. (Here V is measured in gallons and h is measured in feet).
At what rate is the water level increasing when the level equals five feet?
Problem 1. Related Rates (20 points)
Water is pouring into a conical vat at a rate of 300 gallons per minute
(see picture). The volume V of water is related to the water level
h by the formula
. (Here V is measured in gallons and h is measured in feet).
At what rate is the water level increasing when the level equals five feet?
Since
, we must have 
, so 
feet per minute.
Problem 2. Growth and decay (20 points)
The population of a certain town is growing exponentially. In a certain year (call this t=0), the population is 15,000. Forty years later (t=40), the population is 60,000.
a) By what percent is the town growing each year? In other words, what
is the growth rate r (also called k)? Give an exact
answer: something like 
, not like 
.
Since this is exponential growth, we must have 
. We are told that y(0)=15,000 and that y(40)=60,000. Thus
Put another way, the population doubles every 20 years.
b) If this exponential growth continues, what will the population of the town be at t=80? Give an exact answer, and then simplify.
. Another way to see this is that the population doubles every 20 years,
quadruples every 40 years, and so is multiplied by 16 every 80 years.
[Historical note: the town is Austin, and the starting date is 1880. Austin's population has experienced steady exponential growth for the past 150 years].
Problem 3. Velocity and time (20 points)
A car is moving with velocity dx/dt = 50 + 10 t, where x is measured in miles, t in hours, and dx/dt in miles/hour. At time t=0 the car is at milepost x=100. Where is it at time t=2?
. 
, so we must have C=100, so 
, and 
.
Problem 4. L'Hopital's rule (20 points)
Evaluate the following three limits:
a) 
.
b) 
. L'Hopital's rule does not apply here.
c) 
.
d) 
Problem 5. Definite integrals (20 points)
We wish to find the area under the curve 
between x=0 and x=4.
a) Estimate this area by dividing the interval [0,4] into four pieces and adding the areas of the corresponding rectangles.
Since we are dividing into 4 pieces, we have n=4 and 
. Our points are 
, 
, 
, 
and 
. Using upper rectangles, we get 
. [It is also OK to use lower rectangles, in which case we get 
.]
b) Estimate the area by dividing the interval into n pieces.
Leave your answer as a sum, like 
. (No, that's not the right answer). You do NOT need to evaluate this sum.
, and 
. Our sum is 
. [Alternatively, using lower sums, you could write 
].
Problem 6. Indefinite integrals (20 points)
Evaluate the following indefinite integrals
a) 
b) 
where we have used the substitution 
, 
.
c) 
.
d) 
,
where we have used the substitution 
, 
.