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Q1: (6) The sequence {an} is defined by a1 = 1 and a_{n+1}=\left(2 a_{n}+1\right)^{\frac{1}{2}}-\frac{1}{4}\left(2 a_{n}+1\right)^{\frac{1}{4}}, \quad n=1,2, \ldots Show that the sequence is convergent.See Answer
Q2:3. Let f(x)=x²-6. With Po=3 and p₁=2, find p3. a. Use the Secant method.See Answer
Q3:8. Use the Secant method. b. Use the method of False Position. 5. Use Newton's method to find solutions accurate to within 10 for the following problems. a. b. C. d. ¹-2x²-5-0, [1,4] x-cos x = 0, [0, π/2] x+3x²-1=0. [-3,-2] x 0.8-0.2 sin x = 0, [0, π/2]See Answer
Q4:= 1. Use the forward-difference formulas and backward-difference formulas to determine each missing entry in the following tables. a. x f(x) f'(x) 0.5 0.4794 0.6 0.5646 0.7 0.6442 0.0 0.2 0.4 f(x) f'(x) 0.00000 0.74140 1.3718See Answer
Q5:a. f(x) = 2 cos 2x -x b. f(x)=x² Inx+1 5. Use the most accurate three-point formula to determine each missing entry in the following tables. a. x f(x) f'(x) b. x f(x) f'(x) 1.1 9.025013 1.2 11.02318 1.3 13.46374 1.4 16.44465 8.1 8.3 8.5 8.7 16.94410 17.56492 18.19056 18.82091See Answer
Q6:1. If an integer n equals 2.k and k is an integer, then n is even. O equals 2.0 and O is an integer.See Answer
Q7:2. For all real numbers a, b, c, and d, if b = 0 and d = 0, then a/b + c/d = (ad + bc)/bd. a = 2, b = 3, c = 4, and d = 5 are particular real numbers such that b = 0 and d = 0.See Answer
Q8:3. Consider the following algorithm segment: if x.y > 0 then do y := 3. x x := x + 1 end do z :=x.y Find the value of z if prior to execution x and y have the values given below. a. x = 2, y =3 b.x = 1, y = 1See Answer
Q9:1. Solve the following system of linear equations by hand, employing Cramer's rule. Verify your solution employing x = inv(A)*d and x=A\d. x12x2 + 3x3 = 0 -4x1 + x2 - 2x3 = -1 -3x16x2 + 2x3 = 2See Answer
Q10:2. Given the equations 10x1 + 2x2x3 = 27 -3x16x2 + 2x3 = -123/2 x1 + x2 + 5x3 = -43/2 Solve, by hand, using Gauss elimination. Show all steps of the computation. Verify using the function gauss.See Answer
Q11:3. Given the equations 2x2 + 3x3 = 8 4x1 + 6x2 + 7x3 = -3 2x13x2 + 6x3 = 5 Solve, by hand, using Gauss elimination with partial pivoting. Show all steps. Verify using the function gauss_pivot.See Answer
Q12:4. Find, by hand, the exact solution for the system 0.0003x1 + 3x2 = 2.0001 x1 + x2 = 1 Then, solve it using gauss and gauss_pivot. Set format to long. Which function is more accurate to use? Why?See Answer
Q13:5. Apply (by hand) LU decomposition to the system in Problem 2. Check your answer using Matlab. Then, employ back and forward substitution to find the solution for the following d vectors: [0-1 2] and [-2 4 5]T. Verify your answers using Matlab.See Answer
Q14:6. Apply the Jacobi and guass_seidel functions to solve the system of equations in Problem 2. Assume the initial guess is [111]¹.See Answer
Q15:1. Bessel functions often arise in engineering analysis such as the study of electric filters. The following table provides 4 uniformly spaced samples for the zero-order Bessel function of the first kind. X J1(x) 1.8 0.5815 2.0 0.5767 2.2 0.5560 2.4 0.5202 generate the third-order Newton's interpolating polynomial. Use the newtintr function to interpolate at x = 2.1. Validate your answer using Matlab's polyfit and polyval functions.See Answer
Q16:2. Ohm's law states that the voltage, resistance and current in a resistor are related by the linear equation V=RI. In practice, however, real resistors may not always obey this law. Suppose that you performed a precise experiment in the lab and obtained the following data for the resistor. I V -2 -637 -1 -96.5 -0.5 -20.5 0.5 20.5 1 96.5 2 637 Use a fifth-order polynomial to fit the data and compute V for 1 = 0.1. Employ function newtintr. Next, use Matlab's polyfit to generate the interpolation polynomial. Plot the data points and the polynomial, on the same graph, for x-axis limits [-2.2 2.2].See Answer
