**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 , .