MTH 112 – LAB #2 - Elgin
Notes on TECH LAB #2: Piecewise Defined Functions by Mary Ann TuerkMy students in College Algebra struggle with the concept, the notation and the graphs of piecewise defined functions. Then, of course, when they get to calculus, they are still struggling when these functions are used a lot in illustrating limits and differentiability.I use this lab to reinforce my standards for all labs and graded assignments:All graphs must be clearly drawn with a straight edgeAxes must be labeled with nouns and scaled quantitiesAnswers in complete sentences must be grammatically correctThis lab is always started in class. I have them fill out the tables in #1 and #3 in class and walk around and check (or pair check) to be sure these are correct. Students will struggle with writing the correct functions in #2 and #4. Make sure they have a way to check that the functions are correct – tell them if they have the wrong functions, their line segments will not be connected. An in-class tutor is a great help for this activity. The calculator concepts introduced include:How to determine a reasonable window for an application problemWriting a function and then re-naming the variables for entering in a calculatorHow a graphing calculator can be used to graph a piecewise defined function by restricting the domain for each branch. This is not a required learning outcome and I do not stress it or require students to be able to do this. Drawing the graph by hand is acceptable.MTH 112 – TECH LAB #2_________________________DUE: __________ NAMETOPIC: PIECEWISE DEFINED FUNCIONSPURPOSE: To investigate a real world problem with a piecewise defined function as its algebraic model.PROBLEM: For residential customers, Commonwealth Edison charges a base charge of $7.58. This is charged to each customer regardless of the amount of electricity used. In addition each customer pays a charge for each kilowatt hour (kwh) used. For the first 400 kwh, the charge is $ .08275 per kwh. For additional kilowatts over the first 400, the customer pays $ .06208 per kwh.Sample computation of electric bill : Customer used 706 kwh in the month.Total bill = $7.58 + 400 kwh * .08275 + 306 kwh * .06208 = $ 59.68Exercises:Complete the table below. Calculate the total bill for the number of kwh used.kwh usedBase Charge Charge on kwhsTotal Bill0____________________________50____________________________ 100____________________________ 200____________________________ 400____________________________ 2. Let k represent the kwh used in a month and T represent the total bill. If k< 400, express T as a function of k: T = f(k) = ______________________Complete the table below. Calculate the total bill for the number of kwh usedkwh usedBase Chge Chge on 1st Chge on kwh Total Bill 400 kwh over 400450 ________ __________ __________ __________500__________________ __________ __________ 700__________________ __________ __________ 800________ __________ __________ __________ Let represent the total kwh used in a month and T represent the bill. If k > 400, express T as a function of k: T = f(k) = ____________________________Write the function as a single piecewise defined function using conventional function notation:T = f(k) = _____________________________ if x ≤400 ______________________________ if x>400Graph the two parts of the piecewise defined function on a single screen and carefully copy the graph below. Clearly label both axes. Indicate below the viewing rectangle you used on your calculator.You can do your graph by hand OR use your graphing calculator. To generate the graph on a TI-84, enter the equations as follows:Y1 = (1st branch of the function from above) / (x < 400)Y2 = (2nd branch of the function from above) / (x > 400)Be sure to include both sets of parentheses; the inequality operators are under the TEST menu (2nd MATH).Use the values from your tables to determine an appropriate viewing rectangle.XMIN =________ XMAX = ________ YMIN = ________YMAX=________Both pieces of the function are linear. What is the slope of the line when k<400 ? __________ What is the slope of the line when k > 400 ? ___________Interpret the slope in terms of the rates charged per kilowatt hour used (Answer in a complete sentence). ................
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