MTH 124 Goals Focus Session 5 Marginal Analysis Maximizing Pro t cid 136 Use tangent lines to graphically estimate the derivative at a point cid 136 Use the derivative to study marginal cost marginal revenue and marginal pro t cid 136 Use the derivative rules to calculate the derivative of a function cid 136 Understand the Pro t Maximization Rule and know how to maximize the pro t Pre activity Do before class De nitions To answer pro t maximization questions economists use marginal analysis Marginal analysis relies on the tools of calculus Recall that for a Cost function C x the derivative is Marginal Cost C cid 48 x The marginal cost function gives the approximate cost of producing one more item after x items have already been produced For a Revenue function R x the derivative is Marginal Revenue R cid 48 x The marginal revenue function gives the approximate revenue from selling one more item after x items have already been sold For a Pro t function H x the derivative is Marginal Pro t H cid 48 x The marginal pro t function gives the approximate pro t from selling one more item after x items have already been sold Note that marginal pro t is marginal revenue minus marginal cost H cid 48 x R cid 48 x C cid 48 x Pro t Maximization Rule Marginal analysis yields the following rules for maximizing pro t cid 136 If H cid 48 x 0 the company can increase their pro t by producing an additional item If Marginal Revenue Marginal Cost Increase Production cid 136 If H cid 48 x 0 the company s pro t will decrease if they produce an additional item In order to increase pro t the company should decrease production If Marginal Revenue Marginal Cost Decrease Production cid 136 To maximize pro t nd the production level where Marginal Revenue Marginal Cost At this production level we have that Marginal Pro t 0 In a practical business sense we assume implicitly that initially the cost is greater than revenue until the break even point then after that point the revenue is greater than cost 1 MTH 124 Focus Session 5 Marginal Analysis Maximizing Pro t In Class Activity Part I Derivatives Graphically A small company produces hand stitched backpacks The company nance team provides the following plots which represent the monthly revenue R x and cost C x of producing x backpacks each month The company wants to know what monthly production quantity will maximize their pro t 1 a Sketch the tangent lines of C x and R x at x 400 in the gure to the right and label each line C x R x b Use the tangent lines to estimate C cid 48 400 and R cid 48 400 Recall the slope of the tangent line of a function at a point is the value of the derivative of the function at that point 120 000 100 000 80 000 s r a l l o D 60 000 40 000 20 000 0 0 200 400 600 800 1 000 1 200 1 400 Number of backpacks produced monthly 2 Why are your answers to 1 b approximations and not exact answers Explain ZOOM POLL Q1 Based on your answers to the previous question the company should increase production when x 400 A True B False Hint Look on the rst page for a useful rule 2 MTH 124 Focus Session 5 Marginal Analysis Maximizing Pro t The gure below provides monthly pro t H x earned from producing x backpacks each month 3 Follow the steps below to sketch the marginal pro t function H cid 48 x below a Recall that H cid 48 x R cid 48 x C cid 48 x so it immediately follows that H cid 48 400 R cid 48 400 C cid 48 400 Use your answers from question 1 to calculate H cid 48 400 H x 15 000 10 000 5 000 s r a l l o D 0 5 000 0 200 400 600 800 1 000 1 200 1 400 Number of backpacks produced monthly b In the gure above sketch tangent lines at x 400 600 1000 Notice that the slope of your tangent line at x 400 should be what you calculated in part a c Estimate the slopes of the tangent lines at x 600 and x 1000 These values will give you H cid 48 600 and H cid 48 1000 respectively Record your answers below H cid 48 x 100 50 0 d Now use the derivative values of the test points above to sketch a plot of H cid 48 x by plotting those points in gure to the right and then connecting them with a curve line 50 100 0 200 400 600 800 1 000 1 200 1 400 Number of backpacks produced monthly 3 MTH 124 Focus Session 5 Marginal Analysis Maximizing Pro t ZOOM POLL Q2 Based on your plot of H cid 48 x at what production quantity x is pro t increasing the most rapidly A x 0 B x 100 C x 600 D x 1100 ZOOM POLL Q3 Use marginal analysis and the plot of H cid 48 x to determine how many backpacks should the company produce in order to maximize pro t A x 0 B x 100 C x 600 D x 1100 Part II Derivatives Algebraically Another company manufactures surfboards The company s monthly costs and monthly revenue are given by C x 0 045x2 34x 4800 and R x 70x where C x is the monthly cost of producing x surfboards each month and R x is the monthly revenue from selling x surfboards each month The company wants to know what monthly production quantity will maximize their pro t 4 Given the cost and revenue functions above use derivative rules to nd the marginal cost and marginal revenue functions C cid 48 x and R cid 48 x 5 Find the marginal pro t function H cid 48 x ZOOM POLL Q4 What monthly production quantity maximizes pro t i e nd x so that marginal pro t 0 which also means marginal revenue marginal cost A x 0 B x 169 C x 400 D x 631 4