Practice Second Midterm

1. Find the derivatives of the following functions

f(x) = (x3-x)4

g(x) = æ ç è x+1 x-1 ö ÷ ø 5

h(x) =   ____ Ö2x+1 x2-1

k(x) = (x3+2x+1) æ ç è 2+ 1 x2 ö ÷ ø

l(x) = 2x2 - 4 x + 2 Öx

2. For the function

f(x) = x x2+1

(a) Find an equation of the tangent line at the point (0,0).
(b) Find the point(s) on the graph where the tangent line is horiziontal, and write an equation for the tangent line at those points.

3. The marketing department of Telecom corporation has determined that the demand for their cordless phones is

p = -0.02 x+600               (0 £ x £ 30,000)

where p is the phone's unit price in dollars and x is the quantity demanded.

(a) Find the revenue function R(x). If the cost of producing x=10,000 phones is C(10,000)=1,000,000 dollars, what is the corresponding profit P(10,000)?
(b) Find the marginal revenue function R¢(x). Compute R¢(10,000) and interpret your result.
(c) Determine whether the demand is elastic, inelastic or unitary when x=10,000 and x=20,000.

4. Find the absolute maximum value and absolute minimum value, if any, of the function

f(x)=8x- 1 x2

on the interval [1,3].

5. For the function

f(x)= x 1+x3

(a) Find the intervals where it is increasing and the intervals where it is decreasing.
(b) Find the relative extrema, if any.

6. For the function

f(x) = 2x3 -6x2 +6x +1

(a) Find the intervals where f is concave upward and where it is concave downward.
(b) Find the inflection points, if any.

7. The average worker at Wakefield Avionics, Inc., can assemble

N(t)=-2t3 +12 t2 +2t        (0 £ t £ 4)

ready-to-fly radio-controlled model airplanes t hours into the 8 am to 12 noon morning shift. At what time during this shift is the average worker performing at peak efficiency?

8. A man wishes to have an enclosed vegetable garden in his backyard. If the garden is to be a rectangular area of 400 ft², find the dimensions of the garden that will minimize the amount of fencing material needed.

9. A manufacturer of tennis rackets finds that the total cost C(x) (in dollars) of manufacturing x rackets per day is

C(x)=400+4x +0.0001x2

Each racket can be sold at a price of p dollars, where p is related to x by the demand equation

p=10-0.0004 x

If all the rackets that are manufactured can be sold, find the level of production that will yield a maximum profit for the manufacturer.

10. You wish to construct a closed rectangular box that has a volume of 4ft³. The length of the base of the box will be twice as long as its width. The material for the top and bottom of the box costs 30 cents/square foot. The material for the sides of the box costs 20 cents/square foot. Find the dimensions of the least expensive box that can be constructed.

11. The demand equation for the Roland portable hair dryer is given by

p=   ______ Ö225-5x          (0 £ x £ 45)

where x (measured in units of a hundred) is the quantity demanded per week and p is the unit price in dollars.

(a) Is the demand elastic or inelastic when p=8 and p=10?

(b) When is the demand unitary?

(c) If the unit price is lowered slightly from $10, will the revenue increase or decrease?

(d) If the unit price is lowered slightly from $8, will the revenue increase or decrease?

12. Sketch the graph of the function

f(x)=-2x3 +3x2 +12 x+2

13. A truck gets 400/x miles per gallon when driven at a constant speed of x mph (between 50 and 40 mph). If the price of fuel is $1/gallon and the driver is paid $8/hour, at what speed between 50 and 70 mph is it most economical to drive?

14. The Department of Interior of an African country began to record an index of environmental quality to measure progress or decline in the environmental quality of its wildlife. The index for the years 1984 through 1994 is approximated by the function

I(t)= 50 t2 +600 t2+10        (0 £ t £ 10)

(a) Compute I¢(t) and show that I(t) is decreasing on the interval (0,10).

(b) Compute I¢¢(t) and study the concavity of the graph of I on the interval (0,10).

(c) Interpret your results.

15. Two ships leave the same port at noon with ship A sailing south and ship B sailing west. One-half hour later, ship A is 6 miles from the port and running at 18 mph while ship B is 5 miles from the port and running at 15 mph. How fast is the distance between the two ships changing at that instant of time?

16. Simplify the following expressions

æ ç è 8 27 ö ÷ ø -1/3   æ ç è 81 256 ö ÷ ø -1/4

(4x2)(-3x5)2