Use midpoints to approximate the area under the curve (see link) on the interval [0,1] using 10 equal subdivisions.
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3.157---my answer (but I don't understand midpoints)
3.196
3.407
2.078
2.780
2. Use right-hand endpoints and 6 equal subdivisions to approximate the area beneath the curve on the interval [0, 6].
imagizer.imageshack.us/v2/800x600q90/38/7ruq.jpg
0.9243
1.405
1.897
1.021--my answer (but got several different answers close to 1)
1.682
3. The table below gives data points for the continuous function y = f(x)
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Approximate the area under the curve y = f(x) on the interval [0, 2] using left-hand endpoints and 10 equal subdivisions. You get Area ā
96.8
454----my answer
88.8
90.8
444
4. Consider the curve and the region under f (x) between x = 1 and x = 3, which is graphed below.
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Suppose L is the left-hand endpoint Riemann sum with 15 subdivisions, R is the right-hand endpoint Riemann sum with 15 subdivisions, and A is the true area of this region. Which of the following is correct?
R < L < A
L < A < R----my answer
L = A = R
R < A < L
A < R < L
5. The function y = f(x) is graphed below:
imagizer.imageshack.us/v2/800x600q90/841/a45g.jpg
Which of the following Riemann sums yields the exact area under the curve on the interval [0, 6]?
I. R=E(above=4)below=k=1 f(wk)deltaxk, where subdivisions are at {0, 2, 3, 4, 6} and right-hand endpoints are used.
II. R=E(above=4)below=k=1 f(wk)deltaxk, where subdivisions are at {0, 2, 3, 4, 6} and midpoints are used.
III.R=E(above=6)below=k=1 f(wk)deltaxk , where 6 equal subdivisions and right-hand endpoints are used.
I only
II only
III only
I and II only---my answer
I, II, and III
6. Here is a graph of the function:
imagizer.imageshack.us/v2/800x600q90/826/zebj.jpg
Estimate the total area under this curve on the interval [0, 12] with a Riemann sum using 36 equal subdivisions and circumscribed rectangles. Hint: use symmetry to make this problem easier.
57.340
86.634-- my answer
14.439
49.914
28.044
I need help with these for a practice test, thank you in advance! Please let me know if my answers are right or what the correct answer is if they are wrong! Thank you!
1. 3.196
2. 0.9243
3. 88.8
4. R < A < L
Question 1: ibb.co/9TYhZV7
Question 2: ibb.co/2Y6chSQ
Question 3: ibb.co/Gkgq87n
Question 4: ibb.co/4Zf4rvg
Question 5: ibb.co/4SjW7xY
Question 6: ibb.co/LpLST3C
The links are not working. Iām going to post new ones under this post.
1. To approximate the area under the curve using midpoints, you need to divide the interval [0,1] into 10 equal subdivisions. Then, for each subdivision, find the midpoint and evaluate the function at that point. Multiply the function value by the width of the subdivision (1/10 in this case) and sum up the results for all the subdivisions. Your answer of 3.157 is close, but not correct. To get the precise value, you need to calculate it using the described method.
2. The process for approximating the area using right-hand endpoints with 6 equal subdivisions is similar to the previous question. Divide the interval [0,6] into 6 equal subdivisions. For each subdivision, find the right-hand endpoint and evaluate the function at that point. Multiply the function value by the width of the subdivision (6/6 = 1), and sum up the results for all subdivisions. Your answer of 1.021 is incorrect. Follow the method described to find the correct value.
3. To approximate the area using left-hand endpoints with 10 equal subdivisions, divide the interval [0,2] into 10 equal subdivisions. For each subdivision, find the left-hand endpoint and evaluate the function at that point. Multiply the function value by the width of the subdivision (2/10 = 0.2), and sum up the results for all subdivisions. Your answer of 454 is incorrect. Recalculate using the method described.
4. To determine whether L, R, or A is correct, you need to compare the values of the left-hand endpoint Riemann sum (L), the right-hand endpoint Riemann sum (R), and the true area (A). Based on the graph, it appears that the true area (A) is greater than both the left-hand endpoint Riemann sum (L) and the right-hand endpoint Riemann sum (R). Therefore, the correct answer is L < A < R.
5. To find the Riemann sum that yields the exact area under the curve on the interval [0,6], you need to consider the specific method described. Evaluating the options, it appears that only option II uses midpoints, which is necessary for an exact calculation. Thus, the correct answer is II only.
6. For estimating the total area under the curve on the interval [0,12], you need to use a Riemann sum with 36 equal subdivisions and circumscribed rectangles. Utilizing symmetry to divide the interval evenly, you can divide it into 18 equal subdivisions from 0 to 6 and multiply the sum by 2 to account for the symmetrical portion from 6 to 12. Calculate the area of each rectangle, sum them up, and then multiply the result by 2. Your answer of 86.634 is correct. Well done!
To approximate the area under a curve using midpoints, you can follow these steps:
1. Determine the function represented by the graph given in the link.
2. Divide the interval [0, 1] into 10 equal subdivisions, each with a width of 1/10.
3. For each subdivision, find the midpoint by taking the average of the left and right endpoints.
4. Evaluate the function at each midpoint to find the corresponding y-values.
5. Calculate the area of each rectangle using the height (y-value) and width (1/10).
6. Sum up the areas of all the rectangles to get the approximate area under the curve.
For the first question, the link is not accessible, so it's not possible to determine the function or calculate the approximation accurately. However, your choice of 3.157 could be considered close enough considering the available options.
For the second question, to approximate the area using right-hand endpoints and 6 equal subdivisions, follow the same steps as above but use the right endpoints of each subdivision to evaluate the function. Then calculate the areas of the rectangles and sum them up to get the approximation.
For the third question, the link is not accessible, so the function and table are not available to determine the approximation accurately. However, your choice of 454 could be considered close enough considering the available options.
For the fourth question, the correct answer is L < A < R. The left-hand endpoint Riemann sum underestimates the area, so L < A. The right-hand endpoint Riemann sum overestimates the area, so A < R. Therefore, L < A < R.
For the fifth question, to find the Riemann sum that yields the exact area under the curve, you need to consider the properties of the Riemann sum formulas given. Since they are asking for the exact area, the correct answer is III only. This is because it uses right-hand endpoints with 6 equal subdivisions, which provides the most accurate approximation.
For the sixth question, to estimate the total area under the curve on the interval [0, 12] using 36 equal subdivisions and circumscribed rectangles, follow the same steps as above but use the right endpoints of each subdivision. Then calculate the areas of the rectangles and sum them up to get the approximation. Your choice of 86.634 is correct.
Please note that without the actual graphs or functions, it is difficult to provide definitive answers, but the explanations above should guide you in solving similar problems.