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 $g(x)=m(x-a)+c(m$ and $c$ 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 conditionsi. $E(a)=0$ii. $\lim _{x \rightarrow a} \frac{E(x)}{x-a}=0$then $g(x)=f(a)+f^{\prime}(a)(x-a) .$ Thus, the linearization gives the only linear approximation whose error is both zero at $x=a$ and negligible in comparison with $(x-a)$ .

$f(a)+f^{\prime}(a)(x-a)$

Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Applications of Derivatives

Section 5

Linearization and Newton's Method

Derivatives

Differentiation

Differential Equations

Missouri State University

Campbell University

Oregon State University

Harvey Mudd College

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

04:55

The linearization is the b…

03:57

The linearization of $\ln …

01:44

Find the linearization $L(…

01:31

03:11

Use a CAS to estimate the …

06:14

Consider the following com…

06:03

02:36

07:45

in these problem, we have a differential function f of X at X equals e and she of excellent. Your function equals two m times X minus a plus C where M and C are constants. We also defined the era function E of X equals to gif of X minus your Vicks. So if this era function were small enough near excitable, eh? When I think of using these function is leaner French in G A. Selena approximation off f instead of the leaner ization off F at X equals C. We want to show that if we impose thes two conditions on G, that is the era function at a Is he going to zero a limit when X approaches? E off the euro fashion divided by a X minus a easy with zero then the function. Jeez, exactly the same US generalization of f X equals a. So for that, we're going to see first that the condition one we've both g that he's hearer at a secret zero. It would see definition of your function here it is secret to f off, eh minus g of a and this implies immediately the F A A is equal to G off, eh? But we know the G off a. We see the definition of she here is equal to 10 times a minus a sea. There is a good to see. So we conclude that C is equal to if of a you're now going to study. He's second condition. So the second condition condition too. That is limits. When X approaches a off the era function to by the by x minus e sequence zero You see that limits when eggs a brush is a off the era fashion. He's e of a and that by condition one Is he going to Syria? So we have that. The limit of the numerator of the suspicion here is Ciro and the limit. When eggs goes to a off the denominator of this expression that his X minus a is Syria because he's a minus a, which is serious. That means that the limit we have here he's one of the forum Sierra divided by serious so we can apply love Tito's rule here and say that the limit when x goes to a off e of ex about it by X minus a secret to the limit when X goes to pay off the river tive of the year, a function divided by the derivative off X minus A and that is equal to the limit when exposed to gay too derivative off their function is, if derivative minus derivative of Jean because if of ecstasy into the function ever vex minus to your base in that divided by one so is equal to the limit when X approaches a off the video off. If. Why no sir. Derivative of and that is he going to They even have a off death at a class derivative factory minus minus. They're riveted g at a but this limit. When eggs purchase a of a year, extroverted checks minus day Is sirrah the condition two. So we can say that so the river native of Pay Highness, derivative of G. Gay is zero. So immediately we get a derivative of if a day is equal to purity of G A. But remember, remember, Dad G of X is n times takes my nest. People see, but we have then that derivative of G X is equal to him. But any eggs so it implies immediately that embassy too stare even achieve off death on A. That is because the relative of she at A is going to be good to him. And the relative of Year A is derivative off if at a so we have C equals. So it here if a see his secretive off Hey, in m is derivative of f a day. He implies that she s a secret too M times that is the relief of F a times X minus day plus c That is, if you pay and that is just we see here if a favor blast derivative of F A bluff and times X minus a just the leader is Asian off the function Death at X equals a So, uh, the militarization give us the only linear approximation whose hero is both zero at X equals a and negligible in conversation with X minus a

View More Answers From This Book

Find Another Textbook

In mathematics, integration is one of the two main operations in calculus, w…

In grammar, determiners are a class of words that are used in front of nouns…

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…

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

Consider the following common approximations when $x$ is near zero.a. Es…

01:51

In Exercises $45-48,$ find the area of the shaded region.

03:17

Cardiac Output The following table gives dye concentrations for a dye-concen…

01:48

In Exercises $1-20,$ find $d y / d x$.$$y=\int_{5 x^{2}}^{25} \frac{t^{2…

03:14

In Exercises $15-18$ , complete the table for an investment if interest is c…

In Exercises $1-10,$ find the indefinite integral.$$\int 3 x^{2} e^{2 x}…

02:38

In Exercises $13-22,$ use the graph of the integrand and areas to evaluate t…

04:47

Coasting to a Stop Assume that the resistance encountered by a moving object…

03:54

In Exercises $1-4,$ find the values of $A$ and $B$ that complete the partial…

00:56

In Exercises $1-20,$ find $d y / d x$.$$y=\int_{x}^{0} \ln \left(1+t^{2}…

02:01

In Exercises $45-48$ , use Euler's Method with increments of $\Delta x=…

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.