Survey of Calculus Exam #2
Thursday, October 21, 2004
Name ______________________________
You must show all your work on this paper. Solutions without correct supporting work will not be accepted.
You must omit one problem by clearly writing “OMIT” by the problem number. If you do not omit a
problem, I will omit the last one for you. If you omit one part of a problem, you have omitted the entire
problem. Each problem is worth 10 points. You may work the problem you omit for up to 5 points extra credit
if you wish. Good luck!
1. Find the critical points for the function f ( x)  x 4  4 x 3  4 x 2  1 and classify them as relative maxima,
relative minima, or neither.
2. Determine the interval(s) where the graph of f ( x)  x 3  3x 2  24 x  1 is concave up.
3. Given f ( x)  2 x 3  3x 2  12 x  7 on the interval [-3,5], find the absolute maximum point and the absolute
minimum point.
4. Find
dy
using implicit differentiation:
dx
 x2
5. a. Simplify ln 
 y



y 2  3xy  x 2  7
3
b. An initial deposit of $5000 was made into an account that compounds interest continuously. How much
money will be in the account at the end of 8 years?
c. How long does it take $550 to double if it is invested at 4% interest compounded monthly?
6. The revenue from the sale of a luxury product t years from now is given by the equation R(t )  6  8e t .
What is the relative rate of change 3 years from now?
7. A monthly demand function is given by D( p)  18  2 p . If the present price is $4, should the price be
increased in order to increase total revenue?
8. A cable television company currently has 10,000 customers and charges $30 per month. A survey by a
marketing firm indicated that each decrease of $1 in monthly charges will result in 1000 new subscribers.
Determine the monthly charges that will result in a maximum monthly revenue.
9. An air conditioner manufacturer will be increasing production at the rate of 70 air conditioners per week.
Revenue from the sale of x air conditioners is R( x)  100 x  0.001x 2 dollars. Find the rate of change of
revenue with respect to time when the weekly production level is 550 air conditioners.
10. Use the definition of the derivative to find f (x) for f ( x)  3x 2  2 x  1 .
11. a. A company’s cost function is C ( x)  9 x  1100 dollars. Find the marginal average cost function.
b. Evaluate lim
x 7
x 2  2 x  35
x7