Prof. S. Brick Math 120
Summer ’02 Applied Calculus – Final Exam section 11
Show all your work. Justify all your answers. Do the problems in order in your bluebook.
1. Find the equation of the line tangent to y=√x2+ 7 at x= 3. On a single graph sketch
both the original function and the tangent line.
2. Using the left-hand rule with 4 evenly-spaced rectangles set up (but do not compute) the
Riemann sum for R2
1ln(x)dx. Sketch a graph showing the rectangles.
3. Suppose f(5) = 119.1 and f(25) = 31.7. Find two possible values for f(45), one if fis
linear and the other if fis exponential. Explain your calculation.
4. A population of rabbits are introduced to an island. Suppose initially there are 3000
rabbits and that the population grows at an annual rate of 20.25%. Using a non-logistic
exponential model, find the time it takes for the population to reach 100,000. Why would a
logistic model be better to use ?
5. A frozen pizza takes a trip from the freezer, to the oven, and then to your plate. Let
T(t) be its average internal temperature. On a single graph, sketch the curves y=T(t) and
y=T0(t). Be sure to label which is which, as well as the relevant sections of the graphs.
6. Find the inflection points of f(x) given the graph of f0(x) below. (Be sure to copy the
graph into your bluebook). Explain your reasoning.
7. Suppose demand for zombiepills is given by the equation q= 500−10p. Is demand elastic
or inelastic at p= $30 ? How should you change the price if you want to increase revenue ?
8. A driver of a car steps on the brakes. Let v(t) denote its speed in feet per second, t
seconds after the brakes are applied. If v(0) = 100, v(1) = 60, v(1.75) = 20 and v(2) = 0,
then give an underestimate and an overestimate for how far the car travels as it comes to a
stop. What integral would give the exact stopping distance ?
9. The depth of water in a tank oscillates once every 9 hours, with the smallest depth being
2 feet and the largest being 16 feet. Using the sine function, find a possible formula for depth
as a function of time.
10. Find the average value of y=x2over [−1,3]. Sketch a graph representing it. What does
Goldilocks and the 3 Bears have to do with average values ?
11. Use the second derivative test to classify the critical points of y=x3
−75x−231.
12. At a price of $1, sales for math-cola are 1500 per week. An increase in price of a dime
causes a drop of 100 sales. If the equilibrium price is $1, find the consumer surplus (assume
a linear demand).