Intro+to+the+derivative+notes.pdf

D dx h 3 x 2 i d dx h x 2 3 i 2 3 x 2 3 1 2 3 x 1 3 2

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d dx h 3 x 2 i = d dx h x 2 3 i = 2 3 x 2 3 - 1 = 2 3 x - 1 3 = 2 3 1 x 1 3 = 2 3 3 x b. d dx 1 x 2 = d dx x - 2 = ( - 2) x - 2 - 1 = - 2 x - 3 = - 2 x 3
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3. Introduction to the derivative: Limits and Continuity 37 The Constant Multiple Rule. If c is a constant and f is a differentiable function , then d dx ( cf ( x ) ) = c d dx f ( x ) , or ( cf ) 0 = cf 0 The Sum Rule. If f and g are both differentiable , then d dx ( f ( x ) + g ( x ) ) = d dx f ( x ) + d dx g ( x ) , or ( f + g ) 0 = f 0 + g 0 The Difference Rule. If f and g are both differentiable , then d dx ( f ( x ) - g ( x ) ) = d dx f ( x ) - d dx g ( x ) , or ( f - g ) 0 = f 0 - g 0 We are now in a position to compute the derivative of any polynomial function. Example 3.36. d dx ( x 8 + 12 x 5 - 4 x 4 + 10 x 3 - 6 x + 5 ) = = d dx ( x 8 ) + 12 d dx ( x 5 ) - 4 d dx ( x 4 ) + 10 d dx ( x 3 ) - 6 d dx ( x ) + d dx ( 5 ) = = 8 x 7 + 12 ( 5 x 4 ) - 4 ( 4 x 3 ) + 10 ( 3 x 2 ) - 6 · 1 + 0 = = 8 x 7 + 60 x 4 - 16 x 3 + 30 x 2 - 6
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38 J. S´ anchez-Ortega 3.11 Marginal Analysis 1. Marginal Cost Suppose that C ( x ) is the total cost that a company incurs in producing x items . Recall that such a function C is called a cost function . The marginal cost function is the derivative function C 0 of the cost function. Thus C 0 ( x ) measures the rate of change of cost with respect to x . The units of of marginal cost are units of costs per items. Assuming that x items have been already produced, C 0 ( x ) pro- vides an approximation to the cost of the x + 1 item. Example 3.37. Suppose that the cost in dollars to manufacture portable CD players is given by C ( x ) = 150000+20 x - 0 . 0001 x 2 , where x is the number of CD players manufactured. Find the marginal cost function C 0 ( x ) and use it to estimate the cost of manufacturing the 50001st CD player. Solution: Applying the Constant, Sum, Difference and Power Rule we get that the marginal cost function is C 0 ( x ) = 20 - 0 . 0002 x The units of C 0 ( x ) are units of C (dollars in our case) per unit of x (CD players). Thus, C 0 ( x ) is measured in dollars per CD player. An estimation to the cost of the 50 001st CD player is C 0 (50000): C 0 (50000) = 20 - 0 . 0002 · 50000 = $10 per CD player Notice that he cost of the 50001st CD player is the amount by which the total cost would rise if we increased production from 50000 CD players to 50001. The exact cost is C (50001) - C (50000) = ... = $9 . 9999 So, the marginal cost is a good approximation to the actual cost.
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3. Introduction to the derivative: Limits and Continuity 39 2. Marginal Revenue and Profit Recall that a revenue or profit function specifies the total revenue R or profit P as a function of the number of items x . The derivatives, R 0 ( x ) and P 0 ( x ) of these functions are called the marginal revenue and marginal profit function , respectively. They measure the rate of change of revenue and profit with respect to x . The units of marginal revenue and profit are the same as those of marginal cost: dollars (or rands, euros, pesos, etc.) per item. We interpret R 0 ( x ) and P 0 ( x ) as the approximate revenue and profit from the sale of one more item. Example 3.38. You operate an iPod customizing service (a typical cus- tomized iPod might have a custom color case with blinking lights and a personalized logo). The cost (in dollars) to refurbish x iPods in a month is calculated to be C ( x ) = 0 . 25 x 2 + 40 x + 1000 You charge customers $80 per iPod for the work.
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