04-2. Differential Rule

Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. Differentiation3 Basic Rules of Differentiation The Product and Quotient Rules The Chain Rule Marginal Functions in Economics Higher-Order Derivatives Implicit Differentiation and Related Rates Differentials

Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. Marginal FunctionsThe Marginal Cost Function approximates the change in the actual cost of producing an additional unit.The Marginal Average Cost Function measures the rate of change of the average cost function with respect to the number of units produced.The Marginal Profit Function measures the rate of change of the profit function. It approximates the profit from the sale of an additional unit.The Marginal Revenue Function measures the rate of change of the revenue function. It approximates the revenue from the sale of an additional unit.


Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. Cost FunctionsGiven a cost function, C(x), the Marginal Cost Function isthe Average Cost Function isthe Marginal Average Cost Function is


Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. Revenue FunctionsGiven a revenue function, R(x), the Marginal Revenue Function isGiven a profit function, P(x), the Marginal Profit Function isProfit Functions


Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. Elasticity of DemandIf f is a differentiable demand function defined byThen the elasticity of demand at price p is given by Demand is:Elastic if E(p) > 1Unitary if E(p) = 1Inelastic if E(p) < 1


Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. Elasticity of DemandIf the demand is elastic at p, then an increase in unit price causes a decrease in revenue. A decrease in unit price causes an increase in revenue.If the demand is unitary at p, then with an increase in unit price the revenue will stay about the same.If the demand is inelastic at p, then an increase in unit price causes an increase in revenue. A decrease in unit price causes a decrease in revenue.


Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. Example Consider the demand equation

which describes the relationship between the unit price p in dollars and the quantity demanded x of the Acrosonic model F loudspeaker systems. Find the elasticity of demand


Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. ExampleThe monthly demand for T-shirts is given by where p denotes the wholesale unit price in dollars and x denotes the quantity demanded. The monthly cost function for these T-shirts is 1. Find the revenue and profit functions.3. Find the marginal average cost function. 2. Find the marginal cost, marginal revenue, and marginal profit functions.


Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. Solution1. Find the revenue and profit functions.2. Find the marginal cost, marginal revenue, and marginal profit functions.Revenue = xpProfit = revenue costMarginal Cost =


Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. Solution3. Find the marginal average cost function.2. (cont.) Find the marginal revenue and marginal profit functions.Marginal revenue =Marginal profit =


Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative.DerivativeNotationsSecondThirdFourthnth


Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. Implicit Differentiationy is expressed explicitly as a function of x. y is expressed implicitly as a function of x. To differentiate the implicit equation, we write f (x) in place of y to get:


Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. Related RatesLook at how the rate of change of one quantity is related to the rate of change of another quantity.Ex.Two cars leave an intersection at the same time. One car travels north at 35 mi./hr., the other travels west at 60 mi./hr. How fast is the distance between them changing after 2 hours?Note: The rate of change of the distance between them is related to the rate at which the cars are traveling.


Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. Related RatesSteps to solve a related rate problem:1. Assign a variable to each quantity. Draw a diagram if appropriate.2. Write down the known values/rates.3. Relate variables with an equation.4. Differentiate the equation implicitly.5. Plug in values and solve.


Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. Ex.Two cars leave an intersection at the same time. One car travels north at 35 mi./hr., the other travels east at 60 mi./hr. How fast is the distance between them changing after 2 hours?Distance = zxyFrom original relationship


Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. IncrementsAn increment in x represents a change from x1 to x2 and is defined by:Read delta xAn increment in y represents a change in y and is defined by:


Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. DifferentialsLet y = f (x) be a differentiable function, then the differential of x, denoted dx, is such that The differential of y, denoted dy, isNote:


Copyright 2006 Brooks/Cole, a division of Thomson Learning, Inc. ExampleThe total cost incurred in operating a certain type of truck on a 500-mile trip, traveling at an average speed of v mph, is estimated to be Find the approximate change in the total operating cost when the average speed is increased from 55 mph to 58 mph.


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04-2. Differential Rule