Q17:1 3. Consider the function f(x) = Generate five equidistantly spaced points 1+25x² (xi, f (xi)) over the interval: [-1 1]. Fit this data with (a) a fourth-order polynomial, (b) a linear spline and (c) a cubic spline. Assume spline derivatives are zero at the end points. Compare your results graphically (use the x-axis range [-1 1]).See Answer
Q18:X J1(x) 1.8 0.5815 -2 -637 2.0 0.5767 generate the third-order Newton's interpolating polynomial. Use the newtintr function to interpolate at x = 2.1. Validate your answer using Matlab's polyfit and polyval functions. -1 -96.5 2.2 0.5560 2. Ohm's law states that the voltage, resistance and current in a resistor are related by the linear equation V=RI. In practice, however, real resistors may not always obey this law. Suppose that you performed a precise experiment in the lab and obtained the following data for the resistor. -0.5 -20.5 2.4 0.5202 0.5 20.5 1 96.5 2 637 Use a fifth-order polynomial to fit the data and compute V for 1 = 0.1. Employ function newtintr. Next, use Matlab's polyfit to generate the interpolation polynomial. Plot the data points and the polynomial, on the same graph, for x-axis limits [-2.2 2.2]. 1 3. Consider the function f(x) = 1+25x2 Generate five equidistantly spaced points (xi, f (xi)) over the interval: [-1 1]. Fit this data with (a) a fourth-order polynomial, (b) a linear spline and (c) a cubic spline. Assume spline derivatives are zero at the end points. Compare your results graphically (use the x-axis range [-1 1]).See Answer
Q19: EGN 3433 NAME The concentration of a drug in a person's stomach can be described with the following linear ODE. Where ● dc dt = - KA, (c-cs) c is the concentration of the drug in the stomach k is the mass transfer transfer coefficient (4.1) A, is the ratio of the surface area of the stomach lining to its volume Sep 14, 2023 B. What is the forcing function? c, is the concentration of the drug in the blood stream, which can be a function of time. A. What are the independent and the dependent variables? dy C. Rewrite the ODE in the form: t + y = f(t) where y and x are the dependent and dx independent variables, respectively, Given that, k = : 0.9 m/s, A, = .01 m¯¹, what is the value T? Use correct units! 1 EGN 3433 Sep 14, 2023 D. If at time zero, c = 12 mg/ml, what is the amount of time needed for the concentration of the drug in the stomach to be 3 mg/mL? Use separation of variables. Here you can assume, that Cs = .01 mg/mL. In the problem, k 0.9 m/s, A. = .01 m¯¹. = E. Let's say that c¸ is not a constant, but slowly increases with time. For instance, mg cs(t) = 0.001- t +0.002. ml. s mg ml Can you use separation of variables to solve? Why or why not? Suggest an alternative technique that you can use to solve. 2See Answer
Q20:b. Determine A and r. Give details about the implementation of the Neumann boundary condition. c. Show that the eigenvalues λ; of A are real and that it holds A; <0. For the integration in x-direction of (4) we take the 0-method, given by Un+1 = Un + Ax[(1 - 0) (Aun +rn) + 0(Aun+1+Fn+1)], 0 = [0,1]. (5) In here Az is the step size in z-direction. d. Determine the order of the local truncation error of (5). We choose 0 = ³. e. Show that (5) is unconditionally stable for 0= . Is the method super-stable? f. What is the order of the global discretization error of (5) for 0 = ? The absorption of the gas is determined by a =y=1. g. Give a first-order and a second-order accurate (one-sided) finite-difference formula for the computa- tion of a. The corresponding numerical approximations of a are denoted by a₁ and 02. h. Choose Ay = 0.05, Az = 0.02 and = 1. Plot in a single graph the numerical solution (as a function of y) for x =nAz with n = 5, 10, 15, 20, 25. Make tables of a₁ and 0₂. Discuss the results. i. Choose Az = 0.02, v = 1 and define L = 20Ar. We investigate the accuracy in a at x = L. Choose Ay = 0.1, 0.05, 0.025. Make tables of a₁ and a₂ for x = L. Discuss the accuracy of a₁ and ₂2. For this, assume that the global discretization error can be written as CAy". Estimate C and p. Discuss the results. j. Add the software you wrote as appendix/appendices to your report.See Answer

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