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Homework Answers
The area is a overestimate because when we consider the case of
concave up curve then the case will be look like the one in the
following figure. So when we join two points then we overestimate a
small portion of area over the curve.
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Suppose f(x) is continuous and increasing on [a, b] , and concave up on (a, b)...
Suppose f(x) is continuous and increasing on [a, b], and concave up on (a, b). Is...
Suppose f(x) is continuous and increasing on [a, b], and concave up on (a, b). Is M6 (the Midpoint Rule approximation to f(x) dx with n = 6) an over-estimate or an under- estimate? Over-Estimate. Under-Estimate. There is not enough information to decide.
Suppose f(x) is continuous and increasing on [a, b], and concave up on (a, b). Is...
Suppose f(x) is continuous and increasing on [a, b], and concave up on (a, b). Is S. (the Simpson's Rule approximation to S. f(x) dx with n = 6) an over-estimate or an under- estimate? Over-Estimate. Under-Estimate. There is not enough information to decide.
4. Another approximation for integrals is the Trapezoid Rule: integral (a to b)f(x) dx ≈ ∆x/2...
4. Another approximation for integrals is the Trapezoid Rule: integral (a to b)f(x) dx ≈ ∆x/2 (f(x_0) + 2f(x_1) + 2f(x_2) + · · · + 2f(x_n−2) + 2f(x_(n−1)) + f(x_n)) There is a built-in function trapz in the package scipy.integrate (refer to the Overview for importing and using this and the next command). (a) Compute the Trapezoid approximation using n = 100 subintervals. (b) Is the Trapezoid approximation equal to the average of the Left and Right Endpoint approximations?...
If f has a continuous second derivative on [a, b], then the error E in approximating...
If f has a continuous second derivative on [a, b], then the error E in approximating by the Trapezoidal Rule is (b- a 12n rmax x)1. asxsb. JE s Moreover, if f has a continuous fourth derivative on [a, bl, then the error E in approximating by fix) dx Simpson's Rule is b-a)s 180a lrmax (x. asxsb. Use these to find the minimum integer n such that the error in the approximation of the definite integral is less than or...
If f has a continuous second derivative on tə, b), then the error E in approximating...
If f has a continuous second derivative on tə, b), then the error E in approximating f(x) dx by the Trapezoidal Rule is IELS (-a) [max 1f"(x)), a sxs b. 12n2 Moreover, if f has a continuous fourth derivative on (a, b), then the error E in approximating Rx) dx by Simpson's Rule is IES (6-a) [max 1(1)(x)), a sxs b. 1804 Use these to find the minimum Integer n such that the error in the approximation of the definite...
Problem 3. Suppose you are programming the composite trapezoid rule (CTR) to approximate 1(f) =| f(x) dx using the TR with N subintervals, and that you mistakenly forget to weight down the two en...
Problem 3. Suppose you are programming the composite trapezoid rule (CTR) to approximate 1(f) =| f(x) dx using the TR with N subintervals, and that you mistakenly forget to weight down the two endpoints by 3. That is, you have accidentally programmed the quadrature rule where h-%.. (Note: sinoefe C, you know that UIL is bounded.) 1. Find QBADN -OCTRN where QCTRN ) is the approximation to (x) dx computed via the CTR with N subintervals.
Problem 3. Suppose you...
Suppose that f(x) is a continuous function on (2,7), positive on (2,5) and negative on (5,7)...
Suppose that f(x) is a continuous function on (2,7), positive on (2,5) and negative on (5,7) If f(x) dx = 3, then find If(x) dar. () dx = 11 and Ś "S" ) dr = = -3, Ls =) dr = 5, (b) Suppose that f is an even and integrable function. If then find Lºs(z) dr.
3. Suppose we estimateſ f(x)dx using our rules with the same number of subdivisions, n but...
3. Suppose we estimateſ f(x)dx using our rules with the same number of subdivisions, n but only record three of our estimates: Right(n)=1.8569 Mid(n) = 2.3481 Trap(n) = 2.1627. If f(x)is monotone and does not have any inflection points in the interval [A, B], A. Is f(x) increasing or decreasing? B. Is f(x) concave up or down? C. Estimate the value of Left(n)and Simp(n)
Question 14 Suppose f(x) is an decreasing, concave up function and you use numeric integration to...
