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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)...
Suppose f(x) is continuous and increasing on [a, b] , and concave up on (a, b) .ls TO (the Trapezoid 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 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.
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...
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...
7. (15 pts) Numerical Integration. Given a continuous function f (x) on the interval [a, b],...
7. (15 pts) Numerical Integration. Given a continuous function f (x) on the interval [a, b], write the Lagrange form of the quadratic polynomial interpolating f(a), (a b)), f(b). Instead of calculating the integral I(f) Jaf(x)dx we could approximate it via Q(f) = | q(x)dx. Find an expression for this quadrature rule, the so-called Simpson's rule.
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 6 [10 marks a) Let f(x) = x for each xe [a,b]. Show that for...
Question 6 [10 marks a) Let f(x) = x for each xe [a,b]. Show that for any number of subintervals, the global error js(x)dx-SUS J) = 0. [6] Hint: Obtain the local error first and then calculate the global error. SCS ,h) denotes approximation using the composite Simpson's Rule. b) Determine the minimum number of subintervals so that the upper bound of the (absolute) global error for the composite Simpson's Rule applied to ja?-10x”) dx is less than 10%. [41...
Suppose f '(x)=x+ (x - 1). Then f '(x) = x (3x - 2). Over which...
Suppose f '(x)=x+ (x - 1). Then f '(x) = x (3x - 2). Over which interval(s) is the graph off both increasing and concave up? 1.x<0 Il. 0<x< II. <x<1 IV. * > 1 (A) I only (B) II only (C) II and IV (D) I and IV O (E) IY only Suppose f'(x) = x+ (x - 1). Then f"(x) = x (3x - 2). Over which interval(s) is the graph off both increasing and concave up? 1.x<0...
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) .ls TO (the Trapezoid 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 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.
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...
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...
7. (15 pts) Numerical Integration. Given a continuous function f (x) on the interval [a, b], write the Lagrange form of the quadratic polynomial interpolating f(a), (a b)), f(b). Instead of calculating the integral I(f) Jaf(x)dx we could approximate it via Q(f) = | q(x)dx. Find an expression for this quadrature rule, the so-called Simpson's rule.
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 6 [10 marks a) Let f(x) = x for each xe [a,b]. Show that for any number of subintervals, the global error js(x)dx-SUS J) = 0. [6] Hint: Obtain the local error first and then calculate the global error. SCS ,h) denotes approximation using the composite Simpson's Rule. b) Determine the minimum number of subintervals so that the upper bound of the (absolute) global error for the composite Simpson's Rule applied to ja?-10x”) dx is less than 10%. [41...
Suppose f '(x)=x+ (x - 1). Then f '(x) = x (3x - 2). Over which interval(s) is the graph off both increasing and concave up? 1.x<0 Il. 0<x< II. <x<1 IV. * > 1 (A) I only (B) II only (C) II and IV (D) I and IV O (E) IY only Suppose f'(x) = x+ (x - 1). Then f"(x) = x (3x - 2). Over which interval(s) is the graph off both increasing and concave up? 1.x<0...
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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