Prof. S. Brick Math 120
Fall ’02 Applied Calculus; Final Exam section 52
1. You leave home for school, driving slowly at first, but you speed up because you don’t
want to miss Calculus, but then you slow down when you see a police car a few blocks
from school. Sketch a graph of your distance from school as a function of time. Label the
various sections of the graph.
2. Rabbits are brought to an island. Suppose initially there are 5000 rabbits and that the
population grows at an annual rate of 9.5%. Using a non-logistic exponential model, find
when the population reaches 10,000. Why would a logistic model be better ?
3. Find Z3
1
2x−x2dx using the fundamental theorem of calculus, showing all your work.
Sketch a graph and indicate the geometric quantity your answer corresponds to.
4. A freshly baked pizza is removed from the oven and is placed in a room (where 72
degrees is room temperature). Newton’s Law of Cooling says that the differential equation
y0=k·(y−72), where kis a constant, can be used to model the temperature of the pizza.
Is ka positive or negative constant ? Why ?
5. Sketch a graph of a function that is concave down at x= 0 and has an inflection point
at x=π.
6. A ball is tossed up in the air. A total of 4 seconds elapses from the moment it is tossed
until it hits the ground. Sketch a graph of its velocity as a function of time for 0 ≤t≤4.
Assume an upwards velocity is considered positive.
7. Find the equation of the line tangent to y= 6x−x2at x= 1.
8. Using the right-hand rule with 4 evenly-spaced rectangles set up (but do not compute)
the Riemann sum for R2
1√x dx. Sketch a graph showing the rectangles.
9. Find the average value of y= 3x2over [1,3]. What does Goldilocks and the 3 Bears
have to do with average values ?
10. Sketch on a single graph a demand curve and a supply curve, but make them linear
functions. Then shade in the region that corresponds to the consumer surplus. Be sure to
label all the relevant things on the graph.
11. Suppose demand for zombiepills is given by the equation q= 500 −10p. Is demand
elastic or inelastic at p= $10 ? What should you do if you want to increase revenue ?
12. Let f(v) be the fuel efficiency of a car in mpg that is driven at a speed vmph. What
are the units of f0(v) ? Suppose f(20) = 15. What would f0(20) = .5 tell you ?