Q.The demand function for a product is $P = 200 - 4Q$. Find the Total Revenue and Marginal Revenue functions, and the Marginal Revenue at $Q=20$.

Total Revenue is price times quantity; differentiating it with respect to $Q$ gives Marginal Revenue, which is then evaluated at the given output level. …

Given: demand function $P=200-4Q$.

Step 1 — Find Total Revenue: $TR = P\times Q = (200-4Q)Q = 200Q-4Q^2$.

Step 2 — Differentiate $TR$ to find $MR$: $MR = \dfrac{d(TR)}{dQ} = 200-8Q$.

Step 3 — Substitute $Q=20$: $MR(20) = 200-8(20)=200-160=40$.

Check (independent method — direct revenue-difference approximation): $TR(20) = 200(20)-4(400)=4000-1600=2400$; $TR(21)=200(21)-4(441)=4200-1764=2436$; the actual extra revenue from the 21st unit is $2436-2400=36$, reasonably close to the calculus-based $MR=40$ at $Q=20$ (the small gap is …

Forgetting that $MR$ falls TWICE as fast as price as $Q$ rises for a linear demand function (coefficient doubles from $-4$ in $P$ to $-8$ in $MR$) — a student who writes $MR=200-4Q$ (c …