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1.1 Convert the binary number 1010100 to decimal format. Solution 1.2 Convert the binary number 1101.001 to decimal format. Solution The digits to the left of the decimal point are treated exactly as in problem 1.1. Thus, The numbers to […]

1.23 Use the first seven terms in Eq. (1.21) to calculate an estimated value of e. Do the calculation with MATLAB (use format long to display the numbers). Determine the true relative error. For the exact value of e, use […]

1.37 Write a user-defined MATLAB function that adds two integers in binary form according to the algo- rithm of Problem 1.30. Name the function aplusb = addbin(a,b), where the input arguments a and b are the numbers to be added […]

2 The the user-defined function odeRK3 is used in following program (script file) to solve Problem 8.2. clear, clc ODEHW10_2 = @ (x,y) x-x*y/2; a=1; b=3.4; yINI=1; h1=0.8; h2=0.1; [x1, y1] = odeRK3(ODEHW10_2,a,b,h1,yINI); [x2, y2] = odeRK3(ODEHW10_2,a,b,h2,yINI); fplot(‘2-exp((1-x^2)/4)’,[1 3.4]) hold […]

1 10.1 Consider the following first-order ODE: from to with (a) Solve with Euler’s explicit method using . (b) Solve with the modified Euler method using . (c) Solve with the classical fourth-order Runge–Kutta method using . The analytical solution […]

1 10.9 Write the following second-order ODEs as systems of two first-order ODEs: (a), where g, T, and w are constants. (b). Solution (a) Define , . With these definitions, the system of two first-order ODEs is: 1 g —–d2h […]

3 The listing of the user-defined function ODEsHW10_6 (the system of ODEs in Problem 10.6) that is used in the solution is: When the program is executed the numerical solution is displayed in the Command Window and the fol- lowing […]

4 a=0; b=pi; n=100; Ya=1.5; Yb=0; WL=-5; WH=-1.5; When the script file is executed, the following figure is displayed in the Figure Window: This solution agrees with the solution in Example 11-6. 00.5 11.5 22.5 3 3.5 −1 −0.5 0 […]

1 11.1 Consider the following second-order ODE: (a) Using the central difference formula for approximating the second derivative, discretize the ODE (rewrite the equation in a form suitable for solution with the finite difference method). (b) If the step size […]

3 function res = bcfunHW11_27b(ya,yb) Tinf=300; h=100; k=72; When executed, the program produces the following plot: 0123456 x 10-5 624.098 624.1 624.102 624.104 x (m) T (K) 624.106 624.108 624.11 BCa = 0; BCb = -h*(yb(1)-Tinf)/k; res = [ya(2) – […]

3 (c) Using MATLAB built-in functions to solve the ODE. The ODE is transforming into a system of two first-order ODEs: 0 100 200 300 400 T (C) 11.5 22.5 33.5 500 600 r (cm) dT dr —–– w= dw […]

1 ______________________________________________________________________________ (2.1) Apply the intermediate value theorem to show that the polynomial 24x10x)x(f 2−+−= has a root in the interval [3,5]. Solution The intermediate value theorem (see Section 2.2) guarantees that there exists at least one 2.2 Apply the […]

2.36 Rope with a length of 10 m is to be used to enclose a square area with side x and a circular area with radius r. How much rope should be used for the square and how much for […]

2 ______________________________________________________________________________ 2.25 Write a user-defined MATLAB function that evaluates the definite integral of a function by using the Riemann sum (see Eq. (2.7)). For function name and arguments, use I=Rie- mannSum(Fun,a,b). Fun is a name for the function that […]

3.1 The tolerance, , of the solution in the bisection method is given by , where and are the endpoints of the interval after the nth iteration. The number of iterations n that are required for obtaining a solution with […]

1 3.23 Modify the function NewtonRoot that is listed in Fig. 3-11, such that the output will have three arguments. Name the function [ Xs,FXs,iact] = NewtonRootMod ( Fun,FunDer,Xest,Err,imax ) . The first output argument is the solution, the second […]

3.14 Solve the following system of nonlinear equations: (a) Use Newton’s method. Start at , , and carry out the first five iterations . (b) Use the fixed-point iteration method. Start at , , and carry out the first five […]

