what is the relationship between the marginal revenue and the slope of the revenue function?

Marginal revenue is the slope of the revenue function: mathematically, marginal revenue at a quantity qqq is the derivative of total revenue with respect to qqq, which is exactly the slope of the revenue curve at that point. To unpack that a bit:

• Let total revenue be R(q)R(q)R(q), where qqq is quantity sold.
• The marginal revenue function is MR(q)=dRdqMR(q)=\dfrac{dR}{dq}MR(q)=dqdR​.
• Geometrically, dRdq\dfrac{dR}{dq}dqdR​ is the slope of the tangent line to the revenue curve at each quantity.
• So:
  • Where the revenue curve is steeply rising , marginal revenue is large and positive.
  • Where the revenue curve is flat , marginal revenue is zero (revenue is at a maximum there).
  • Where the revenue curve is sloping downward , marginal revenue is negative (total revenue falls when you sell more).

For a simple example story: imagine your total revenue as a hill you’re climbing as you sell more units. At first, the hill goes up sharply, so each extra step (each extra unit sold) adds a lot of height—that “height gain per step” is your marginal revenue. As you get closer to the top, the hill flattens: each extra step adds less height, until you reach the top (slope and marginal revenue are zero). If you keep going past the top, you actually start going downhill: each extra step lowers your height—this is like marginal revenue turning negative and total revenue falling when quantity increases further.