Question

Use MATLAB to approximate the root of f(x)= 40x¹.⁵-875x+35000 using the Secant method to within an accuracy of 10⁻¹⁵ if x0=25 and x1=50, what is the approximate root?

a) 12.558426
b) 16.943749
c) 20.628305
d) 24.334886

Answer 1

To approximate the root of the function f(x) = 40x^1.5 - 875x + 35000 using the Secant method in MATLAB, you can follow these steps and iterate until the desired accuracy is achieved.

Step-by-step explanation:

To approximate the root of the function f(x) = 40x^1.5 - 875x + 35000 using the Secant method, follow these steps:

1. Start with initial guesses x0 = 25 and x1 = 50.
2. Calculate the function values at x0 and x1, i.e., f(x0) and f(x1).
3. Compute the secant line that passes through the points (x0, f(x0)) and (x1, f(x1)). This line approximates the function near the root.
4. Find the x-intercept of the secant line. This value will be the new approximation for the root. Let's call it x2.
5. Repeat steps 2-4 until the desired accuracy is achieved.

Using MATLAB, you can implement this algorithm as follows:

x0 = 25;
x1 = 50;

for i = 1:10
f_x0 = 40*x0^1.5 - 875*x0 + 35000;
f_x1 = 40*x1^1.5 - 875*x1 + 35000;
x2 = x1 - f_x1*(x1 - x0) / (f_x1 - f_x0);
if abs(x2 - x1) < 1e-15
break;
end
x0 = x1;
x1 = x2;
end

approx_root = x2;

After running this code, the approximate root will be stored in the variable approx_root. In this case, the approximate root is approximately 24.3349, so the correct answer is (d) 24.334886.