Final answer:
The Newton-Raphson method is an iterative numerical technique used to approximate the roots of a real-valued function. By applying the update rule to the function 10^x + x - 4, one iterates to find a real root of the equation.
Step-by-step explanation:
To find by Newton-Raphson method, a real root of the equation 10x+x-4=0, we first need to understand the iterative formula of the Newton-Raphson method. This method is used to find successively better approximations to the roots (or zeroes) of a real-valued function.
The Newton-Raphson update rule is given by:
xn+1 = xn - f(xn) / f'(xn)
First, we need to identify the function we're interesting in, which is:
f(x) = 10x + x - 4
Next, we compute its derivative, f'(x):
f'(x) = 10x * ln(10) + 1
The next step is to choose an initial guess x0 for the root. The choice of x0 can sometimes be guided by graphing the function or by physical intuition. After the initial guess, apply the update rule repeatedly to get x1, x2, x3, etc., each an approximation to the real root, until the value of x stabilizes within a desired range of accuracy.
This iterative process will converge to a real root of the equation, provided the initial guess is sufficiently close to the actual root and that f'(x) does not equal zero.