![Theorem 4.4 Let f e C4[a,b], n be even, h = (b - a)/n, and x; = a + jh, for each j = 0, 1,...,n. There exists a ue (a, b) for](http://img.homeworklib.com/questions/3df21200-bc70-11ea-9f8f-f9ca3f257961.png?x-oss-process=image/resize,w_560)
![Theorem 4.5 Let f e C?[a, b], h = (b - a)/n, and x; = a + jh, for each j = 0,1,...,n. There exists au € (a,b) for which the C](http://img.homeworklib.com/questions/3e4a3eb0-bc70-11ea-abef-a7b74d798c3a.png?x-oss-process=image/resize,w_560)
Homework Answers
![..}d2 = 0,3 [at 1+0.5+ 2 CO-9001+ 0.8333 + 07692 +0.7143 to 6667 + 0.6250 +0.5882 +0.5556 to 5263) [o I = 3! [1.5+2 (6-1877)]](http://img.homeworklib.com/questions/3fe689a0-bc70-11ea-8845-c9fc92786603.png?x-oss-process=image/resize,w_560)
=0.6931![[3 ] Midpoint sule we have ² 1 dx, xn=2, xo=1, T Let n=4, we get xn= not nh .-) 2 = 1 +4h =) 1=an a h= 114 The subintervals c](http://img.homeworklib.com/questions/40924980-bc70-11ea-b585-1726a8e2836c.png?x-oss-process=image/resize,w_560)
![Also, ļ fw dz = ge / dx = [ log x] = log a- log 1 = log 2-0 a dog a =10 69311 True relue Error Estimation Treepezoidal sule e](http://img.homeworklib.com/questions/413309e0-bc70-11ea-acec-1fd976a0a06c.png?x-oss-process=image/resize,w_560)
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121, C. Consider the integral - dr. Use the theorems from Section 4.4 to get upper...
3. Assume we have Simpson's Rule: to = a, 13 = , h = (b-a)/2 =...
3. Assume we have Simpson's Rule: to = a, 13 = , h = (b-a)/2 = a +h. (20) + 47(01) + f(x)]- ()where do < < Let fe .b), be even, h= (b-a)/n, and = a + jh, for each j = 0,1...... Show that there exists a l E (a,b) for which the Composite Simpson's rule for n subintervals can be written with its crror term as n/2 bar (n/2) - 1 f(a) +2 =1 (12) + 4...
Numerical Methods Consider the integral 2 (a) [16 marks] Use the composite Simpson's rule with four intervals to calculate (by hand) approximate value of the integral Calculate the maximum value o...
Numerical Methods
Consider the integral 2 (a) [16 marks] Use the composite Simpson's rule with four intervals to calculate (by hand) approximate value of the integral Calculate the maximum value of the error in your approximation, and compare it with the true error. (b) 19 marks] Determine the number of subintervals n and the step size h so that the composite Simpson's rule for n subintervals can be used to compute the given integral with an accuracy of 5 ×...
13. Let f(x)and consider the integral 1= | f(x) dr. 0 (a) Use the composite trapezoidal...
13. Let f(x)and consider the integral 1= | f(x) dr. 0 (a) Use the composite trapezoidal rule with h = 0.25 to approximate 1. 13. Let f(x) e and consider the integral -I:f( 1e)dr. (a) Use the composite trapezoidal rule with h 0.25 to approxinate 1. (b) Calculate the bound on the absolute error for the Trapezoidal rule.
class : numerical analysis I wish if it was written in block letter Sorry I can't...
class : numerical analysis
I wish if it was written in block letter
Sorry I can't read cursive
= Problem 2: Let I(f) = S• f (x)dx. We are interested in approximating this integral within a certain error tolerance. First some notation. Let n be a positive integer and define xj = a + j xh where h (b − a)/n. Recall that the Midpoint rule approximates the integral of f by a Riemann sum that evaluates the function at...
Use Matlab code Consider the following function sin(x) Using the following parameters in your functions: -func:...
Use
Matlab code
Consider the following function sin(x) Using the following parameters in your functions: -func: the function/equation that you are required to integrate -a, b: the integration limits n: the number of points to be used for the integration I:Integral estimate a) Write a function capable of performing numerical integration of h(x) using the composite trapezoidal rule. Use your function to integration the equation with 9 points. Write a function capable of performing numerical integration of h(x) using the...
