Page - 169 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

6.8 Exercises 169 6.8 Exercises Exercise6.1:HandCalculationsfor the TrapezoidalMethod Compute by hand the area composed of two trapezoids (of equal width) that ap- proximates the integral ∫3 1 2x 3dx.Makea test function thatcalls thetrapezoidal function in trapezoidal.py and compares the return value with the hand- calculatedvalue. Filename:trapezoidal_test_func.py. Exercise6.2:HandCalculationsfor the Midpoint Method Computebyhandtheareacomposedof tworectangles(ofequalwidth) thatapprox- imates the integral ∫3 1 2x 3dx.Makea test functionthatcalls themidpointfunction inmidpoint.pyandcompares the returnvaluewith thehand-calculatedvalue. Filename:midpoint_test_func.py. Exercise6.3:ComputeaSimple Integral Applythetrapezoidalandmidpointfunctionstocomputethe integral ∫6 2 x(x− 1)dxwith2 and100subintervals.Compute the error too. Filename:integrate_parabola.py. Exercise6.4:Hand-Calculationswith Sine Integrals We consider integrating thesine function: ∫b 0 sin(x)dx. a) Let b = π and use two intervals in the trapezoidal and midpoint method. Compute the integral by hand and illustrate how the two numerical methods approximate the integral.Comparewith the exactvalue. b) Doa) whenb=2π. Filename:integrate_sine.py. Exercise6.5:Make Test Functions for theMidpoint Method Modify the filetest_trapezoidal.pysuch that the three tests are applied to the functionmidpoint implementing themidpointmethodfor integration. Filename:test_midpoint.py. Exercise6.6:ExploreRounding ErrorswithLargeNumbers The trapezoidal method integrates linear functions exactly, and this prop- erty was used in the test function test_trapezoidal_linear in the file test_trapezoidal.py. Change the function used in Sect.6.6.2 to f(x) = 6 · 108x − 4 · 106 and rerun the test. What happens? How must you change the test tomakeituseful?Howdoestheconvergencerate testbehave?Anyneedfor adjustment? Filename:test_trapezoidal2.py.