5/14 MATH 153 UNIT 2 PROJECT PART I You own a T-shirt company that has done so well that the class has decided to expand its operations. You plan to open a store near campus, in competition with several other stores. You must therefore sell your shirts at or near the prevailing price of $15 each. You can make a profit as long as the selling price exceeds the average cost of producing each shirt. To determine profitable production levels, you need to know your anticipated average cost, which will be a function of your production level. (points) (5) 1. You need to rent production and office space at $700 per month. Materials are $4.00 per shirt. You estimate the advertising cost of selling x shirts to be $0.03x2. Write a function C(x), that gives the total cost of producing x shirts per month. (5) 2. The average cost per shirt is the total cost of producing and selling the shirts, divided by the number of shirts sold. Write a function A(x) that gives the average cost per shirt. What kind of function is A(x)? (5) 3. Provide a printout of the graph of A(x). Choose an appropriate window and label your graph completely. (5) 4. Complete the following table: Monthly production, x 100 shirts 10 1 0.1 Average cost per shirt, A(x) (Think of 0.1 shirts per month as 1 shirt every 10 months.) What physical information is provided by each row in the table? (5) 5. What does the table above tell you about the average cost at low production levels? How is your answer reflected in the graph of A(x)? (5) 6. What happens to the average cost as you increase production levels? Support your answer with a table and an analysis of the function. (5) 7. You can make a profit as long as you can sell the shirts for more than it costs to produce them. Because of the nearby competition, your selling price is fixed at $15. Write an inequality that can be used to determine your profitable levels of production. (10) 8. Show a graphical solution to your inequality in #7 with everything labeled. Use the answer to explain the profitable levels of production. PART II The goal for Part II is to use the basic function f ( x) 2 x to create two new functions g ( x) and h( x) by performing a series of transformations. At each step you will write the new function in terms of f ( x ) , and you will sketch the resulting transformation. 1. Creating g ( x) : (2) a. Sketch the graph of f ( x) 2 x . (2) b. Shift the graph of f ( x ) to the left by 2 units. Write a new function, g ( x) , in terms of f ( x ) , and sketch the resulting function on the same grid as a). 1 (2) c. Compress your result from b) vertically by a factor of . Write the new function, 2 g ( x) , in terms of f ( x) , and sketch the resulting transformation on the same grid as a).. (2) d. Reflect the result from c) across the x-axis. Write the new function, g ( x) , in terms of f ( x ) , and sketch the resulting transformation on the same grid as a).. (2) e. You now have an equation for g ( x) in terms of f ( x ) . Write an equation for the function g ( x) in terms of x. 2. Creating h( x) : (2) a. Sketch the graph of f ( x) 2 x . (2) b. Stretch the graph of f ( x) vertically by 2 units. Write a new function, h( x) , in terms of f ( x ) , and sketch the resulting transformation on the same grid as a). (2) c. Reflect the result from b) across the x-axis. Write the new function, h( x) , in terms of f ( x) , and sketch the resulting transformation on the same grid as a). (2) d. You now have an equation for h( x) in terms of f ( x) . Write an equation for the function h( x) in terms of x . (8) 3. Take the equation for g ( x) from 1e) and the equation for h( x) from 2d) and show algebraically that g ( x ) h( x ) . PART III Average rate of change of exponential functions near zero. (5) 1. Given f (x) 2 x , find the average rate of change of f (x) from x = 0 to x = 0.1, so that y x = 0.1. Recall that the average rate of change is . Show your steps and write your x answer on the first line of the table below. (5) 2. Continue finding the average rate of change using 0 as your initial x-value and letting x get smaller. Write the results in the table below and copy to your paper. Interval [0, 0.1] [0, 0.01] [0, 0.001] [0, 0.0001] Average rate of change 0.1 0.01 0.001 0.0001 (2) 3. Using your observations from the table, complete this sentence on your paper: As x gets y smaller and smaller, gets . x (5) 4. Make a similar table for g(x) 3x , using the same intervals, and copy to your paper. (2) 5. Using your observations from this new table, complete this sentence on your paper: As y x gets smaller and smaller, gets . x (5) 6. For both functions it seems that as x gets smaller, the average rate of change approaches a particular decimal. One of the decimals is greater than 1 and the other is less than 1. Using trial and error, try to find an exponential function whose average rate of change, using the same intervals as above, approaches 1. You should present at least 3 other exponential functions and their tables and each table should be accompanied by an observation similar to those in parts 3 and 5. (5) 7. Write your conjecture about the exponential function whose average rate of change on the interval [0, x], as x gets smaller and smaller, comes the closest to 1