3.12 9019:26XNumerical Method...3.6 Determine the positive root of the polynomial x³ +0.6x² +5.6-4.8.(a) Plot the polynomial and choose a point near the root for the first estimate of the solution. Using New-ton's method, determine the approximate solution in the first four iterations.(b) From the plot in part (a), choose two points near the root to start the solution process with the secantmethod. Determine the approximate solution in the first four iterations.3.7 The equation 1.2x³ + 2x²-20x-10 = 0 has a root between x = -4 and x = -5. Use these valuesfor the initial two points and calculate the next four estimates for the solution using the secant method.3.8 Find the root of the equation √x + x² = 7 using Newton's method. Start at x = 7 and carry out thefirst five iterations.3.9 The equation x³-x-et-2 = 0 has a root between x = 2 and x = 3.(a) Write four different iteration functions for solving the equation using the fixed-point iteration method.(b) Determine which g(x) from part (a) could be used according to the condition in Eq. (3.30).(c) Carry out the first five iterations using the g(x) determined in part (b), starting with x = 2.3.10 The equation f(x) = x²-5x1/3 + 1 = 0 has a root between x=2 and x = 2.5. To find the root byusing the fixed-point iteration method, the equation has to be written in the form x = g(x). Derive two pos-sible forms for g(x) - one by solving for x from the first term of the equation, and the next by solving forx from the second term of the equation.(a) Determine which form should be used according to the condition in Eq. (3.30).(b) Carry out the first five iterations using both forms of g(x) to confirm your determination in part (a).3.11 The equation f(x) = 2x³-4x² - 4x-20 = 0 has a root between x = 3 and x = 4. Find the root byusing the fixed-point iteration method. Determine the appropriate form of g(x) according to Eq. (3.30).Start the iterations with x = 2.5 and carry out the first five iterations.3.12 Determine the positive root of the equation cosx-0.8x² = 0 by using the fixed-point iterationmethod. Carry out the first five iterations.3.13 Solve the following system of nonlinear equations:-2x³+3y² +42 = 05x² + 3y³-69 = 0(a) Use Newton's method. Start at x = 1, y = 1, and carry out the first five iterations.(b) Use the fixed-point iteration method. Use the iteration functions yx = (³x² + 42) 1/³. Start at x = 1, y = 1, and carry out the first five iterations.3.14 Solve the following system of nonlinear equations:Chapter 3 Solving Nonlinear Equationsx²+2x+2y²-26= 040- (-5x3+69) 132x³y2 + 4y - 19 = 0(a) Use Newton's method. Start at x = 1, y = 1, and carry out the first five iterations.(b) Use the fixed-point iteration method. Start at x = 1, y = 1, and carry out the first five iterations.andProblems to be programmed in MATLABSolve the following problems using the MATLAB environment. Do not use MATLAB's built-in functions forsolving nonlinear equations.3.15 In the program of Example 3-1 the iterations are executed in the for-end loop. In the loop, the anon-ymous function F is used twice (once in the command FxNS = F (xNS) and once in the command ifF (a) *FxNS

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3.12 9019:26XNumerical Method...3.6 Determine the positive root of the polynomial x³ +0.6x² +5.6-4.8.(a) Plot the polynomial and choose a point near the root for the first estimate of the solution. Using New-ton's method, determine the approximate solution in the first four iterations.(b) From the plot in part (a), choose two points near the root to start the solution process with the secantmethod. Determine the approximate solution in the first four iterations.3.7 The equation 1.2x³ + 2x²-20x-10 = 0 has a root between x = -4 and x = -5. Use these valuesfor the initial two points and calculate the next four estimates for the solution using the secant method.3.8 Find the root of the equation √x + x² = 7 using Newton's method. Start at x = 7 and carry out thefirst five iterations.3.9 The equation x³-x-et-2 = 0 has a root between x = 2 and x = 3.(a) Write four different iteration functions for solving the equation using the fixed-point iteration method.(b) Determine which g(x) from part (a) could be used according to the condition in Eq. (3.30).(c) Carry out the first five iterations using the g(x) determined in part (b), starting with x = 2.3.10 The equation f(x) = x²-5x1/3 + 1 = 0 has a root between x=2 and x = 2.5. To find the root byusing the fixed-point iteration method, the equation has to be written in the form x = g(x). Derive two pos-sible forms for g(x) - one by solving for x from the first term of the equation, and the next by solving forx from the second term of the equation.(a) Determine which form should be used according to the condition in Eq. (3.30).(b) Carry out the first five iterations using both forms of g(x) to confirm your determination in part (a).3.11 The equation f(x) = 2x³-4x² - 4x-20 = 0 has a root between x = 3 and x = 4. Find the root byusing the fixed-point iteration method. Determine the appropriate form of g(x) according to Eq. (3.30).Start the iterations with x = 2.5 and carry out the first five iterations.3.12 Determine the positive root of the equation cosx-0.8x² = 0 by using the fixed-point iterationmethod. Carry out the first five iterations.3.13 Solve the following system of nonlinear equations:-2x³+3y² +42 = 05x² + 3y³-69 = 0(a) Use Newton's method. Start at x = 1, y = 1, and carry out the first five iterations.(b) Use the fixed-point iteration method. Use the iteration functions yx = (³x² + 42) 1/³. Start at x = 1, y = 1, and carry out the first five iterations.3.14 Solve the following system of nonlinear equations:Chapter 3 Solving Nonlinear Equationsx²+2x+2y²-26= 040- (-5x3+69) 132x³y2 + 4y - 19 = 0(a) Use Newton's method. Start at x = 1, y = 1, and carry out the first five iterations.(b) Use the fixed-point iteration method. Start at x = 1, y = 1, and carry out the first five iterations.andProblems to be programmed in MATLABSolve the following problems using the MATLAB environment. Do not use MATLAB's built-in functions forsolving nonlinear equations.3.15 In the program of Example 3-1 the iterations are executed in the for-end loop. In the loop, the anon-ymous function F is used twice (once in the command FxNS = F (xNS) and once in the command ifF (a) *FxNS < 0). Rewrite the program such that the anonymous function F is used inside the loop onlyonce. Execute the new program and show that the output is the same as in the example.3.16 Write a MATLAB user-defined function that solves for a root of a nonlinear equation f(x) = 0 usingthe bisection method. Name the function Xs - BisectionRoot(Fun, a, b). The output argument Xs isthe solution. The input argument Fun is a name for the function that calculates f(x) for a given x (it is adummy name for the function that is imported into BisectionRoot); a and b are two points thatbracket the root. The iterations should stop when the tolerance in f(x) (Eq. (3.5)) is smaller than 0.000001.The program should check if points a and b are on opposite sides of the solution. If not, the programshould stop and display an error message. Use BisectionRoot to solve Problem 3.2.3.17 Determining the square root of a number p, √p, is the same as finding a solution to the equationf(x)=x²-p=0. Write a MATLAB user-defined function that determines the square root of a positivenumber by solving the equation using Newton's method. Name the function Xs = SquareRoot(p). The out-put argument xs is the answer, and theroot is determined.The program should include the following features:

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3.12 Determine the positive root of the equation cosx-0.8x² = 0 by using the fixed-point iteration method. Carry out the first five iterations.