CML
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

• Notes
• 43

This preview shows pages 38–42. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

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
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.

This preview has intentionally blurred sections. Sign up to view the full version.

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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern