Problem

To approximate the reciprocal of a number $a$ wit…

02:53

Problem 54 Easy Difficulty

Let $a>0$ be given and suppose we want to approximate $\sqrt{a}$ using Newton's method.
a. Explain why the square root problem is equivalent to finding the positive root of $f(x)=x^{2}-a$
b. Show that Newton's method applied to this function takes the form (sometimes called the Babylonian method)
$$
x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{a}{x_{n}}\right), \text { for } n=0,1,2, \ldots
$$
c. How would you choose initial approximations to approximate $\sqrt{13}$ and $\sqrt{73} ?$
d. Approximate $\sqrt{13}$ and $\sqrt{73}$ with at least ten significant digits.

Answer

Related Courses

Calculus 1 / AB

Calculus Early Transcendentals

Chapter 4

Applications of the Derivative

Section 8

Newton's Method

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Video Transcript

54th 40 f of x equals X squared minus a If B squared is the root off X squared minus eight and B squared minus equals zero. Okay, it means that b squared equally then full square root toe me either So either b equals square root of a or Sorry, not eso either be equals the square root who came and by a mind square root of a If the is bigger than than zero, you know B equals square root of e okay, on toe port number be from Newton's method X n plus one equals xn minus X and where minus a over two X m which is equal to two x and square miles X and square miles A it over two x in Finally, we can see that this is equal to X m squared plus a over two X m uh, kicks in. Thus a over X and okay on to problem number c. Okay, green three squared equal mine, which is less than 15. Also, four squared equals 16 which is bigger than than 15. We would choose somewhere between them and closer to four. Around 3.644 73 as it's square they could 64 which is less than 73 nine square. They called 81 which is bigger than 73. Who would choose somewhere between nine, which is around 8.5. Okay, now on to problem number D. They've given function. Half of ax equals stacks squared minus 15. Newton's formula is X n plus one, which is equal to X n minus f of x n over death of accent. Okay, its course. Our function. As we said, X squared minus 15 Have national X equals two x and X Note equals four. You can now, Cal Keurig, you think the table on the iterations? All of what does that do for extreme? And it's and Rex, That's Measurex. Okay, you know wolf 38 Toe. 3.6 to 5 a few defenders thickness three 3.6 unifies 0.37 point 21 for separation. 3.6055 you point 0000 which is almost 207.2111 and the final iteration. 3.60555 1275 This is negative. 1.77 Multiply 10 Power native 15. 7.2111 Okay, our second function f X equals X squared 9 17 f dash of X equals wax and explode equals truth way can follow the same table format to opt in. Find the answers for the x zero who have negative 18 4.125 0.1568 point 25 Her desperation food on 231 your 0.0 003 8.246 and our final attrition. 4.1231056 1.918 times 10 Power 90 13 88.24 Okay. Oh.

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