# Consider the following economic example suppose that

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Consider the following economic example. Suppose that the Variable Cost of pro- ducing a level of output Q is given by the equation VC = f ( Q ) = Q 2 , where VC is in \$ and Q is in kilograms. Then the equation for Marginal Cost (see “The Rules”, in Module 5) is MC = 2 Q , since Average Variable Cost AVC = VC/Q = Q 2 / Q = Q, and MC has the same vertical intercept (=0) as and twice the slope of the AVC function. If we know that the MC function has the linear form MC = 2 Q , we can retrace our steps to arrive at the VC function: VC = Q 2 . This reverse process is essentially what is involved in integration : if MC is the derivative (or slope) of the VC function, then VC is an integral of the MC function. It is not necessarily “ the ” integral of the MC function. What we can- not do is go from the MC curve to the Total Cost (TC) curve, because Total Cost = Variable Cost + Fixed Costs , TC = VC + FC. Since Fixed Costs are constant , they affect Total Cost but not Marginal Cost. Hence, we can get from MC to VC, but then we need MATH MODULE 10: CALCULUS RESULTS FOR THE NON-CALCULUS SPEAKER M10-3 2 1 2 3 4 6 1 1 2 3 4 9 (a) (b) y = 2 x y = x 2 FIGURE M.10-1

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to add the constant FC (what mathematicians call “the constant of integration”) to VC to get TC. To summarize our sketch so far, if you did some of the exercises in Module 5, you have already been doing calculus, for a special class of functions, namely linear Average and Marginal functions and quadratic Total functions. Table M.10-1 provides some basic formulas for calculating derivatives, which cover virtually all nonlinear and linear functions one can expect to find in an undergraduate economics course. The double- starred (**) questions in the text give you a chance to apply some of these formulas. Partial Derivatives: In both consumption theory and production theory, we make use of partial derivatives. These are discussed in detail in Appendixes 3 and 9, and you should study both Appendixes to get a sense of how they are utilized. The crucial factor to remember, however, is that partial derivatives (in the forms you are likely to encounter them in undergraduate study) have a different notation, but are no more difficult to manipulate, than “regular” derivatives, provided that you treat all variables except the specific independent variable with respect to which you are calculating the (partial) derivative as constants, using the mathematical equivalent of the economist’s ceteris paribus assumption. 1.2 DERIVATIVES, SLOPES, MAXIMA AND MINIMA Economists love maximizing and minimizing, or at least they assume that rational eco- nomic agents do. Economists tend to assume, for instance, that consumers want to max- imize their “utility” or level of satisfaction (subject to a budget constraint), and that M10-4 MATH MODULE 10: CALCULUS RESULTS FOR THE NON-CALCULUS SPEAKER Function: Function: Derivative: Derivative: y = f ( x ) Example dy/dx = y’ = f’ ( x ) Example 1. y = a y = 4 y = 0 y’ = 0 2. y = ax y = 12 x y’ = a y’ = 12 3. y = ax n y = 4 x 3 y’ = nax n – 1 y’ = 12 x 2 y = 3 x 4 y’ = 12 x 3 4. y = f ( x ) + g ( x ) y = 4 x 3 + 3 x y’ = f ’ ( x ) + g’ ( x ) y’ = 12
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