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MAT 168 written project on 4.2-4.4 Name: PURPOSES: • to estimate area using various approximation methods • to compute area exactly using a limit of Riemann sums and using the Fundamental Theorem of Calculus • to understand the meaning of the definite integral GRADING: 30 points available by the rubric on the next page Questions: 1. What is your student ID number? Find the first 5 non-zero digits, and label them A,B,C,a,b. Then construct the function f(x) = 2A(x - B)(x + C) and the interval [-a,b] by filling in these digits. If you have fewer than 5 nonzero digits in your ID number, email Dr. Kohl. For example, if your ID number is 10234056, then your function is f(x) = 2*1(x-2)(x+3) = 2(x-2)(x+3) and the interval is [-4, 5]. Write your ID, the function, and the interval on your paper. 2. Sketch a graph of the function over the interval and shade the region whose net area you will find. Make sure the window size is appropriate to the function and the interval. You may print the graph from a website/software if you like, but tell me what you used. Use some colors! Choose an appropriate scale in both dimensions so that the shape entire region is shown clearly. 3. Do you expect the net area to be positive, negative, or zero? Why? 4. Estimate the area of that region using (a) 4 right endpoints, (b) 4 left endpoints, and (c) 4 midpoints. Make sure your final answer is exact to 3 decimal places. 5. Set up the definite integral notation to denote the exact net area of the region. Do not do any evaluation here. 6. Calculate the integral (net area) using a limit of Riemann sums. Leave the answer in exact form—no decimals! (This problem is worth the most, but it’s the only time you’ll have to compute area using a limit of Riemann sums as in Section 4.3 so be sure to take this problem seriously!) 7. Use the Fundamental Theorem of Calculus to find the net area (integral), making sure it agrees with your answer above. Leave the answer in exact form—no decimals! 8. Which of the estimates from 4(a)-4(c) was the most accurate? Was it an overestimate or an underestimate? How large is the error? 9. If the upper endpoint (x=b) is moved slightly to the right (to some x=c), do you expect the new integral over [a,c] to be more or less than the integral over [a,b] ? Why? Submit your neatly-written solutions including your name! Scans should be made as a single PDF with bright white background, cropped to page size and without significant shadows. Submit the blank rubric with your document as the final page. STUDENT ID= 10010436 CATEGORY 4 points 3 points 2 points 1 point Mathematical Errors double points with a max of 2 pts if #6 not worked using the required approach 90-100% of the steps and solutions have no mathematical errors. 8 points Almost all (85-89%) of the steps and solutions have no mathematical errors. 6 points Most (75-84%) of the steps and solutions have no mathematical errors. 4 points More than 25% of the steps and solutions have mathematical errors. 2 points Mathematical Terminology and Notation Correct terminology and notation are always used, making it easy to understand what was done. Correct terminology and notation are usually used, making it easy to understand what was done. Correct terminology and notation are used, but it is sometimes not easy to understand what was done. There is little use, or a lot of inappropriate use, of terminology and notation. Strategy/ Procedure Typically, uses an efficient and effective strategy to solve the problem(s). All work is shown. Typically, uses an effective strategy to solve the problem(s). All work is shown. Sometimes uses an effective strategy to solve problems, but does not do it consistently. Most work is shown. Rarely uses an effective strategy to solve problems. Most work is shown. Neatness and Organization The work is presented in a neat, clear, organized fashion that is easy to read. The work is presented in a neat and organized fashion that is usually easy to read. The work is presented in an organized fashion but may be hard to read at times. The work appears sloppy and unorganized. It is hard to know what information goes together. Accuracy, neatness, and attractiveness of Plot A ruler and graph paper (or software) are used. The curve and shaded region are plotted correctly and are easy to see. Exceptionally well designed, neat. Uses colors that go well together make the graph more readable. The curve and shaded region are plotted accurately and are easy to see. Neat and relatively attractive. A ruler and graph paper (or graphing computer program) are used to make the graph more readable. All curves, lines, shaded regions, and points are plotted correctly and are neatly drawn but the graph appears quite plain. The curve, shaded region, and/or points are not plotted correctly. Appears messy and "thrown together" in a hurry. Lines are visibly crooked. Communication Gives full and correct reasoning that is easy to understand. Uses complete sentences. Gives correct reasoning, but some small details are missing. Uses complete sentences. Most reasoning is correct. Even if correct, leaps are made in the arguments. Uses complete sentences. Very little to no reasoning given. Uses incomplete sentences. Peer reviews (2 required): 2 points (forfeited by not submitting original version on time)