Text: Numerical Integration & Differentiation Q2. Consider the function f(x) = 5 + 3cos(x) 1=0 : 0.2 : 6. (a) Use MATLAB to plot the f(x) to get the graphs shown in Figure (2) using plot and stem commands. Explain their differences. Continuous: Using plot command. Discrete: Using stem command. Figure (2) Integrate analytically I(x) for (x=0 to x=3). Integrate numerically I(x) for (x=0 to x=3) with: Single application of the trapezoidal rule. Multiple application of the trapezoidal rule with n-5. Simpson's 1/3 rule. Simpson's 3/8 rule. Multiple application of Simpson's rules with n-5. Compute preliminary numerical using MATLAB with trapz command.
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The continuous plot will show a smooth curve while the discrete plot will show individual points connected by lines. The difference between the two is that the continuous plot is better for visualizing the overall shape of the function while the discrete plot is Show more…
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1. Evaluate the integral of the data that is tabulated below, with: a) the trapezoidal rule b) and Simpson's rules: X | -2 | 0 | 2 | 4 | 6 | 8 | 10 f(x) | 35 | 5 | -10 | 2 | 5 | 3 | 20 2. The function f(x) = 2e^-1.5x can be used to generate the following table of unevenly spaced data: X | 0 | 0.05 | 0.15 | 0.25 | 0.35 | 0.475 | 0.6 f(x) | 2 | 1.8555 | 1.5970 | 1.3746 | 1.1831 | 0.9808 | 0.8131 3. Use Romberg integration of order h^8 to evaluate: ∫[0 to 3] xe^x dx Compare ε_a ε_r Obtain an estimate of the integral of Problem 3, but use Gauss-Legendre formulas with two, three and four points. Calculate ε_t for each case based on the solution analytics. 4. Use numerical integration to evaluate the following: a. ∫[2 to ∞] dx / x(x+2) b. ∫[0 to ∞] e^-y sin^2 y dy c. ∫[0 to ∞] 1 / ((1+y^2)(1+y^2/2)) dy d. ∫[-2 to ∞] ye^-y dy e. ∫[0 to ∞] (1/√2) e^(-x^2/2) dx
Sri K.
a) Consider the following data points: x = {0,2,4,6,8}; y = {0,1,3,3,2}. Calculate the first derivatives, y'(x), at these points, using the finite divided difference method. b) Consider the following data points: x = {0,1,3,5,8}; y = {3,5,2,4,1}. Integrate from x=0 to x=5 by multiple application of the trapezoidal rule. c) Use the Simpson's 1/3 rule to numerically integrate a function f = ln(x/2) from x=20 to x=60.
Adi S.
Trapezoidal Simpson's Rule for Definite Integral Trapezoidal and Simpson's Rule are numerical methods used to approximate definite integrals. These methods divide the interval of integration into smaller subintervals and use a combination of linear and quadratic approximations to estimate the area under the curve. The Trapezoidal Rule approximates the area using trapezoids, while Simpson's Rule uses quadratic approximations. Both methods are useful when the exact value of the integral is difficult or impossible to find analytically.
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