Final answer:
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:
- Start with initial guesses x0 = 25 and x1 = 50.
- Calculate the function values at x0 and x1, i.e., f(x0) and f(x1).
- Compute the secant line that passes through the points (x0, f(x0)) and (x1, f(x1)). This line approximates the function near the root.
- Find the x-intercept of the secant line. This value will be the new approximation for the root. Let's call it x2.
- 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.