Consider the approximation of an integral I L f (x) dx by the trapezoidal or midpoint...

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Consider the approximation of an integral I L f (x) dx by the trapezoidal or midpoint rule which is the average of the left and right Riemann sums and generally provides a more accurate solution. The algorithm for the trapezoidal rule is given as follows: h 12 k=1 [f(xk) + f (xk+1)] where the data array x = [x1, x2,,XN,XN+1] is the discretized vector of the interval [a, b] with N divisions, and h = (ba)/N is the uniform step size. a) Based on what is learned in class, write a user-defined MATLAB function (numerical solver) that approximates numerical integration using the trapezoidal rule. A partially completed m-file ftrap (fun, a, b, N) is provided for you to complete. b) An athlete starts running from rest (velocity, v = 0). His acceleration as a function of time is given by the following plot. Apply the numerical solver ftrap to determine the maximum velocity of the athlete, which occurs when a d = 0 at t = 4s. Use N = 20 divisions in the trapezoidal rule. The velocity can be calculated using basic concepts of physics and calculus. adv-adt = = dt 5.5 5 4.5 4 3.5 Acceleration [m/s] 3 2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 Time [s] 2.5 3 3.5 4 Note that a(2.5) 5.4 m/s. In the range t = [0,2.5), the acceleration function is a(t) 2.16 t m/s, and in te [2.5,4], we have a(t) = -3.6(t - 4) m/s. Pay attention to how the time domains are defined to understand how to define the acceleration function correctly in MATLAB. Consider the approximation of an integral I L f (x) dx by the trapezoidal or midpoint rule which is the average of the left and right Riemann sums and generally provides a more accurate solution. The algorithm for the trapezoidal rule is given as follows: h 12 k=1 [f(xk) + f (xk+1)] where the data array x = [x1, x2,,XN,XN+1] is the discretized vector of the interval [a, b] with N divisions, and h = (ba)/N is the uniform step size. a) Based on what is learned in class, write a user-defined MATLAB function (numerical solver) that approximates numerical integration using the trapezoidal rule. A partially completed m-file ftrap (fun, a, b, N) is provided for you to complete. b) An athlete starts running from rest (velocity, v = 0). His acceleration as a function of time is given by the following plot. Apply the numerical solver ftrap to determine the maximum velocity of the athlete, which occurs when a d = 0 at t = 4s. Use N = 20 divisions in the trapezoidal rule. The velocity can be calculated using basic concepts of physics and calculus. adv-adt = = dt 5.5 5 4.5 4 3.5 Acceleration [m/s] 3 2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 Time [s] 2.5 3 3.5 4 Note that a(2.5) 5.4 m/s. In the range t = [0,2.5), the acceleration function is a(t) 2.16 t m/s, and in te [2.5,4], we have a(t) = -3.6(t - 4) m/s. Pay attention to how the time domains are defined to understand how to define the acceleration function correctly in MATLAB.