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R(x) = xp(x) is the revenue function. R (x) is the marginal revenue P(x) = R(x)  C(x) is the profit function P (x) is the marginal profit. *V   SB <9 8@` 33___PPT10i.yY@)f+D=' = @B +"I PH00(     xd0e0e ?Rectangle 2d  0 K ECONOMICS      xf0e0e ?Rectangle 33{ 0 &Suppose C(x) is the total cost that a company incurs in producing x units of a certain commodity. The function C is called a cost function. L 8/ @`  F ?Text Box 7 p`@ i Example 8 2    33___PPT10i.yY@)f+D=' = @B +"  P(  $   x  0e0e ?Rectangle 3/{ 0 XDIf the number of items produced is increased from x1 to x2, then the additional cost is "C = C(x2) - C(x1) and the average rate of change of the cost is: 2&      2 @`x  s <A ??Object 2q {      ~D 0e0e ?Rectangle 6d  0 N AVERAGE RATE     Fz?Text Box 7 p`@ i Example 8 2    33___PPT10i.yYf+D=' = @B +" p(     x90e0e ?Rectangle 3/{ 0 dThe limit of this quantity as "x ! 0, that is, the instantaneous rate of change of cost with respect to the number of items produced, is called the marginal cost by economists:V    @`x  s <A ??Object 2 O  !   ~0e0e ?Rectangle 6d  0 O MARGINAL COST    F@h?Text Box 7 p`@ i Example 8 2    33___PPT10i.yYpf+D=' = @B +"} |d(      x`i0e0e ?Rectangle 3/{ 0 TnAs x often takes on only integer values, it may not make literal sense to let "x approach 0. However, we can always replace C(x) by a smooth approximating function as in Example 6.^ Z ""K""""" "        7   @`   ~j0e0e ?Rectangle 5d  0 K ECONOMICS     Fk?Text Box 6 p`@ i Example 8 2    33___PPT10i.yY0g+D=' = @B +" (     xj0e0e ?Rectangle 3/n 0 Taking "x = 1 and n large (so that "x is small compared to n), we have: C (n) H" C(n + 1)  C(n) Thus, the marginal cost of producing n units is approximately equal to the cost of producing one more unit [the (n + 1)st unit].H   """"""" " """""" "$$$$ " $ " $ " $ " $ " $ " $ " $ " $ " $$$$%$$$$M$$$$$$("( @`   ~@n0e0e ?Rectangle 5d  0 K ECONOMICS     F`o?Text Box 6 p`@ i Example 8 2    33___PPT10i.Y`ڦ&+D=' = @B +"  (      x0e0e ?Rectangle 33w 0 It is often appropriate to represent a total cost function by a polynomial C(x) = a + bx + cx2 + dx3 where a represents the overhead cost (rent, heat, and maintenance) and the other terms represent the cost of raw materials, labor, and so on. K   K  @`   ~0e0e ?Rectangle 5d  0 K ECONOMICS     F?Text Box 6 p`@ i Example 8 2    33___PPT10i.Y&+D=' = @B +" $j(  $& $  x 0e0e ?Rectangle 33{ 0 ZThe cost of raw materials may be proportional to x. However, labor costs might depend partly on higher powers of x because of overtime costs and inefficiencies involved in large-scale operations.L 24R @` $  ~@0e0e ?Rectangle 5d  0 K ECONOMICS    $ F`?Text Box 6 p`@ i Example 8 2    33___PPT10i.Y&+D=' = @B +" ((  (~ (  x0e0e ?Rectangle 33n 0 |For instance, suppose a company has estimated that the cost (in dollars) of producing x items is: C(x) = 10,000 + 5x + 0.01x2 Then, the marginal cost function is: C (x) = 5 + 0.02xe   = X +  @` (  ~0e0e ?Rectangle 5d  0 K ECONOMICS    ( F ?Text Box 6 p`@ i Example 8 2    33___PPT10i.YQ&+D=' = @B +"  0,(  ,^ ,  x 0e0e ?Rectangle 34n 0 The marginal cost at the production level of 500 items is: C (500) = 5 + 0.02(500) = $15/item This