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...
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?...
Using n=6 approximate the value of ∫_(-1)^2▒√(e^(-x^2 )+1) dx using Trapezoid rule. (6) Using n 6 approximate the value of L3 Ve-x2 + 1 dx using Trapezoid rule. 15Marks
Problem 2 (1) Approximate If = 1.4" dx using the composite trpezoidal rule with uniform partition Rr(f;P), where h = 1/2. (2) Find the true value of the definte integral using the antiderivative of f(x) = 4". (3) Does your approximation overestimate or underestimate the true value? Use the graph to explain the overesimation or underestimation geometrically.
2 Problem 3 (25 points) Let I = ïrdz. a) [by hand] Use a composite trapezoidal rule to evaluate 1 using N = 3 subintervals. b) MATLAB] Use a composite trapezoidal rule to evaluate I using N - 6 subinterval:s c) by hand] Use Romberg extrapolation to combine your results from a) and b) and obtain an improved approximation (you may want to compare with a numerical approximation of the exact value of the integral 2 Problem 3 (25 points)...
3. Suppose we want to use the ri-term trapezoid rule to approximate Sinde (a) (3 points) Make a graph of y= between = 2 and 3 = 4. Draw on your graph the trapezoids used to apply the Trapezoidal Rule with n = 3. (So, your graph should have 3 trapezoids.) (b) (2 points) Does the Trapezoidal Rule overestimate or underestimate the value of justify your answer. 1 dx? No need to (c) (5 points) For the Trapezoidal Rule, the...
10. Trapezoidal Rule is used to approximate the integral f(a) dx using 1- (yo +2y1 + 2y2 + x-na b-a + 2yn-1 +%),where Use this approximation technique to estimate the area under the curve y = sinx over。 a. π with n 4 partitions. x A 0 B: @ Δy B-A b. The error formula for the trapezoidal rule is RSL (12ba)1 where cischosen on the interval [a, b] to maximize lf" (c)l. Use this to compute the error bound...