Deriving the MR curve

Q: How do I derive the MR curve from the demand curve?

A: In class, we said that the marginal revenue curve was twice as steep as the demand curve. Remember that slope of the demand curve is change in P/change in Q. Also, the demand curve and the marginal revenue curve will have the same intercept on the price axis. Here is an example:

if we have demand as P=40-2Q, then we will have marginal revenue as P=40-4Q. (Remember that this works only when you solve for P in your demand equation.)

Again, you don't NEED to know this for EC10, but...why does this work?

So, total revenue = price times quantity. So, using the above example:
TR = PQ = (40-2Q)Q = 40Q-2Q^2

As mentioned before, marginal revenue is the derivative of total revenue. Hence:
MR = 40-4Q. Just as above! You can see from the structure of the problem why this would always be the case.