Differentiation Purpose to determine instantaneous rate of change

Differentiation • Purpose- to determine instantaneous rate of change Eg: instantaneous rate of change in total cost per unit of the good We will learn • Marginal Demand, Marginal Revenue, Marginal Cost, and Marginal Profit

Marginal Cost : MC(q) • What is Marginal cost? The cost per unit at a given level of production EX: Recall Dinner problem C(q) = C 0 + VC(q). = 9000+177*q 0. 633 MC(q)- the marginal cost at q dinners MC(100)- gives us the marginal cost at 100 dinners This means the cost per unit at 100 dinners How to find MC(q) ? We will learn 3 plans

Marginal Analysis • First Plan • Cost of one more unit •

Marginal Analysis • Ex. Suppose the cost for producing a particular item is given by where q is quantity in whole units. Approximate MC(500). •

Marginal Analysis • Second Plan • Average cost of one more and one less unit •

Marginal Analysis • Ex. Suppose the cost for producing a particular item is given by where q is quantity in whole units. Approximate MC(500). •

Marginal Analysis • Final Plan • Average cost of fractionally more and fractionally less units difference quotients • • Typically use with h = 0. 001

Marginal Analysis • Ex. Suppose the cost for producing a particular item is given by where q is quantity in whole units. Approximate MC(500). • In terms of money, the marginal cost at the production level of 500, $6. 71 per unit

Marginal Analysis • Use “Final Plan” to determine answers • All marginal functions defined similarly •

Marginal Analysis • Graphs D(q) is always decreasing All the difference quotients for marginal demand are negative MD(q) is always negative

Maximum revenue Marginal revenue 0 Marginal Analysis • Graphs q 1 R(q) is increasing • difference quotients for marginal revenue are positive • MR(q) is positive R(q) is decreasing • difference quotients for marginal revenue are negative • MR(q) is negative

Marginal Analysis • Graphs

Derivatives-part 2 • Difference quotients • • Called the derivative of f(x)

Derivatives • Ex 1. Evaluate if

Derivatives • Differentiating. xls file • Graphs both function and derivative • Can evaluate function and derivative

Derivatives • Differentiating. xls

Derivatives • Use Differentiating. xls to graph the derivative of on the interval [-2, 8]. Then evaluate.

FUNCTION & DERIVATIVE function derivative 300 250 200 f(x) f'(x) 150 100 50 0 -4 -2 0 2 4 x 6 8 10

Differentiating. xslm Key points 1)Formula for the function in x 2)Plot Interval is ESSENTIAL 3)You can use the computation cells to evaluate the f(x) & f’(x) at different values 4)If using Office 2007, save it as a macro enabled file

Derivatives • Properties If If (c is a constant) then (m is a constant) then

Derivatives • Tangent line approximations • Useful for easy approximations to complicated functions • Need a point and a slope (a derivative) • Use y = mx +b

Derivatives • Ex. Determine the equation of the tangent line to at x = 3. • Recall and we have the point (3, 14) • Tangent line is y = 5. 5452 x – 2. 6356

Derivatives • Project (Marginal Revenue) - Typically - In project, - units are

Recall: Revenue function-R(q) • Revenue in million dollars R(q) • Why do this conversion? Marginal Revenue in dollars per drive 24

Derivatives • Project (Marginal Cost) - Typically - In project, - units are

Derivatives • Project (Marginal Cost) - Marginal Cost is given in original data - Cost per unit at different production levels - Use IF function in Excel

Derivatives • Project (Marginal Profit) MP(q) = MR(q) – MC(q) - If MP(q) > 0, profit is increasing - If MR(q) > MC(q), profit is increasing - If MP(q) < 0, profit is decreasing - If MR(q) < MC(q), profit is decreasing

Derivatives • Project (Marginal Cost) - Calculate MC(q) Nested If function, the if function using values for Q 1 -4 & 6 - IF(q<=800, 160, IF(q<=1200, 128, 72)) In the GOLDEN sheet need to use cell referencing for IF function because we will make copies of it, and do other project questions =IF(B 30<$E$20, $D$20, IF(B 30<$E$22, $D$21, $D$22))

Recall -Production cost estimates • • Fixed overhead cost - $ 135, 000 Variable cost (Used for the MC(q) function) 1) First 800, 000 - $ 160 per drive 2) Next 400, 000 - $ 128 per drive 3) All drives after the first 1, 200, 000$ 72 per drive

Derivatives • Project (Maximum Profit) - Maximum profit occurs when MP(q) = 0 - Max profit occurs when MR(q) = MC(q) - Estimate quantity from graph of Profit - Estimate quantity from graph of Marginal Profit

Derivatives-change • Project (Answering Questions 1 -3) 1. What price? $285. 88 2. What quantity? 1262(K’s) units 3. What profit? $42. 17 million

Derivatives • Project (What to do) - Create one graph showing MR and MC - Create one graph showing MP - Prepare computational cells answering your team’s questions 1 - 3

Marginal AnalysisØ where h = 0. 000001 Ø MR(q) = R′(q) ∙ 1, 000 Ø Ø Marketing Project

Marginal AnalysisØ where h = 0. 000001 Ø In Excel we use derivative of R(q) Ø R(q)=aq^3+bq^2+cq Ø R’(q)=(a*3*q^2+b*2*q+c)/1000 Ø Marketing Project

Marginal Analysis (continued)Marginal Revenue and Cost $600 MR(q) $ Per Drive $400 $200 $0 0 400 800 1 200 1 600 2 000 2 400 2 800 -$200 -$400 -$600 q (K's) Marketing Project

Marginal Analysis Ø MP(q) = MR(q) – MC(q) Marginal Profit MP(q) $ Per Drive $400 $200 $0 0 400 800 1 200 1 600 2 000 -$200 -$400 q (K's) Ø We will use Solver to find the exact value of q for which MP(q) = 0. Here we estimate from the graph Marketing Project

Profit Function Ø The profit function, P(q), gives the relationship between the profit and quantity produced and sold. Ø P(q) = R(q) – C(q)

Goals • 1. What price should Card Tech put on the drives, in order to achieve the maximum profit? • 2. How many drives might they expect to sell at the optimal price? • 3. What maximum profit can be expected from sales of the 12 -GB? • 4. How sensitive is profit to changes from the optimal quantity of drives, as found in Question 2? • 5. What is the consumer surplus if profit is maximized? 38

Goals-Contd. • 6. What profit could Card Tech expect, if they price the drives at $299. 99? • 7. How much should Card Tech pay for an advertising campaign that would increase demand for the 12 -GB drives by 10% at all price levels? • 8. How would the 10% increase in demand effect the optimal price of the drives? • 9. Would it be wise for Card Tech to put $15, 000 into training and streamlining which would reduce the variable production costs by 7% for the coming year? 39