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1 Derivatives Difference quotients are used in many business situations, other than marginal analysis (as in the previous section)
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2 Derivatives Difference quotients Called the derivative of f(x) Computing Called differentiation
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3 Derivatives Ex. Evaluate if
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4 Derivatives Numerical differentiation is used to avoid tedious difference quotient calculations Differentiating.xls file (Numerical differentiation utility) Graphs both function and derivative Can evaluate function and derivative
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5 Derivatives Differentiating.xls
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6 Derivatives Use Differentiating.xls to graph the derivative of on the interval [-2, 8]. Then evaluate.
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7 Important If f '(x) is constant, the displayed plot will be distorted. To correct this, format the y-axis to have fixed minimum and maximum values. Eg: Lets try to plot g(x)=10x in [-2,8]
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8 Derivatives Properties If then
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9 Derivatives Tangent line approximations Useful for easy approximations to complicated functions Need a point and slope (derivative) Use y = mx +b
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10 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 = 5.5452x – 2.6356 The slope of the graph of f at the point (3,14)
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11 Derivatives Project (Marginal Revenue) - Typically - In project, - Why ?
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12 Recall:Revenue function-R(q) Revenue in million dollars R(q) Why do this conversion? Marginal Revenue in dollars per drive
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13 Derivatives Project (Marginal Cost) - Typically - In project, -
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14 Derivatives Project (Marginal Cost) - Marginal Cost is given in original data - Cost per unit at different production levels - Use IF function in Excel
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15 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
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16 Derivatives Project (Marginal Revenue) - Calculate MR(q) -
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17 Derivatives Project (Marginal Cost) - Calculate MC(q) - IF(q<=500,115,IF(q<=1100,100,90))
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18 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
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19 Derivatives Project (Maximum Profit) - Create table for calculations
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20 Derivatives Project (Answering Questions 1-3) 1. What price? $167.70 2. What quantity? 575,644 units 3. What profit? $9.87 million
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21 Derivatives Project (Answering Question 4) 4. How sensitive? Somewhat sensitive -0.2% -4.7%
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22 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- 4
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