Math Assignment -

Q1. Let f(x) = -x³-cos(x), and p₀ = 1. Use Newton's method to find p₂. Could p0=0 be used?

Q2. Perform two iterations by Newton's method and the secant method to each of the following:

a. e^x + 2^-x + 2 cos (x) - 6 = 0, for 1 ≤ x ≤ 2.

b. x³ + 3x² - 1 = 0, [-3, -2].

Which of the two methods has better approximation of the root, justify your answer.

Q3. The following model is given at the beginning of the chapter of the book of Burden & Faires regarding the birth rate of certain population. Find an approximate for λ, accurate to within 10^-4, for the population equation:

1,564,000 = 1,000,000e^λ + (435,000/λ) (e^λ - 1),

discussed in the introduction to this chapter. Use this value to predict the population at the end of the second year, assuming that the immigration rate during this year remains at 435,000 individuals per year. Where λ is the constant birth rate of the population. You may use the implented Newton's method and the secant method using MATLAB to carry out the required result.

Q4. Let f (x) = 3^3x+1 - 7.5^2x.

Plot f(x) to find initial approximations to roots of f. Use Newton's method to find one root of f to within 10^-16.

Q5. The power of a signal being measured by a tower is decreasing over time according to this model:

P(t) = e^-tcost(t)

Starting at t_o =0s, find the time that the signal will have lost all power. Apply one of the studied methods to carry out your result.

Q6. Determine the real roots of f(x) = -0.6x² + 2.4x + 5.5.

(a) Graphically.

(b) Using three iterations of the bisection method to determine the highest root. Employ initial guesses of x_l = 5 and x_l = 10. Compute the estimated error ∈_a and the true error ∈_t after each iteration.

Q7. The velocity v of a falling parachutist is given by

v = (gm/C)(1-e^-(c/m)t)

where g = 9.8 m/s². For a parachutist with a drag coefficient c = 15 Kg/s, compute the mass m so that the velocity is v = 35 m/s at t = 9 s. Use the false-position method to determine m to a level of ∈_s = 0.1%.

Q8. Write the C++ code to calculate the following:

a) Find the roots of the equation x³ - 4x - 9 = 0 using Bisection method within 10^-6.

b) Find the roots of the equation x³ - 4x - 9 = 0 usin Newton's method within 10^-6.

c) Calculate the relative error in each iteration for both methods.

d) Your code compare between the number of iterations in each method and tell us which the fastest.