Final answer:
To find the positive roots of the equation 10esinx = x^2 - 5x + 4, we can use Newton’s method or the bisection method. Newton's method involves iteratively updating a guess using the equation x1 = x0 - f(x0)/f'(x0), while the bisection method involves dividing an interval in half until finding a subinterval where f(a) and f(b) have opposite signs.
Step-by-step explanation:
Newton’s Method:
To find the positive roots of the equation 10esinx = x^2 - 5x + 4, we can use Newton’s method. First, let's rearrange the equation to have 0 on one side: 10esinx - x^2 + 5x - 4 = 0. We can then use Newton’s method to approximate the roots by iteratively updating our guess. Starting with an initial guess, we can use the formula: x1 = x0 - f(x0)/f'(x0), where f(x) is the equation and f'(x) is its derivative. Repeat this process until the desired accuracy is achieved.
Bisection Method:
To find the positive roots using the bisection method, we need to find an interval [a, b] such that f(a) and f(b) have opposite signs. Since the equation 10esinx - x^2 + 5x - 4 does not have a simple closed form solution, we can use a numerical approach. We can start by choosing an initial interval [a, b] such that f(a) and f(b) have opposite signs. Then, repeatedly divide the interval in half and check which subinterval satisfies the condition f(a) and f(b) have opposite signs. Continue this process until the desired accuracy is achieved.