Quantitative Methods Answers

Solve the following prblem using fixed point iteration method (approx￾imate to three decimal place).

a. sin x =x + 1/x − 1

b. 3x + sin x = e^x

Two different moulds grow at different rates. The mass of the first mould (in grams) is well described by the function m1(t) = 20 log(t + 2) where the time t is measured in hours. The second mould grows according to m2(t) = t 2 . Question 3 continues on the next page.

(a) Write a MATLAB program using array operations to generate a table (with headings) of the amount of each mould each hour starting at time t = 0 up to a maximum time entered by the user. Run your program with the maximum time set to 10.

(b) Write a separate MATLAB program using the plot command to graph the amount of the two moulds on the same axes for 0 ≤ t ≤ 10. Make sure you label your axes.

(c) Use the graphical output from your MATLAB program in part (b) and the ginput command to estimate the time when the amounts of the moulds are equal.

f(x)=sin(√x)-x .Using simple fixed-point iteration determine the real root of the function below with initial guess x_i=0.5 and iterate until the estimated error 𝜀_a falls below the level of 𝜀_s=0.005%.

Determine the real root of 𝑓(𝑥)=𝑥⁴−8𝑥³−35𝑥²+450𝑥 −1001. Using false-position method to locate the root. Employ initial guesses of 𝑥_l=4.5 and 𝑥_u=6 and iterate until the estimated error 𝜀_a falls below the level of 𝜀_s=1.0%

Determine the real root of 𝑓(𝑥)=𝑥⁴−8𝑥³−35𝑥²+450𝑥 −1001. Using false-position method to locate the root. Employ initial guesses of 𝑥_l=4.5 and 𝑥_u=6 and iterate until the estimated error 𝜀_a falls below the level of 𝜀_s=1.0%

Determine the real root of 𝑓(𝑥)=𝑥⁴−8𝑥³−35𝑥²+450𝑥 −1001. Using false-position method to locate the root. Employ initial guesses of 𝑥_l=4.5 and 𝑥_u=6 and iterate until the estimated error 𝜀_a falls below the level of 𝜀_s=1.0%

1.Consider the equation xe^x = cos x

(a) Apply the intermediate value theorem to show that the function has a root in the interval

[0, 1].

(b) Find the real root using the secant method. Start with the two points, x1 = 0 and x2 = 1

and carry out the first four iterations.

(c) Find the real root using the Newton-Raphson method. Start with an initial approximation,

x0 = 0.5 correct to two decimal places.

2.Consider the initial value problem

dy = t(y + t) − 2, y(0) = 2. It is derivative of y respect to t

dt

(a) Use Eulers method with step sizes h = 0.3, h = 0.2 and h = 0.15, compute the approximations to y(0.6).

(b) Use the fourth order Runge-Kutta method Compute y(0.4) with h = 0.2.

Find a root of the equation cos x- e^x=0 correct to three decimal places in the 

interval (0,1) after four iterations by Secant method

 Use appropriate Lagrange interpolating polynomials of degrees one, two, and three to approximate each of the following: a. f (8.4) if f (8.1) = 16.94410, f (8.3) = 17.56492, f (8.6) = 18.50515, f (8.7) = 18.82091 b. f  −1 3  if f (−0.75) = −0.07181250, f (−0.5) = −0.02475000, f (−0.25) = 0.33493750, f (0) = 1.10100000 c. f (0.25) if f (0.1) = 0.62049958, f (0.2) = −0.28398668, f (0.3) = 0.00660095, f (0.4) = 0.24842440 d. f (0.9) if f (0.6) = −0.17694460, f (0.7) = 0.01375227, f (0.8) = 0.22363362, f (1.0) = 0.65809197