Math 120 Prof. Brick
section 21 Summer 03
Applied Calculus Exam 1
Do the problems in order in your bluebook. Show your work.
1. The population of WeLoveMath grows at an annual rate of 9.2%. If initially there were
100,000 inhabitants, how long before the population reaches a half of a million ? And find
the equivalent continuous growth rate of the population.
2. Approximate the instantaneous rate of change of y=x2+1
xat x= 2 by using the
intervals [2,2.5] and [2,2.1]
3. Find the equation of the line tangent to y=√xat x= 16 given that the derivative is
y0=1
2√x. Graph both the original function and the tangent line on a single graph.
4. The depth of water in a tank oscillates once every πhours. The smallest depth is 2.4
feet and the largest is 9.4 feet. Using a sine function of the form y=A·sin(B·x) + Cto
find a formula for the depth as a function of time.
5. You decide to sell coffee during lunchtime. If you charge a dollar, you sell 100 cups,
while each dime increase in price results in 5 fewer sales. Assuming the change in demand
is proportional to the change in price, find the equation of the demand curve and graph it
(using quantity as input and price in cents as output).
6. You are driving slowly to your favorite class (Math 120) when you realize that you have
forgotten your calculator. You turn around and drive very fast back home, where you get
the calculator. But your Mom tells you to take your little brother with you and drop him
off at his school. The bad news is that his school is miles in the opposite direction and
you are not allowed to drive fast with him in the car. So you take your brother to school
slowly, muttering under your breath all the while, drop him off and turn around, driving
slowly until you pass your home. Then you drive to Calculus quickly. Let f(t) be the
distance you are from home, where tis time. Sketch a graph of f(t). Label the various
sections of the graph.
7. How is the graph of y= 4 + (7 ·f(x+ 2)) gotten from that of y=f(x) ?
8. A manufacturer sells dongles for $20 each. They have fixed costs of $50,000 and a
break-even point of 4000 dongles. Find the (linear) equations of the cost function and the
revenue function. Graph both on a single graph.
9. Suppose f(27) = 40 and f(35) = 7. Estimate for what value xis f(x) = 100 if fis
linear ? if fis exponential ? (Your answer can be of the bounds on x– the more accurate,
the more credit.)