You should introduce the numerical tables provided and explain what data is in them. Tables are provided below and should be copied and pasted into your paper. Note that I used the standard formulas to calculate the approximate value and then found the error by computing the approximate value minus the actual integral. . You should include a brief discussion of the 5 ways to estimate a definite integral numerically. You should include a graph of the function. You could use DESMOS to get a good graph of the function. Analysis: (These do not need to be done in this order if you want to make your paper flow better.) o Are your Mid(n) and Trap(n) under and over estimates. Discuss both rules in relation to each interval given. What property of the function is it that lets you know this? Explain why thinking about the rectangles or trapezoids and where they sit in relation to the function. (See the lecture video on the trapezoid rule and watch the end..) Think about how I computed 0.707 using my calculus 1 skills to choose the intervals for the tables. o Which of the 5 methods appears to be most accurate and why do you think that is the case? Look at the absolute values of the error given in the tables. Use the tables to get your initial answer but then use how that method computes the approximation to explain why it is the best. o Is the error of the Mid(n) roughly 1/2 of the Trap(n) error in absolute value? Explain and use specific examples to back up your conclusion. o How does the error change when n is doubled? For example, is the error approximately % of the error for the previous n? Figure out an approximate factor. Do this for all 5 rules separately. Give specific examples. Tell me what n you are looking at when answering the question and which rule you are thinking of to back up your general claims. (don't just tell me the error is smaller... be specific, like the error is cut in about half or we gained two decimal places of accuracy each time we double n.) Write a conclusion giving key ideas you wrote about. The Function: f(x) = e-x on the interval [0,5]. The Tables on [0, 0.707]: n 2 4 8 20 40 80 n 2 4 8 20 40 80 n CINA 2 4 8 20 40 80 Left 0.665474 0.63748 0.621777 0.611817 n 2 4 8 20 40 80 0.608407 0.606686 Trap 0.595945 0.602715 0.604394 0.604864 0.604931 0.604947 L error: 0.0605214 0.032527 0.0168238 The Tables on [ 0.707,5]: Left L error: 1.02147 1.30274 0.696482 0.415208 0.465034 0.183761 0.139506 0.349683 0.0684093 0.219472 0.314651 0.0333769 0.249545 0.297756 0.0164823 0.265203 Trap 0.651682 0.370953 0.30227 0.284578 0.282098 0.28148 0.00686362 0.00345414 0.00173266 T Error -0.00900812 -0.00223776 -0.000558562 -0.0000893309 -0.0000223313 -5.58274 x 10-6 Right 0.526415 0.567951 0.587012 0.597911 0.601454 0.603209 Mid 0.609486 0.606074 0.605232 0.604998 0.604964 0.0000111658 0.604956 2.79138 x 10-6 Right 0.00062446 0.0454248 Mid T Error 0.370408 0.0902252 0.0896796 0.233587 0.0209962 0.27062 0.00330361 0.279618 0.000823991 0.280862 0.000205879 0.281171 R Error -0.0785376 -0.0370025 -0.0179409 -0.00704228 -0.00349881 -0.00174382 Simp 0.60528 0.604972 0.604954 0.0000446682 0.604953 M Error 0.00453261 0.00112063 0.00027939 R Error -0.280649 -0.235849 -0.141768 -0.0618021 -0.0317289 -0.0160706 S Error 0.000326526 0.0000190307 1.16941x 106 2.97966 x 10-8 0.604953 1.86104 x 107 -9 0.604953 1.16295x107 -10 S Error M Error -0.191049. Simp 0.434871 0.153597 -0.0476872 0.277377 -0.0038965 -0.0106543 0.279376 -0.00165563 0.281232 -0.000412233 0.281271 -0.000102954 0.281274 -0.00189826 -0.0000418466 -2.55043 x 10 -1.58416 x 10 -6 -7