Exercise 6: Given the table of the function f(x)-2" 2 X 0 3 2 f(x) 1 2 4 8 a) Write down the Newton polynomial...
Exercise 6: Given the table of the function f(x)-2" 2 X 0 3 2 f(x) 1 2 4 8 a) Write down the Newton polynomials P1(x), P2(x), Pa(x). b) Evaluate f(2.5) by using Pa(x). c) Obtain a bound for the errors E1(x), E2(x), Es(x) Exercise 7: Consider f(x)- In(x) use the following formula to answer the given questions '(x) +16-30f+16f,- 12h a) Derive the numerical differentiation formula using Taylor Series and find the truncation error b) Approximate f'(1.2) with h-0.05...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one end of a string with length 1 and can swing in any direction. In fact, it moves on a sphere, a subspace of R3 1 0 φ g 2.1 Use the spherical coordinates (1,0,) to derive the Lagrangian L(0,0,0,0) = T-U, namely the difference of kinetic energy T and potential energy U. (Note r = 1 is fixed.) 2.2 Calculate the Euler-Lagrange equations, namely...
3. Assume we have Simpson's Rule: to = a, 13 = , h = (b-a)/2 = a +h. (20) + 47(01) + f(x)]- ()where do < < Let fe .b), be even, h= (b-a)/n, and = a + jh, for each j = 0,1...... Show that there exists a l E (a,b) for which the Composite Simpson's rule for n subintervals can be written with its crror term as n/2 bar (n/2) - 1 f(a) +2 =1 (12) + 4...
Numerical Methods
Consider the integral 2 (a) [16 marks] Use the composite Simpson's rule with four intervals to calculate (by hand) approximate value of the integral Calculate the maximum value of the error in your approximation, and compare it with the true error. (b) 19 marks] Determine the number of subintervals n and the step size h so that the composite Simpson's rule for n subintervals can be used to compute the given integral with an accuracy of 5 ×...
13. Let f(x)and consider the integral 1= | f(x) dr. 0 (a) Use the composite trapezoidal rule with h = 0.25 to approximate 1. 13. Let f(x) e and consider the integral -I:f( 1e)dr. (a) Use the composite trapezoidal rule with h 0.25 to approxinate 1. (b) Calculate the bound on the absolute error for the Trapezoidal rule.
class : numerical analysis
I wish if it was written in block letter
Sorry I can't read cursive
= Problem 2: Let I(f) = S• f (x)dx. We are interested in approximating this integral within a certain error tolerance. First some notation. Let n be a positive integer and define xj = a + j xh where h (b − a)/n. Recall that the Midpoint rule approximates the integral of f by a Riemann sum that evaluates the function at...
Use
Matlab code
Consider the following function sin(x) Using the following parameters in your functions: -func: the function/equation that you are required to integrate -a, b: the integration limits n: the number of points to be used for the integration I:Integral estimate a) Write a function capable of performing numerical integration of h(x) using the composite trapezoidal rule. Use your function to integration the equation with 9 points. Write a function capable of performing numerical integration of h(x) using the...
Exercise 6: Given the table of the function f(x)-2" 2 X 0 3 2 f(x) 1 2 4 8 a) Write down the Newton polynomials P1(x), P2(x), Pa(x). b) Evaluate f(2.5) by using Pa(x). c) Obtain a bound for the errors E1(x), E2(x), Es(x) Exercise 7: Consider f(x)- In(x) use the following formula to answer the given questions '(x) +16-30f+16f,- 12h a) Derive the numerical differentiation formula using Taylor Series and find the truncation error b) Approximate f'(1.2) with h-0.05...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one end of a string with length 1 and can swing in any direction. In fact, it moves on a sphere, a subspace of R3 1 0 φ g 2.1 Use the spherical coordinates (1,0,) to derive the Lagrangian L(0,0,0,0) = T-U, namely the difference of kinetic energy T and potential energy U. (Note r = 1 is fixed.) 2.2 Calculate the Euler-Lagrange equations, namely...
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