M403K 3rd Midterm Solutions

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 tex2html_wrap_inline67 . (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 tex2html_wrap_inline67 , we must have tex2html_wrap_inline75 , so tex2html_wrap_inline77 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 tex2html_wrap_inline87 , not like tex2html_wrap_inline89 .

Since this is exponential growth, we must have tex2html_wrap_inline91 . 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.

tex2html_wrap_inline99 . 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?

tex2html_wrap_inline115tex2html_wrap_inline117 , so we must have C=100, so tex2html_wrap_inline121 , and tex2html_wrap_inline123 .

Problem 4. L'Hopital's rule (20 points)

Evaluate the following three limits:

a) tex2html_wrap_inline125 .

b) tex2html_wrap_inline127 . L'Hopital's rule does not apply here.

c) tex2html_wrap_inline129 .

d) tex2html_wrap_inline131

Problem 5. Definite integrals (20 points)

We wish to find the area under the curve tex2html_wrap_inline133 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 tex2html_wrap_inline143 . Our points are tex2html_wrap_inline145tex2html_wrap_inline147tex2html_wrap_inline149tex2html_wrap_inline151 and tex2html_wrap_inline153 . Using upper rectangles, we get tex2html_wrap_inline155 . [It is also OK to use lower rectangles, in which case we get tex2html_wrap_inline157 .]

b) Estimate the area by dividing the interval into n pieces. Leave your answer as a sum, like tex2html_wrap_inline161 . (No, that's not the right answer). You do NOT need to evaluate this sum.

tex2html_wrap_inline163 , and tex2html_wrap_inline165 . Our sum is tex2html_wrap_inline167 . [Alternatively, using lower sums, you could write tex2html_wrap_inline169 ].

Problem 6. Indefinite integrals (20 points)

Evaluate the following indefinite integrals

a) tex2html_wrap_inline171

b) tex2html_wrap_inline173

where we have used the substitution tex2html_wrap_inline175tex2html_wrap_inline177 .

c) tex2html_wrap_inline179 .

d) tex2html_wrap_inline181 ,

where we have used the substitution tex2html_wrap_inline183tex2html_wrap_inline185 .