meters).
(a) Compute
D
C
.
(b) How
D
C
is affected by changes in T and S ?
(c) Does sound travel faster as depth increases? Explain.
(d) Does
D
C
increase or decrease as depth increases? Explain.
3. The speed of a cyclist riding a bike is given by
c
R
S133.3
where S is the cyclist’s speed in(in miles per hour), R is the rate of pedaling (in revolutions per
minute), and c is the number of teeth on the rear cog.
(a) Use linear differentials to estimate in the change in speed (dS) as it relates to changes in R
and c.
(b) Estimate the cyclist’s change in speed resulting from an increase in 2 cog teeth and
increasing her pedaling 5 rpm when R = 85 and c = 15 teeth. Interpret the sign and magnitude
of your answer.
(c) Calculate her actual speed(s) before and after the gear change.
Before (R = 85, c = 15) After (R = 90, c = 17)
(d) How does your answer in part b relate to the results in part c?
4. Find the directional derivative of
zxyxzyxf 12),,(
2
at the point (2,1,3) in the
direction of
kjiv 22
.
5. Find the maximum rate of change of
yyxyxf 4),(
2
at the point (2,1). In which
direction does it occur?
6. Determine and classify the critical points of
33
86),( yxyxyxf
. State the value of f for
any local maximum or minimum.
