Newton's method is a very easy algorithm to apply to a function that produces more and more accurate approximations of an actual root. All you need is the equation for the function, an equation for its derivative, and a first approximation for the root.
In the case of the above questions, we will need to know the actual root to at least 4 (or 5) decimal places to decide when we have obtained an accurate enough approximation using Newton's Method. We can use a calculator for that, as well as for applying Newton's Method as follows:
1) f(x) = x4 - x - 10. Enter this function into Y1 in a TI calculator. Find f'(x) by taking the derivative. Enter into Y2.
Find the zero of f(x) using CALC ZERO. x ~ 1.8555. We are told the root is between 1 and 2, so let's choose x0 = 2 as our first approximation.
Newton's Method is given by the formula xn+1 = xn - (f(xn))/(f'(xn)). (What this does is follow the tangent line to the curve until the tangent line hits the x-axis. We have a new x-value, and we iterate the process).
I will show the calculations for getting x1 here, then explain calculator steps to continue the process:
x1 = 2 - 4/31 = 1.8709...
In your calculator store this number to A by typing STO→ ALPHA A. Then type A - Y1(A)/Y2(A). ENTER.
x2 = 1.8557... STO> ALPHA A again. Hit ENTRY twice to recall A - Y1(A)/Y2(A). ENTER again.
x3 = 1.8555... which is accurate to the 4 decimal places required by the question.
Btw, there are several online Newton's Method calculators that will do the calculation for you, and even show you the steps involved. Here's just one of those: https://keisan.casio.com/exec/system/1244946907
You would do well to memorize the formula for Newton's Method and know how to apply it. It would also be useful for you to look at a picture and see why it is that this formula will give you the tangent line zero. You might also consider why the tangent line zero will always give you a better approximation of the actual root.
Dinesh R.
thank you bradford sir02/17/21