Problem

(a) Show that the function defined by $ f(x)…

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Problem 83 Hard Difficulty

Prove Taylor's Inequality for $ n = 2, $ that is, prove that if $ \mid f''' (x) \mid \le M \text { for }
\mid x - a \mid \le d, $ then

$ \mid R_2 (x) \mid \le \frac {M}{6} \mid x - a \mid^3 $ for $ \mid x - a \mid \le d $

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Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

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Video Transcript

to prove the problem is proof Taylor's inequality for in musical tube. That is true that if there are derivative ABS, value is less than I am for absolutely. Of Oxman's A is less than the then at two x soiree of tracks is less than m over six times as value of Xmas eight to the power of three. Mm. As word of axeman is a is last D. So from here behalf ex prime from from X This is the last time, um, for X is greater than a and the last time they plus the than behalf integral from a two x What exit is spirit between A and a plastic us from from from two. He this last time integral from eight X Um, mhm. So on behalf years, this is of Hank's from from minus half A from prom is less than X months a and, um then we have integral of have x cramp from from h x r b u T t his last integral from a two x have from from okay class. Um, he minus a it's so we have at the tax prom minus have a problem is less than from? From a hams X minus. Mm. Class, Um, over two times X minus. Hey, square on behalf integral of I ftp from from eight x did He is less than 18 year old from a two ex prime a as I have from from eight times three months. Eight plus I'm over two times. He might stay square tiki than behalf have x minus f a. His life must, uh, have a prime times x minus a us f a from from 8/2 times Xmas a square us mm over six times X minus A to the power of three. Since our two X is equal to our fax minus f a minus, have a prom xmas a plus minus five apron, prime over two times X space square sofa. Half our two acts his last time I'm over six x minus A skill Uh huh. X is greater than a and less than plastic. A similar argument using us from from from is greater than negative arm. When X is between negative 50 plus eight and a we can prove r two X is greater than negative of three factorial times. Thanks. One stay. Q. So we have from these two results and the value of our two this last time. M over six times, absolutely of X minus Space cube.

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Problem

(a) Show that the function defined by $ f(x)…

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Calculus: Early Transcendentals

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Watch More Solved Questions in Chapter 11

Video Transcript

James Stewart

Calculus: Early Transcendentals

Grace He

Catherine Ross

Kayleah Tsai

Michael Jacobsen

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Calculus: Early Transcendentals

Grace He

Catherine Ross

Kayleah Tsai

Michael Jacobsen

James StewartCalculus: Early Transcendentals

Grace He

Catherine Ross

Kayleah Tsai

Michael Jacobsen

James Stewart

Calculus: Early Transcendentals