Question 14 Suppose f(x) is an decreasing, concave up function and you use numeric integration to compute the integral of f over the interval [0, 1]. Put the values of the approximations using n = 100 for the left end-point rule (L100), right end-point rule (R100), and Simpson's rule (S100) from the least to the greatest. a) OS100, L100, R100 b) L100, S100, R100 R100, L100, S100 $100, R100, 2100 R100, S100, L100 O None of the above. Review La...
Eunsol Kwon e. Jix) is concave up on [U, OJ Since the grapn is increasing at...
Eunsol Kwon e. Jix) is concave up on [U, OJ Since the grapn is increasing at x = U, the left-most point of this interval Consider the following integral: [x" dx for n € N = {1,2, 3, --}. Now consider the following three statements: is a nonsensical statement. equals the exact area bounded by f(x) = x",x=0, x= 1, and the x-axis. Then Select one: a. All of L., II., and Ill are true. b. II. is the only...
Suppose f(x) is continuous and increasing on [a, b], and concave up on (a, b). Is M6 (the Midpoint Rule approximation to f(x) dx with n = 6) an over-estimate or an under- estimate? Over-Estimate. Under-Estimate. There is not enough information to decide.
Suppose f(x) is continuous and increasing on [a, b], and concave up on (a, b). Is S. (the Simpson's Rule approximation to S. f(x) dx with n = 6) an over-estimate or an under- estimate? Over-Estimate. Under-Estimate. There is not enough information to decide.
If f has a continuous second derivative on [a, b], then the error E in approximating by the Trapezoidal Rule is (b- a 12n rmax x)1. asxsb. JE s Moreover, if f has a continuous fourth derivative on [a, bl, then the error E in approximating by fix) dx Simpson's Rule is b-a)s 180a lrmax (x. asxsb. Use these to find the minimum integer n such that the error in the approximation of the definite integral is less than or...
If f has a continuous second derivative on tə, b), then the error E in approximating f(x) dx by the Trapezoidal Rule is IELS (-a) [max 1f"(x)), a sxs b. 12n2 Moreover, if f has a continuous fourth derivative on (a, b), then the error E in approximating Rx) dx by Simpson's Rule is IES (6-a) [max 1(1)(x)), a sxs b. 1804 Use these to find the minimum Integer n such that the error in the approximation of the definite...
Problem 3. Suppose you are programming the composite trapezoid rule (CTR) to approximate 1(f) =| f(x) dx using the TR with N subintervals, and that you mistakenly forget to weight down the two endpoints by 3. That is, you have accidentally programmed the quadrature rule where h-%.. (Note: sinoefe C, you know that UIL is bounded.) 1. Find QBADN -OCTRN where QCTRN ) is the approximation to (x) dx computed via the CTR with N subintervals.
Problem 3. Suppose you...
Suppose that f(x) is a continuous function on (2,7), positive on (2,5) and negative on (5,7) If f(x) dx = 3, then find If(x) dar. () dx = 11 and Ś "S" ) dr = = -3, Ls =) dr = 5, (b) Suppose that f is an even and integrable function. If then find Lºs(z) dr.
3. Suppose we estimateſ f(x)dx using our rules with the same number of subdivisions, n but only record three of our estimates: Right(n)=1.8569 Mid(n) = 2.3481 Trap(n) = 2.1627. If f(x)is monotone and does not have any inflection points in the interval [A, B], A. Is f(x) increasing or decreasing? B. Is f(x) concave up or down? C. Estimate the value of Left(n)and Simp(n)
Question 14 Suppose f(x) is an decreasing, concave up function and you use numeric integration to compute the integral of f over the interval [0, 1]. Put the values of the approximations using n = 100 for the left end-point rule (L100), right end-point rule (R100), and Simpson's rule (S100) from the least to the greatest. a) OS100, L100, R100 b) L100, S100, R100 R100, L100, S100 $100, R100, 2100 R100, S100, L100 O None of the above. Review La...
Eunsol Kwon e. Jix) is concave up on [U, OJ Since the grapn is increasing at x = U, the left-most point of this interval Consider the following integral: [x" dx for n € N = {1,2, 3, --}. Now consider the following three statements: is a nonsensical statement. equals the exact area bounded by f(x) = x",x=0, x= 1, and the x-axis. Then Select one: a. All of L., II., and Ill are true. b. II. is the only...
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