Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Newton's method is a method where we're going to guess the first approximation to a solution, and then we're going to think about if we get the linearization at that location. So if we guessed X not here, and we found it's linearization, that new linearization would intersect or X axis somewhere. If we then use that new X intercept and found the linearization again, now we're going to intersect the X axis even closer, IE our linearizations are getting closer and closer and closer to the root that we're actually seeking. So if we did X2 and then found its linearization, there would be our X3. Newton's method is used by many calculators and or computers to come up with approximations quickly. So if we look at and we use the first approximation to get the second, the 2nd to get the 3rd, and so on. This is called a regressive equation. So X to the north plus one equal X to the north minus F of X to the north divided by F prime X to the north if the first derivative doesn't equal 0. So if we want to use Newton's method to estimate the one real solution of X ^3 plus three X + 1 = 0, we're going to start with X sub not equaling zero. And then we're going to go out and find X2. So our first thing, we're going to have Y equal X ^3 + 3 X plus one. We're going to have Y prime equaling three X ^2 + 3. So using that recursive equation, X to the m + 1 is going to be X to the north minus the original equation at the location of X sub N divided by the derivative at the location of X sub N. So if we thought about X sub not and we wanted to find X sub one, we'd put zero in every time we saw our N. But we were given in our directions that our X sub 0 is going to be 0. So we get 0 - 0 ^3 + 3 * 0 + 1 / 3 * 0 ^2 + 3. So our X sub one is just going to be negative 1/3 our X sub two. Now we're going to put in one for our N everywhere we see. So we get 1 + 1, which would give me the X sub 2. So X sub 1 -, X sub 1 ^3 + 3, X sub 1 + 1 over 3X sub 1 ^2 + 3. So we'd have negative 1/3 minus negative 1/3 ^3 + 3 * -1 third plus one all over three times negative 1/3 ^2 + 3. Simplifying that up, we're going to get negative 1/3 minus negative. 127th -1 + 1 / -1 third squared is going to be 1. Ninth 1/9 * 3 is going to be 1/3 + 3. Continuing to simplify this up, negative 1/3 -, -, a negative. Let's just make that a positive 127th. 1/3 + 3 is going to give us 10 thirds. So complex fraction, we're going to invert and multiply. So we're going to get 190th, getting a common denominator of 90. So we're going to get -30 ninetieth plus 190th or -29 ninetieths and that's going to be approximately -.3222. So the directions specified find X sub two, and that'd be negative .3222 01 more 2 if we wanted to get to 5 decimal places. Our next example, the same kind of thing, we're going to start with Y equaling X to the 4th -2. So Y prime is going to be 4X cubed. So our XN sub n + 1 is going to equal X sub n -, X sub N to the 4th -2 / 4 X sub n ^3. Those X sub NS are just saying that we're starting at some location. We're doing a recursive equation. So in this one it says we're going to let X sub not equaling -1. So if we let N BE00 plus 1X sub 0 -, X sub 0 to the 4th -2 / 4 X sub zero cubed, our X sub zero was -1 minus -1 to the 4th power -2 / 4 * -1 ^3 so -1 - -1 to the 4th is 1 - 2 / -4 so -1 - -1 / -4 which be would be -1 - 1/4 or -5 fourths. So then if I wanted X sub two, I would think of that as 1 + 1. So my N would now be 1. So now we're going to do the same thing. We're going to have our X sub 1 minus our X sub 1 to the 4th -2 / 4 X sub one cubed. So our X sub one was -5 fourths minus -5 fourths to the 4th -2 / 4 * -5 fourths and we would simplify that up and get something -5 fourths minus. Let's see if we thought about doing -25 to the 4th power. Oops -5 to the 4th power, we'd get 625 / 4 to the 4th power, which would be 256 -, 2 over. That was a cube back here. Where did my cube go? We'd have -5 ^3 so -125 / 4 ^3, which would be 64. This is minus that whole term -5 fourths. So we get -5 fourths -625 / 256 -, 2 would give us 113 / 256 and then 125 / 64 * 4 is going to give us -125 / 16. So -5 fourths, the negative and the negative is going to make it a positive. If I take 113 / 256 and I divide it by 125 sixteenths, we should get 113 / 2000. And then if we combine that with the -5 fourths, we should get -2 three. Oops, 2387 over 2000. If we plug that into our calculator, that's approximately -1.1935. So we've now found X sub 2. Thank you and have a wonderful day.