Question

# The monthly demand for a certain brand of perfume is given by the demand equation p=100e^-0.0002x+150...

The monthly demand for a certain brand of perfume is given by the demand equation

p=100e^-0.0002x+150

where p denotes the retail unit price (in dollars) and x denotes the quantity (in 1-oz bottles) demanded.

a. Find the rate of change of the price per bottle when x = 1000 and when = 5 2000.

b. What is the price per bottle when x =1000? When x = 2000?
1. (10 pt) For the data given:
(i)
State and solve part (a).

(ii) Write a complete sentence explanation of what (a) tells you. Be sure that
your explanation includes units, and contains the phrase “for each
additional bottle you plan to produce and sell, you will have to lower your
price by…”.

(iii) State and solve (b).

(iv) Write a complete sentence explanation of what (b) tells you.

2. (10 pt) For this problem:
(i)
Explain the domain of the function. (HINT: Consider which x-­‐values would

possibly make no sense here in the formula.)

(ii) Write down the first and second derivatives of the price function.

(ii)Explain why you would expect the first derivative to always be negative.
[NOTE: This has NOTHING to do with how you calculate derivatives.
So start with the phrase “If you want to sell more bottles, you have to …..?
(iii) Draw the signs diagram for the first and second derivatives.

(iv)If you were to sketch this function, what would be its y-­‐intercept? Explain
what did that mean about the maximum price you can charge?

(v)Find the limit of our function as x goes to infinity. Then explain in
complete sentences what this means.

(vi) So what is the horizontal asymptote for the graph of our demand
function?

(vii) Now draw axes, label them including units, and sketch this function.
Choose a scale such that you can be sure to include all of the information
contained in #1. Label the points you found in #1.(b), the intercept from
2.(iv), and the asymptote from 2.(vi).

(viii) Finally, explain how you used the signs diagram from 2.(iii) is to get the
shape of the curve in your sketch. Be sure your explanation includes the
phrases “concave (up?)/(down?)…” and “(in?)/(de?)creasing”.