Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

The linearization is the best linear approximation Suppose that $y=f(x)$ is differentiable at $x=a$ and that $g(x)=g(x)=$ $m(x-a)+c$ is a linear function in which $m$ and $c$ are constants. If the error $E(x)=f(x)-g(x)$ were small enough near $x=a$ we might think of using $g$ as a linear approximation of $f$ instead of the linearization $L(x)=f(a)+f^{\prime}(a)(x-a) .$ Show that if we impose on $g$ the conditions1. $E(a)=0$2. $$\lim _{x \rightarrow a} \frac{E(x)}{x-a}=0$$then $g(x)=f(a)+f^{\prime}(a)(x-a) .$ Thus, the linearization $L(x)$ gives the only linear approximation whose error is both zero at $x=a$ and negligible in comparison with $x-a$.

Show that $c=f(a)$ and $m=f^{\prime}(a)$

Calculus 1 / AB

Chapter 3

Derivatives

Section 9

Linearization and Differentials

Missouri State University

Campbell University

Harvey Mudd College

Baylor University

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

04:55

The linearization is the b…

08:16

The Linearization is the B…

03:57

The linearization of $\ln …

01:44

Find the linearization $L(…

01:31

07:16

Linear Approximation Let $…

02:30

Linear approximation and t…

03:11

Use a CAS to estimate the …

06:03

02:36

Okay, so we have a function on like himself attacks and we know that have prime of exists. In other words, F is differential today, and we have that she of x is a linear approximation. Two. Yes. Okay, so it's just in times X minus a plus thing. Yes, they c c and em and see your constant. So it's just it's just some line that's supposed to be approximating f OK. And so we're going to define the error, which is Ethel X minus G of X. And now, But there is small near a we can think of actually about using the linear approximations of F instead of the one you're ization. Okay, so what we actually want to show is that if GI is a good enough approximation, it essentially is the legalization of that at that point. So we're going to impose the condition that, um well, first of all, the air at hey is zero. And then secondly, the limit, his ex goes to a, uh v of X over X minus A Is your, uh okay, So what we want to show is that Jia Vex is actually equal to roll in here approximation of slips that's actually do it. All right, so let's look it. What is the same he of a zero We ve is just today minus jia. But this tells us that FAA equals g ave. But if we look at G g ave is m times a minus eight zero So see, this is actually telling us that seem is equal to as a bell. It's coming from the first condition in the second condition is telling us Is that okay? The limit sexy. Purchase A of Candia Becks is death of X minus J of X over X minus A. Well, this is the limit is experts say f of X, uh, times your plot start minus jean, which is in times X minus a of course, Death of a right plus Eve. What we just found that she was f obey. Next my essay. Okay, And then you just take this limit. So see, that's the limit. Experience is a of and write this as FX mama's f Obey. It's over X minus a and then minus the limit. His exit Purchase A in times X minus a over X minus a Remember, this is all supposed to be zero. Well, this guy is the derivative of that today. And then this guy that cancels this is just em. That means that in is that today? But now look at what that means about jean ends of prime of a so f prime of a times X minus A and C is up today. So imposing these two conditions allowed us to conclude that G really is the linear ization of s at a single day.

View More Answers From This Book

Find Another Textbook

In mathematics, precalculus is the study of functions (as opposed to calculu…

In mathematics, a function (or map) f from a set X to a set Y is a rule whic…

The linearization is the best linear approximation Suppose that $y=f(x)$ is …

The Linearization is the Best Linear Approximation Suppose that $y=f(x)$ is …

The linearization of $\ln (1+x)$ at $x=0$ Instead of approximating ln $x$ ne…

Find the linearization $L(x)$ of $f(x)$ at $x=a.$$f(x)=x+\frac{1}{x}, \q…

Find the linearization $L(x)$ of $f(x)$ at $x=a$$$f(x)=x+\frac{1}{x} \qu…

Linear Approximation Let $f$ be a function with $f(0)=1$ and $f^{\prime}(x)=…

Linear approximation and the second derivative Draw the graph of a function …

Use a CAS to estimate the magnitude of the error in using the linearization …

02:23

a. Find the open intervals on which the function is increasing and decreasin…

02:12

Estimate the allowable percentage error in measuring the diameter $D$ of a s…

02:18

Each figure shows the graph of a function over a closed interval D. At what …

05:31

05:35

Verify that the given point is on the curve and find the lines that are (a) …

11:32

$\begin{array}{l}{\text { a. Horizontal tangent lines }} \\ {\text { tangent…

01:04

In Exercises $1-8,$ given $y=f(u)$ and $u=g(x),$ find $d y / d x=$ $d y / d …

01:46

Each function $f(x)$ changes value when $x$ changes from $x_{0}$ to $x_{0}+d…

09:54

Assume that functions $f$ and $g$ are differentiable with $f(2)=3$$f^{\p…

01:53

Use the definitions of right-hand and left-hand limits to prove the limit st…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.