lab8calc1
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273
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Mathematics
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Feb 20, 2024
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MATH 273.003 Lab Report #8 1 Name: Luke Rosendorf 1. (1 Point) Use the SageMath computation cell above Exercise 1 in the online version of this lab to evaluate the Riemann sum for f(z) = 2¢7° + 5 on [1,3] with 4 rectangles using right-hand endpoints. Record below both the value of Az in this example and the Riemann sum approximation to the area under the curve. Reimann sum apporx=.745. Delta(x)=1/2 2. (7 Points) Consider the function f(z) = 2273 + { on the interval [1,3] for the interval. Use the SageMath interactive cell below Exercise 2 in the online version of this lab to answer the following questions. (a) Complete the table below with the values of the requested left Riemann sum, right Riemann sum, and middle Riemann sums. Produce a printout for each of the left, right, and middle Riemann sums for n = 93 and attach it to this lab report. left Riemann sums n = n =25 n = 49 n = 81 n = 93 1.3491 1.1913 1.1512 1.1352 1.1320 right Riemann sums n = n =25 n =49 n = 81 n = 93 9211 1.0372 1.0726 1.0876 1.0906 middle Riemann sums n=29 n =25 n =49 n =81 n =93 1.0993 1.1095 1.1107 1.1110 1.1110 (b) Does the list of middle Riemann sums appear to be approaching a limit? What is your best guess for the value of this limit? Record your answer below. Yes, the middle Riemann appears to approach a limit of 1.1110
MATH 273.003 Lab Report #8 2 (c) Generate 5 different random Riemann sums, using 29 rectangles in each. Record these randomized Riemann sums in your lab report. Produce a printout of one of these random Riemann sums and attach it to this lab report. Sum 1 Sum 2 Sum 3 Sum 4 Sum 5 1.0970 1.1353 1.1219 1.1048 1.1274 (d) Compute the average of the five random sums you computed above. How does this average compare to your previous estimate from part (b)? Average=1.11728 The average of these randomized Reimann sums and the limit of the middle reimann sums are very close 3. (2 Points) Consider the function f(z) = 2272 + ; on the interval [1, 3] for the interval. (a) Find an antiderivative F(x) of f(z). In other words, find a function F'(x) so that F'(z) = f(z) for all z (in the interval [1,3]). Check that your antiderivative is correct with the SageMath cell below Exercise 3 on the online version of this lab by using the command diff (F,x), which will compute the derivative of F'(x). F(x)=-1/xA2+(1/9)x | (b) Find the value of F(3) — F(1) and record what you found in the report below. How does the answer that you got here compare to the estimate of the limit of Riemann sums that you made above? F(3)-F(1)+10/9 or 1.111 The value is equal to the estimated limit of the middle Riemann sums
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