Problem

Use a CAS to estimate the magnitude of the error …

05:02

Problem 58 Hard Difficulty

Use a CAS to estimate the magnitude of the error in using the linearization in place of the function over a specified interval I. Perform the following steps:
a. Plot the function $f$ over $I$ .
b. Find the linearization $L$ of the function at the point $a$ .
c. Plot $f$ and $L$ together on a single graph.
d. Plot the absolute error $|f(x)-L(x)|$ over $I$ and find its maximum value.
e. From your graph in part (d), estimate as large a $\delta>0$ as you can, satisfying
$$|x-a|<\delta \quad \Rightarrow \quad|f(x)-L(x)|<\varepsilon$$
for $\varepsilon=0.5,0.1,$ and $0.01 .$ Then check graphically to see if your $\delta$-estimate holds true.

$f(x)=\frac{x-1}{4 x^{2}+1}, \quad\left[-\frac{3}{4}, 1\right], \quad a=\frac{1}{2}$

Answer

$$
L(x)=x-\frac{3}{4}
$$

Related Courses

Calculus 1 / AB

Thomas Calculus

Chapter 3

Derivatives

Section 9

Linearization and Differentials

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Video Transcript

Okay, So dysfunction f of X is X minus one over, uh, for X squared plus one. And they plotted over the interval. Three forced one. You never sending aid to be one half. Okay, so I have this plotted here. Yeah, Um, Right. Recorded. Okay. So have applauded Stine and bases to find Elin. You're ization. Okay, so we need to find the jury. So the derivative is going to be for X squared. Plus one times one minus X minus One time's a Becks all over for ex squared spine squared. And so if we evaluate this on one half, get us to quit computation that this is morning. We also see that f of one half its negative one for it. So received the linear ization cans. No x one times X minus a half minus the fourth, which is x minus three fourths. There's one relation. So then, see says exploit absolute value of f of X minus l love X. Okay, So I have plotted here. And if you look, we're trying to find the max, it's pretty clear that the max occurs. Yeah, occurs at X equals negative three fourths in the max here is about Syria. Point nine six. Okay. And then e. We want to look at these Absalon values by Absalon. It's your point. Learn that slimness said your point zero one and we want to find a Delta's that if X is within delta, eh? That our air is actually must not respond. So I just handsome pestilent nations is preserved point five looks like Delta. You claim one will work. Where's your point? One delta equaling zero twenty five will work. And if your Absalon equals zero point zero one looks like Toto's your point one will work, okay?

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