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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:**- The hypothetical computer, Shelly-32 has word lengths of 32 bits (32 binary digits).See Answer**Q3:**1. Let f(x)=x²-6 and po= 1. Use Newton's method to find p2.See Answer**Q4:**3. Let f(x)=x²-6. With Po=3 and p₁=2, find p3. a. Use the Secant method.See Answer**Q5:**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**Q6:**7. Repeat Exercise 5 using the Secant method. 21 21See Answer**Q7:**= 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**Q8:**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**Q9:**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**Q10:**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**Q11:**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**Q12:**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**Q13:**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**Q14:**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**Q15:**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**Q16:**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**Q17:**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**Q18:**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**Q19:**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**Q20:**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

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