Lines and Rates Activity 1 Week #2 This activity will use slopes to calculate marginal profit, revenue and cost of functions. What is Marginal? Marginal cost is the cost added by producing one additional unit of a product or service. Marginal revenue is the revenue gained by producing one additional unit of a product or service. Marginal profit is the profit earned when one additional unit is produced and sold. Note: While we have defined marginal and we will be graphing; an important skill to have in both business and mathematics; we have not done a good job. Business courses that you take later will help you gain a better understanding of marginal. All of these can be found by calculating the slope of the function s curve at a specific point on the curve. This is also what is known as the instantaneous rate of change of a function at a specific point. The profit function for a business is given by the equation P x 4x 2 16x 7, where x is the number of items sold in thousands and P x is the profit in thousands of dollars. Below is the graph of this profit function. P x 4x 2 16x 7 pg. 1
We can use the graph of the profit function to calculate marginal profit (instantaneous rate of change) in two ways. The first way is to use two points, one on each side of the value, and calculate the average rate of change between those two points. The two points that you choose should be as close to the value as possible. 1. Explain why it is better to have the points close to the value. The second way is to use the slope of the tangent line at the value. A tangent line is a line that touches a curve at a point without crossing over the curve. This means that the curve and the line share one point in common. The one point in common Curve Tangent line 2. Using the graph of the profit function on page 1 and average rates of change, calculate the marginal profit at five (5) different points. 3. Using the graph of the profit function on page 1 and tangent lines, calculate the marginal profit at the same five (5) points. 4. Which method is more accurate? Explain. 5. Which method is easier? Explain. 6. For all five points, interpret the value you found. Make sure to use correct units in your description. 7. If you owned this business, how many units would you want to sell? Explain. 8. Describe how the marginal profit changes throughout the graph. Your response should include a discussion about slopes. Creating the Marginal Profit Graph Add a second sheet to your Excel workbook. Label cell A1 x Label cell B1 marginal profit Using column A, starting in cell A2, enter the x values from the five points you selected in questions 2. For example, if you chose the point (1, 5) you would enter 1 as the x value. pg. 2
Using column B, starting in B2, enter the corresponding marginal profit values you calculated at each point. Highlight cells A2 through B6 and create a scatterplot. If needed, refer to Week 3 Excel printout. Add a trendline to your scatter plot. Make sure to check the box next to Display Equation on chart. Again if needed, refer to Week 3 Excel printout. The equation displayed on the graph is the marginal profit function. 9. What do the values on the x-axis represent? Explain. Be sure to include the units. 10. What do the values on the y-axis represent? Explain. Be sure to include the units. 11. Is the trendline the exact marginal profit function? Explain. 12. Explain how you would find the marginal profit for 1900 units. 13. What is the marginal profit for 1900 units? 14. Explain how you would find the marginal profit for 2700 units. 15. What is the marginal profit for 2700 units? A fast-food restaurant has determined that the monthly revenue for its hamburgers 1 2 is R x 3x x. To 20,000 the right is the graph of this revenue function. Marginal Revenue pg. 3
16. Using the graph of the revenue function on page 3 and average rates of change, calculate the marginal revenue at five (5) different points. 17. Using the graph of the revenue function on page 3 and tangent lines, calculate the marginal revenue at the same five (5) points. 18. For all five points, interpret the value you found. Make sure to use correct units in your description. 19. If you owned this business, how many hamburgers would you want to sell? Explain. 20. Describe how the marginal revenue changes throughout the graph. Your response should include a discussion about slopes. Creating the Marginal Revenue Graph Add a third sheet to your Excel workbook. Label cell A1 x Label cell B1 marginal revenue Using column A, starting in cell A2, enter the x values from the five points you selected in questions 16. Using column B, starting in B2, enter the corresponding marginal revenue values you calculated at each point. Highlight cells A2 through B6 and create a scatterplot. If needed, refer to Week 3 Excel printout. Add a trendline to your scatter plot. Make sure to check the box next to Display Equation on chart. Again if needed, refer to Week 3 Excel printout. The equation displayed on the graph is the marginal revenue function. 21. What do the values on the x-axis represent? Explain. Be sure to include the units. 22. What do the values on the y-axis represent? Explain. Be sure to include the units. 23. Is the trendline the exact marginal revenue function? Explain. 24. Explain how you would find the marginal revenue for 6000 hamburgers. 25. What is the marginal profit for 6000 hamburgers? 26. Explain how you would find the marginal revenue for 54,000 hamburgers. 27. What is the marginal revenue for 54,000 hamburgers? pg. 4
More Marginal Profit 2 Given the profit function for producing and selling x unit is P x 0.5x 150x 1500 the graph of this profit function.. Below is 28. You have now used both methods for calculating marginal quantities for two different situations. Which one is the best method? Explain. Your explanation must include what you mean by best and why it is the best. 29. Use the best method to calculate the marginal profit for three points. 30. On a fourth sheet in Excel, create a scatterplot and a trendline for the marginal profit using the three points in question 29. Be sure to display the equation on your graph. 31. Should you pick more than three points? How many points should you pick to estimate the marginal profit? Explain both answers. 32. Now choose three different points and find the marginal profit for them. Using all six points, create a scatterplot and trendline. Be sure to display the equation of the graph. 33. Compare the graphs and equations. Be sure to discuss the similarities and differences. 34. Which graph is more accurate? Explain. pg. 5
35. What does the slope of the marginal profit function represent? pg. 6