Expert: Paul Klarreich - 2/26/2011

Question
Hello! I am studying Business Calculus and right now my class is working on taking the derivatives of functions containing 'e^x' and logarithms. I am having trouble with a particular problem involving finding the relative maximum of a function (of course, this is where the functions derivative is equal to zero), but I keep getting stuck. Here is the question:

Research finds that the number of units of a product sold closely follows D(p) = 3000(p + 10)e^{-0.4 p}, where p is the unit price and q = D(p) is the number of units sold.

(a) Find the revenue as a function of p.
R(p)= p(3000(p+10)e^(-.4p)) <---I had no problem with this part

(b) What price should be charged in order to obtain the largest possible revenue?



Part b is where my trouble lies. I think I am getting the derivative wrong. Here is the derivative I computed for the revenue function:
3000(p+10)-.4e^(-.4p)  <---It just doesn't seem right.

Could you help with computing this derivative? Thank you and I appreciate your time.

Best,
Marissa

Answer
Questioner: Marissa
Country: United States
Category: Calculus
Private: No
Subject: Calculus Derivative, Relative Maximum
Question: Hello! I am studying Business Calculus and right now my class is working on taking the derivatives of functions containing 'e^x' and logarithms. I am having trouble with a particular problem involving finding the relative maximum of a function (of course, this is where the functions derivative is equal to zero), but I keep getting stuck. Here is the question:

Research finds that the number of units of a product sold closely follows D(p) = 3000(p + 10)e^{-0.4 p}, where p is the unit price and q = D(p) is the number of units sold.

(a) Find the revenue as a function of p.
Questioner: Marissa
Country: United States
Category: Calculus
Private: No
Subject: Calculus Derivative, Relative Maximum
Question: Hello! I am studying Business Calculus and right now my class is working on taking the derivatives of functions containing 'e^x' and logarithms. I am having trouble with a particular problem involving finding the relative maximum of a function (of course, this is where the functions derivative is equal to zero), but I keep getting stuck. Here is the question:

Research finds that the number of units of a product sold closely follows D(p) = 3000(p + 10)e^{-0.4 p}, where p is the unit price and q = D(p) is the number of units sold.

(a) Find the revenue as a function of p.
R(p)= p(3000(p+10)e^(-.4p)) <---I had no problem with this part

(b) What price should be charged in order to obtain the largest possible revenue?



Part b is where my trouble lies. I think I am getting the derivative wrong. Here is the derivative I computed for the revenue function:
3000(p+10)-.4e^(-.4p)  <---It just doesn't seem right.

Could you help with computing this derivative? Thank you and I appreciate your time.

Best,
Marissa
<---I had no problem with this part

(b) What price should be charged in order to obtain the largest possible revenue?



Part b is where my trouble lies. I think I am getting the derivative wrong. Here is the derivative I computed for the revenue function:
3000(p+10)-.4e^(-.4p)  <---It just doesn't seem right.

Could you help with computing this derivative? Thank you and I appreciate your time.

Best,
Marissa
.........................................................
If you have:

R(p)= p(p+10)e^(-.4p)  (never mind the 3000)

then that is:

R(p)= (p^2+10p)e^(-.4p)

and you need the product rule to compute the derivative.

R' = (p^2+10p)e^(-.4p)(-0.4) + (2p+10)e^(-.4p)

Set that = 0, factor out the e^(-.4p), which is never zero, and solve.

R' = [ (p^2+10p)(-0.4) + (2p+10) ]e^(-.4p)

Good luck.