MTH 243 CALCULUS FOR MGMT LIFE/SOCIAL SCI MIDTERM FORMAL LAB NAME: This formal lab is an open-note, open-book, individual assignment. You cannot use online or software resources beyond the D2L course website and your calculator. However, you may work on this assignment with other students. Read all problems carefully. Please show all work and write clearly. Problems with work shown can receive partial credit. However, problems with little to no work can receive zero points. Indicate final answers with a box or circle. To submit your work properly, please follow the instructions found in the D2L Assignments page. INTRODUCTION: PRODUCTION, PRICE, DEMAND, REVENUE, & PROFIT A technology startup's market research department is tasked with determining the market viability of a new smartphone device. After suitable testing on the interest in a new smartphone, the research department determines the following price-demand equation: x = 3.2 x 10 - 500p where r is the amount of units (smartphones) in demand at price p (in dollars). For example, if the price of the new smartphone is set at p = $100, then the amount of new smartphones in demand should be: x = 3.2 x 10 - 500(100) = 3150000 units In addition, the financial department provides the cost function measured in dollars: C(x) = 85x + 50000 where r is the number of smartphones produced. Note that $50000 is the fixed costs of production (main- tenance, overhead, etc.) and $85 is the cost (labor, materials, marketing, transportation, storage, etc.) per smartphone. See EXAMPLE 3 on page 164 in the course textbook for a similar problem with a worked solution. Page 1 of 3 MTH 243 Calculus for Momt Life/Social Sci Midterm Formal Lab CS Scanned with CamScanner1. (10pts) Find the expression for the price p in terms of the demand a from the price-demand equation shown in the introduction. Then, determine the domain of the price-demand equation from your work. Note that both price p and demand a must be non-negative (greater than or equal to zero). 2. (10pts) Assume that the startup will be able to sell all smartphones produced. The revenue function R(x) can be described in words as: R(x) = (number of smartphones sold)(the price per smartphone) Use your work in PROBLEM 1 to find the expression for R(x). 3. (5pts) Use your work in PROBLEM 2 to find the marginal revenue R'(r). Then compute and interpret the following: R' (500000), R' (1600000), and R'(2300000). Marauder 91008 - POL (erellob ni) a corig In busmohan Jednordy time) esta MTH 243 Calculus for Mymt Life/Social Sci Midterm Formal Lab Page 2 of 3 CS Scanned with CamScanner4. (10pts) The break-even points occur at some production level(s) r such that the revenue R(x) is equal to the cost C(x). That is, the break-even points a occur such that R(x) = C(x). Use your work in the previous problems to find these break-even points. Hint: use the Quadratic Formula to solve your equation. 5. (5pts) The profit function is defined to be the revenue minus the cost at the production level z. That is, the profit function P(x) = R(x) -C(x). Use your previous work to find P(x), then sketch the graph of this function. Hint: use the domain found in PROBLEM 1. 6. (10pts) From your graph in the previous problem, you should be able to observe the point where the profit is maximized (i.e. the apex point). What is the slope of the tangent line at this point? Then, find the marginal profit function P'(x). Use the slope of the apex point and the marginal profit P'(x) to find the actual apex point (x, y). Hint: you will setup an equation to solve for the x-coordinate of the aper point. Page 3 of 3 MTH 243 Calculus for Mgmt Life/Social Sci Midterm Formal Lab ned with CamScanner