π/2 (6 3 cos x) dx 0 (a) Derive the formula for multi-segment (evenly spaced) left-hand rectangle...
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 ×...
Help PLEASE! Project. Approximating In 2 and π by numerical integration. 1. In Calculus 1 you learned that the natural logarithm of the number 2 is the value of the integral da The value of In 2 correct to 15 decimal places is In2 0.693147180559945 Use the error formula for the Composite Simpson's rule to determine how many correct decimal places you can obtain when you partition [1,2] into 4, n 8, and12 equal-length subintervals. 2. In Calculus you learned...
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...
MATLAB Create a function that provides a definite integration using Simpson's Rule Problem Summar This example demonstrates using instructor-provided and randomized inputs to assess a function problem. Custom numerical tolerances are used to assess the output. Simpson's Rule approximates the definite integral of a function f(x) on the interval a,a according to the following formula + f (ati) This approximation is in general more accurate than the trapezoidal rule, which itself is more accurate than the leftright-hand rules. The increased...
Section A Q1 0 Using the following Taylor series expansion: f(x+h) = f(x)+hf'(x)+22 h 3! f"(x)+ (+0) (1.1) 4! show that the central finite difference formula for the first derivative can be written as: f'(x)= f(x+h)-f(x-1) + ch" +0(hº) (1.2) 2h Determine cp and of the derived equation. [4 marks] Consider the function: f(x) = sin +COS (1.3) 2 2 Let x =ih with n=0.25, give your answer in 3 decimals for (ii) to (vi): (ii) Evaluate f(x) for i...