{Use of Tech} Write the formula for Newton’s method and use th...

Question

{Use of Tech} Write the formula for Newton’s method and use the given initial approximation to compute the approximations x₁ and x₂.

f(x) = x² - 6; x₀ = 3

Accepted Answer

Welcome back, everyone. In this problem, we want to find the first approximations X1 and X2, to the root of the function F of X equals X minus 5 using Newton's method starting with X0 equals 2. A says X1 is 2.2500, while X2 is 2.2361. B says there are 2.3421 and 2.2274 respectively. C. 2.1109 and 2.5580, and D 2.9600 and 2.2331. Now, how can we find our approximations by using Newton's method? What do we know about Newton's method? Well, recall that the iterative formula for Newton's method tells us that XN + 1 is equal to XN minus F of XN divided by the first derivative of F of XN. So, what we'll need to find here, if we can, we already have FFX. If we can find the first derivative, then we can apply Newton's method depending on the approximation we, we need, OK? And we already know that we're starting with X0 equals 2. So that will be helpful to help us find X1. Now, we know that the first derivative or the derivative of X2 minus 5 is equal to 2 X. So let's start with X1, OK? Now applying Newton's method to X1. Is going to be equal to X minus F of X0. Divided by the derivative of F of X not. Now, here, OK, we can already apply our value for FFX and the derivative of FFX, OK. So we know here that X0 is 2, so we're gonna be finding F of 2 for F of X0. And F of 2 equals 22 minus 5, which is 4 minus 5-1. Likewise, the derivative of F of 2, remember that its derivative is 2 X, so that's 2 multiplied by 2, which equals 4. So what this tells us then is that by Newton's method, X1 is gonna be equal to X0 2 minus F of X0 -1, OK, divided by If the derivative of F of X not 4. So this is gonna be 2 + 14, 22 minus 1/4 works out to 2 + 1/4, which equals 2.4 or 2.25. So X1 is 2.25. We figured out the first value. Let's move on to the 2nd. Now, for X2, OK, again, by Newton's method, X2 will be equal to X1. Minus F of X1. And up properly here. Divided by the derivative of F of X1, and we know what X1 is already. Earlier we found that X1 equals 2.25. So now That means F of X1. It's going to be F of 2.25. So that's gonna be equal to 2.25 squared minus 5. 2.25 squared is 5.0625, and when we subtract 5 from that, we get 0.0625. On the other hand, F of X1 is gonna be F of 2.25. 2 multiplied by 2.25. Which equals 4.5. So now that we have these values for F of X1 and F of X1, then we can substitute them into the formula for X2. So X2 is going to be 2.25 minus 0.0625 divided by 4.5, and that would be equal to 2.2361. So, X2 is 2.2361, X1 is 2.25, uh 00, OK, to four decimal places or to 5 significant figures, which means then that A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

{Use of Tech} Write the formula for Newton’s method and use the given initial approximation to compute the approximations x₁ and x₂.

f(x) = x² - 6; x₀ = 3

Key Concepts

Newton's Method

Derivative

Initial Approximation

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