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Problem 32 Hard Difficulty

(a) Use Newton's method with $ x_1 = 1 $ to find the roof of the equation $ x^3 - x = 1 $ correct to six decimal places.
(b) Solve the equation in part (a) using $ x_1 == 0.6 $ as the initial approximation.
(c) Solve the equation in part (a) using $ x_1 = 0.57 $.
(You definitely need a programmable calculator for this part.)
(d) Graph $ f(x) = x^3 - x - 1 $ and its tangent lines at $ x_1 = 1 $, $ 0.6 $, and $ 0.57 $ to explain why Newton's method is so sensitive to the value of the initial approximation.

Answer

a. 1.324718
b. 1.324718
c. 1.324718
d. see graph

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Video Transcript

question 32 we're gonna start out with the vets equals X cubed My X minus one equals zero. Therefore, a crime of X equals three X squared minus one. So x of ink was one equals X minus Mike ex huge minus X minus one over three X squared minus one. So first, all they ask us to f me X one equals one. That's a two plug in one for my exit. One Well being 1.5 and these were all approximately 51.34 That's a four. Then when I plug in 1.34 gives me one point three 25200 And when I plug in that number, I get one point 3 to 4 718 which is the same as accepts. Six. So I stopped there. I'm gonna do X of one is approximately 26 or it's except two. Then is when I plug in 0.6. I get 17.9 except three is approximately 11. My nine poor exit four having why nine a fly by 20 Next five i we three by 909 six. Three point 6 to 4 996 Exit seven is a practically 2.5 055 89 Exit B is approximately one point 8 to 0 129 Next of nine is approximately one point for six 10 for four. Exit 10 is approximately home. 0.339 323 Ex of a mother Come on 11 1.3. All right, we'll sort of you. Except 11. 1.3 to 4913 Well, is approximately except 13. And it's approximately 1.324718 So I get to that are the same. So I stopped. And now the problem that goes on and on, my friend, the song that doesn't end. So I'm gonna start with X one equals approximately 10.57 And you think since it was closed 2.6, it would be similar. All right. It fools us negative. 50 were once it's by 455 naked of 36. We 11 34 293 four. Negative 24 My 08 two 094 65 Negative. 16 My 06 33 87 six Negative. 10 7 to 1 for a three except seven negative. Seven. Boy 16 5534 And again, I'm just putting these numbers like value in for my ex value. You could do this in excel or in a graphing calculator. Eight. Negative or 801 70 or Exit nine Negative. And they all seem to be getting smaller and smaller, though. Okay, minutes her ball. But at this point, I was all in who free, free or 25 10 negative two. White 19 36 seven or six of 11 negative. 1.496 867 Well, negative 0.99 75 46 13 Negative 0.49 6305 Exit 14. Maybe new. 2.89 And who's getting smaller? Fall in fall in there. Jumped negative. 2.89 or 16 You next 15 and negative. 1.96 79 62 14. Negative 2.89 or crazy Here. 16. Negative. 1234 1355 17. Negative 87 01 87 Thanks. Negative point to forward 99 for nine ex of 19 1.1 negative 1.19 2 to 19 My knee Negative 0.7 one night two. Exit. 21 0.355 213 Thanks. A 22 negative 1.75 33 to 2 two two Exit 23 Negative one point on 89 4 to 0 Negative 79 123 Exit 25 0.377 What 844 except 26 negative. One point Really 7872 x 27. Okay, you're on these in a cell. It's not cheap year Negative. 1.32 every 50 28 Exit 20 naked here might be bi 1919 except 29 Negative. 20095 30 Like the song that doesn't end no negative One way 119 creates it. Very one was back down. Negative. Wait 65 or two. No. One. 32 one boy, Why? For a 70 one to the next page. That's up 33. 1.360 Why 34 on 32 58 to 8. 1.3 4719 See that we're getting closer Exit 36 132 4718 and that's an equal X of 37. We're done. Beds were red rectangle. Oh, and be the graph. How could I forget? Be the graph The heat grab. Let's get you by hand. So here's our and that looks like here's one our candidate minds trying to go. Oh, dear, that's awful, Mom, Take it away. Take it away. Takes everything away, but my horrible purple line. So a tangent lines and then here at 0.5 seven and wait 60 as good as it's gonna get.