Prof. S. Brick Math 120
Summer ’02 Applied Calculus; Exam 1 section 11
1. You are driving slowly to your favorite class – Math 120, of course – when you realize
that you have forgotten your bag lunch. You turn around and start to drive very fast
back home, but a police car pulls you over. You manage to use Calculus to convince the
policeman to refrain from giving you a ticket, though it does take you some time. You
then drive slowly home where you very quickly get the lunch. Before you can leave to go
to your Calculus class, your Mom tells you take your little sister to her Ballet class, which
is in the opposite direction from school. You drive very fast to the Ballet class, teasing
your sister and eating your lunch, and drop her off, practically without even coming to a
complete stop. You then drive even faster towards school. You get halfway there (halfway
as measured from the Ballet class) but your luck fails you as the very same policeman from
earlier pulls you over for speeding. You start ranting and raving about Ballet, Calculus
and chocolate-covered space aliens, when all of sudden and for no apparent reason, he
takes you away for observation (we won’t even speculate on why). Let f(t) be the distance
you are from your favorite place in the world: your Calculus class, where tis time with
t= 0 being the time you start this adventure and t=Abeing the time you are taken
away (note that f(t) is not the distance you have travelled). Sketch a graph of f(t) for
0≤t≤A. Label the various sections of the graph.
2. You make math T-shirts. You have fixed costs of $100 and variable costs of $6 per shirt.
Find the cost equation. If you sell the shirts for $10 each, find the revenue function. Graph
both of them on a single graph and find the break-even point. Label it on the graph.
3. You are selling iced cappuccino’s. If you charge one dollar, you end up making 200
sales every week. Each dime increase in price results in 10 fewer sales. Assuming it is
linear, find and graph the demand curve (with quantity on the horizontal axis and price
in dollars).
4. Suppose f(10) = 80 and f(40) = 20. Find two possible values for f(70), one if fis
linear and the other if fis exponential.
5. The half-life of Mobilium is 7 hours. How long before 18 grams decays into 2 grams
6. Let C(r) be the total cost of paying off a car loan where ris the interest rate in
percentage points. What are the units of C0(r) ? What would C0(10) = $700 mean ?
7. Suppose you know for a fact that the derivative of f(x) = √xis f0(x) = 1
2√x. Find
the equation of the tangent line at x= 9.
8. Suppose M R(100) = 40 and M C(100) = 24. Estimate the additional profit made by
increasing the production to a level of q= 106.
9. Sketch a graph of a (single) function with f(2) = −10, f0(2) = +4 and f00(2) <0 and
f(6) = 3, f0(6) = −5 and f00 (6) >0.