Exam given November 22, 2000

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  .