2 Con=w/(360*L*E*I); y=@(x) Con*x*(7*L^4-10*L^2*x^2+3*x^4); yd=@ (x) Con*(7*L^4-10*L^2*3*x^2+3*5*x^4); ydd=@ (x) Con*(-10*L^2*3*2*x+3*5*4*x^3); When the script file is executed the following results are displayed in the Command Window: Part (a) xMax_a = 2.0773 yMax_a = 0.0090 Part (b) xMax_b = 2.0773 yMax_b = […]

1 4.15 Carry out the first three iterations of the solution of the following system of equations using the Gauss–Seidel iterative method. For the first guess of the solution, take the value of all the unknowns to be zero. Solution […]

1 4.1 Solve the following system of equations using the Gauss elimination method: Solution Step 1: Write the system of equations in matrix form: Using the first element of the matrix as a pivot, and . 2x1x2x3 –+1= x12x2x3 ++ […]

3 0 0 0 -1 0 0 0.707 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0.707 0 0 0 0.707 0 -0.5 -1 0 0 0 0 0 0 0 0 0 0 […]

1 4.28 Write a user-defined MATLAB function that calculates the condition number of an matrix by using the infinity norm. For the function name and arguments use c = CondNumb_Inf(A), where A is the matrix and c is the value […]

1 5.1 Show that the eigenvalues of the identity matrix are the number 1 repeated n times. Solution The eigenvalues of any matrix are determined from the solution of the characteristic equation, Eq.(4.93): nn× det a λI–[]0= For the case […]

2 >> sigma=[40 20 -18; 20 28 12; -18 12 14]; >> e = MaxEig(Sigma) e = 56.4898 To check the answer, the eigenvalues of the matrix are calculated with MATLAB’s built-in function eig: >> eig(Sigma) ans = -7.2389 32.7493 […]

1 6.14 Determine the fourth-order Newt on’s interpolating polynomial that passes through the data poi nts given in Problem 6.13. Use the polynomial to calculate the power at a wind speed of 26 mph. Solution Write the data in column […]

1 6.32 Use MATLAB’s built-in functions to determine the coefficients of the third-order polynomial, (where x is the number of years after 1981) that best fits the data in Problem 6.31. Use the polynomial to estimate the percent of computer […]

3 y=[0 3 4.5 5.8 5.9 5.8 6.2 7.4 9.6 15.6 20.7 26.7 31.1 35.6 39.3 41.5]; n=length(x); [a,Er] = CubicPolyFit(x,y) When the program is executed, the following plot is displayed in the Figure Window: 0 1 2 3 4 […]

3 (b) Quadratic function that fits the data has the form: The following program written in a script file uses the user-defined function QuadFit from Problem 5.20 to find the quadratic function that best fits the data. It also plots […]

1 6.1 The following data is given: (a) Use linear least-squares regression to determine the coefficients m and b in the functi on that best fit the data. (b) Use Eq. (6.5) to determine the overall error. Solution (a) In […]

1 8.1 Given the following data: find the first derivative at the point . (a) Use the three-point forward difference formula. (b) Use the three-point backward difference formula. (c) Use the two-point central difference formula. Solution (a) The three point […]

1 8.35 The following data for mean velocity near the wall in a fully developed turbulent pipe air flow was measured [J. Laufer, “The Structure of Turbulence in Fully Developed Pipe Flow,” U.S. National Advisory Committee for Aeronautics (NACA), Technical […]

3 >> format compact >> x=[0 1 2 3 4]; >> y=[0 1 2 3 4]; >> f=[0 3 14 7 5 8 10 14 12 10 2 7 8 9 7 Note that in the MATLAB script, i is […]

1 8.15 Use Lagrange polynomials to develop a difference formula for the second derivative of a function that is specified by a discrete set of data points with unequal spacing. The formula determines the second derivative at point using points […]

1 9.9 The area of the shaded region shown in the figure can be calculated by: Evaluate the integral using the following methods: (a) Simpson’s 1/3 method. Divide the whole interval into four subintervals. (b) Simpson’s 3/8 method. Divide the […]

1 9.1 The function is given in the following tabulated form. Compute with and with . (a) Use the composite rectangle method. (b) Use the composite trapezoidal method. (c) Use the composite Simpson’s 3/8 method. Solution (a) For the composite […]