Derivatives of Polynomials

Derivatives of Polynomials Discussion Board

One common business variable is the net profit margin (NPM). The NPM is the net profit divided by revenue. It shows the profit in dollars as a ratio to the revenue in dollars.

Part I: Why would this be important to business owners and analysts?

Part II: The American ChainMail Corporation has net profits given by the function

in hundreds of dollars, where x is the number of full suits of armor sold. Each full suit of armor costs $5,000. The revenue function is R(x) = 50x, (not R(x) = 5000x, since the revenue function must also be in hundreds of dollars). Complete the profit function by choosing a constant from the following:

If your last name starts with the letter… Choose a fixed cost between…

A-F 5000-10000

G-K 15000-20000

L-N 25000-30000

0-R 35000-40000

S-U 45000-50000 (use these numbers)

V-Z 55000-60000

NOTE: The value of c will actually be negative in the above function.

Label your first post with your last name and your assigned figure for P so that your classmates immediately know the basis of your calculations (e.g., Brook c = 7500, but Brook would use c = 75 in the profit function above since this function is in hundreds of dollars; i.e.,

Part III: Build the function for the net profit margin, , and apply the

quotient rule to find the rate of change when 40 suits of armor are sold. Then, estimate the

number of suits of armor that need to be sold so that the NPM is increasing at a rate of 10%. Please round your answer to the nearest whole suit of armor. Share and compare calculations with your classmates.

Derivatives of Polynomials

1. Find the equation of the line tangent to

at the point where

2. For the above function, what does the derivative at tell you

about the direction at that point? Is it increasing, decreasing, or neither? Why?

3. For the function ,

a. Sketch the graph of

b. Give the local extrema (peaks and valleys; maximum and minimum) for

4. If a company’s sales (in millions of dollars) for time months is given by

a. Find

b. Find

c. What does this tell you about the company’s position in the month of

May (i.e., )?

5. If the derivative can be thought of as a marginal revenue function for x units (in hundreds of items) sold, and the revenue for a company is given by the function

,

a. Sketch the graphs of the functions and

b. Find the number of units sold at which the marginal revenue begins to increase.

1

6. The altitude (in feet) attained by a model rocket t seconds into flight is given by the function

a. Find the maximum altitude attained by the rocket.

b. Why does it not make sense to use this function after

seconds? Use the graph of

h (t)

1 3 2

that is given.

h (t)

3

t 4 t 250

200

150

100

50

20 t 2, for t 2.

t

-8 -6 -4 -2 2 4 6 8 10 12 14 16 18

-50

2