gives the rate at which costs are increasing with respect to the production level when x = 500 and predicts the cost of the 501st item.ja  >""!""]1 @` ,  ~ 0e0e ?Rectangle 6d  0 K ECONOMICS    , F ?Text Box 7 p`@ i Example 8 2    33___PPT10u.Y &+D=' = @B +"  P0(  0r 0  x@0e0e ?Rectangle 3/{ 0 |The actual cost of producing the 501st item is: C(501)  C(500) = [10,000 + 5(501) + 0.01(501)2]  [10,000 + 5(500) + 0.01(500)2] =$15.01 Notice that C (500) H" C(501)  C(500) ' 0*# "           @` 0  ~<0e0e ?Rectangle 5d  0 K ECONOMICS    0 F?Text Box 6 p`@ i Example 8 2    33___PPT10i.Yp'+D=' = @B +"  p4(  4p 4  x0e0e ?Rectangle 3/n 0 ,Economists also study marginal demand, marginal revenue, and marginal profit which are the derivatives of the demand, revenue, and profit functions. 0   @` 4  ~0e0e ?Rectangle 5d  0 K ECONOMICS    4 F ?Text Box 7 p`@ i Example 8 2    33___PPT10i.Yp'+D=' = @B +"G  NF8.(  8$ 8  x0e0e ?Rectangle 2[  0 XMARGINAL COST FUNCTION   8  x0e0e ?Rectangle 3\t 0  Recall that if C(x), the cost function, is the cost of producing x units of a certain product, then the marginal cost is the rate of change of C with respect to x. In other words, the marginal cost function is the derivative, C (x), of the cost function.  /MC @` 33___PPT10i.-Ex+D=' = @B +"  <(  < <  x0e0e ?Rectangle 2[  0 QDEMAND FUNCTION   <  x@0e0e ?Rectangle 3\tm 0 Now, let s consider marketing. Let p(x) be the price per unit that the company can charge if it sells x units. Then, p is called the demand function (or price function), and we would expect it to be a decreasing function of x.   ((Ak @` 33___PPT10i.-Ex+D=' = @B +" @(  @o @  x 0e0e ?Rectangle 3\t 0 If x units are sold and the price per unit is p(x), then the total revenue is: R(x) = xp(x) This is called the revenue function.` & $$*$$$$!$$$$$$$$$$% @`$ @  ~0e0e ?Rectangle 5[  0 RREVENUE FUNCTION   33___PPT10i.-Ex+D=' = @B +"& -%D (  D D  x0e0e ?Rectangle 3t 0 FThe derivative R of the revenue function is called the marginal revenue function. It is the rate of change of revenue with respect to the number of units sold.FT P $$B$P @`- D  ~0e0e ?Rectangle 4[  0 [MARGINAL REVENUE FUNCTION   33___PPT10i.n~+D=' = @B +" H(  H. H  x0e0e ?Rectangle 3 t 0 bhIf x units are sold, then the total profit is P(x) = R(x)  C(x) and is called the profit function. The marginal profit function is P , the derivative of the profit function. ,G)$ @`, H  ~0e0e ?Rectangle 5[  0 ZMARGINAL PROFIT FUNCTION   33___PPT10i.-Ex+D=' = @B +"   0L(  L L  x0e0e ?Rectangle 2d  0 KSummary   L  x0e0e ?Rectangle 33{ 0 XC(x) is cost function; C(x)/x is the average cost C (x) is the marginal cost p(x) is the demand function which is the price per unit if we sell x units. R(x) = xp(x) is the revenue function. R (x) is the marginal revenue P(x) = R(x)  C(x) is the profit function P (x) is the marginal profit. *P   MA :7 -@` 33___PPT10i.yY@)f+D=' = @B +" #PP(  P P  x0e0e ?Rectangle 2d  0 KSummary   P  x@0e0e ?Rectangle 33{ 0 jC(x) is the cost function C(x)/x is the average cost C (x) is the marginal cost p(x) is the demand function which is the price per unit if we sell x units. R(x) = xp(x) is the revenue function. 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