MLC at Boise State Polynomials Activity 2 Week #3

Transcription

1 Polynomials Activity 2 Week #3 This activity will discuss rate of change from a graphical prespective. We will be building a t-chart from a function first by hand and then by using Excel. Getting Started with a Function Profit In manufacturing, profit is often modeled using a quadratic equation. For this instance, assume we are manufacturing cell phones. Let n be the number of cell phones manufactured each month. If a month exists where we manufacture zero cell phones, we still have expenses to be paid. These expenses are called fixed costs. This implies that the company will lose money and the profits will be negative when zero cell phones are manufactured. As we begin to manufacture and sell cell phones, we will start to bring in revenue. At first the company will be working at a loss. At some point, the company will create enough cell phones that the company will start to make a profit. This point where loss turns to profit (or profit turns to loss) is call the break-even point. We will be looking at break-even points next week. As the company makes more and more cell phones, they will need to sell the phones for lower and lower prices. It is easy to see that when the sale price of the phone becomes less than the cost of manufacturing the phone, the company will again have negative profits. Because of fixed costs, this actually occurs long before the sale of the cell phone becomes less than the manufacturing cost. The following function estimates the monthly profit P for a cell phone manufacturing company based on the number of n cell phones they manufacture each month. P n n 25n Create a t-chart; make sure to justify your choice of n values. Choose three values for n, making sure that the values are between and including 0 and 12,000. pg. 1

2 Plotting the Data by Hand You will be plotting the data on the grid shown below, graphing the number of cell phones on the x-axis (number of cell phones is the domain and the independent variable). You will be graphing profit on the y-axis (profit is the range and the dependent variable). Remember that the domain is the set of all possible values of an independent variable of a function and the range is the set of all possible values for a dependent variable of a function. First: Label the n-axis (normally this is the x-axis but we are using n to represent number of cell phones) and the P-axis (normally this is the y-axis but we are using P to represent profit) of your grid. Let s start with the n-axis. What is your smallest n-value? What should be your smallest n-value on your n-axis? (Make sure that your smallest value is on the n-axis. You need to have your n-axis go beyond the smallest value.) What is your largest n-value? What should be your largest n-value on your n-axis? (Make sure that your largest value is on the n-axis. You need to have your n-axis go beyond the largest value.) Label these values on your grid below. What is your smallest P-value (Profit value)? Think about this, can your profit be negative? What should be your smallest P-value on your graph? (Make sure that your smallest P- value is on the P-axis. You need to have your P-axis go beyond the smallest value.) What is your largest P-value (Profit value)? What should be your largest P-value on your P-axis? (Make sure that your largest value is on the P-axis. You need to have your P-axis go beyond the largest value.) Label these values on your grid below. Second: Now that your grid is labeled on the n and P-axis, plot the data from your t-chart onto the grid. pg. 2

3 1. What type of graph does your plot resemble? (ie linear, quadratic, cubic, logarthimic, etc.) Explain. Creating the Graph Using Technology Using Excel and the printout from lab day, complete the following: Place the three values you have for n plus three more between 0 and 12,000 in column A starting in cell A2. In cell B1, type your equation, P n n 25n Doing this gives you the ability to 500 recall the equation that represents the values below. In cell B2, type =-1/500*A2^2+25*A then hit enter. 2. Is the value in the box the same as the value you found for your t-chart? If not, determine where the error occurred. Click on cell B2 and hover the mouse over the box in the lower right corner of the cell. Left click and drag the mouse down the column until the cell next to your last n value is highlighted. The cells should populate with the values from your t-chart. Check these values against your t-chart. You will want to create a graph in Excel using these values. Highlight the cells to include columns A and B as well as the rows 2 through 7. Once highlighted, go up to the top, click on Insert and under the charts, find scatter plot then click on it. You should now see a scatter plot of your six points. Right click on one point on the graph and find Add trendline in the pop up menu, click it. Make sure that you are choosing a polynomial with order 2 (in mathematics we say degree 2). Hopefully from your handmade scatter plot, you found that the data resembled a quadratic which is a polynomial of degree 2. pg. 3

4 Rate of Change from a Graphical Perspective Recall that the average rate of change of a function is the slope of the line that contains two points on the graph of the function. This line is called the secant line. x and 3. What is the formula for calculating the slope of a line through the two points 1, y 1 x? 2, y 2 Choose two points from your t-chart in Excel and calculate the average rate of change between them. 4. Explain what this value represents. Your explanation must include units such as miles per hour. Choose another set of points, they must be different, and calculate the average rate of change. 5. Explain what this value represents. Your explanation must include units such as miles per hour. Excel will allow you to draw the line with the slopes you calculated, experiment with it. 6. Describe the places on the graph where the average rate of change is positive. What does this represent? Your explanation should include both what it represents in the context of the problem and algebraically. 7. Describe the places on the graph where the average rate of change is negative. What does this represent? Your explanation should include both what it represents in the context of the problem and algebraically. pg. 4

5 8. Is there a place where the average rate of change is zero? What does this represent? Your explanation should include both what it represents in the context of the problem and algebraically. Recall that instantaneous rate of change at a point is the same as the slope of the tangent at that point. Make sure that you discuss what a tangent line is before continuing. This is also the same as the slope of the curve at that point. Estimate the instantaneous rate of change at any three points on the graph. To do this, click on Insert in the ribbon at the top of Excel then find Shapes. Under shapes, look for a line that you can add to your graph by clicking on it. Go to your graph and hover over the point that you are trying to find the instantaneous rate of change of Click, hold and drag the mouse to create a tangent line, making sure that the line you create is about six inches long. You will then need to move the center of the line to the point that you started at by click, hold and dragging it. See picture below. 9. How would you calculate the instantaneous rate of change at the points you chose? Find each value. Place each value in a text box next to it s corresponding point. Explain what each slope represents. Your answer must include units. 10. How many cell phones are being produced when the profit is increasing at the fastest rate? 11. How many cell phones are being produced when the profit is increasing at the slowest rate? 12. How many cell phones are being produced when the profit is decreasing at the fastest rate? 13. How many cell phones are being produced when the profit is decreasing at the slowest rate? 14. If you were in charge of this company, how many cell phones would you produce and why? pg. 5

6 15. Why would the CEO of the company never want to produce more than 11,000 cell phones? 16. Explain how slope and rate of change can be useful in making business decisions. Exercises For both of the functions below, P is the profit in dollars and n is the number of cell phones manufactured: a. determine the domain and range keeping in mind what values make sense for this real world situation, b. create a t-chart and create a scatter plot in Excel, c. create the polynomial of degree 2 (quadratic), d. find the average rate of change between two sets of data points and explain what they represent, e. find the instantaneous rate of change between two sets of data points and explain what they represent. f. What number of cell phones would the CEO consider producing? You must justify your answer. 17. P n n 52n P n n n pg. 6