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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
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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
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