QuestionFind $\delta$ such that if $|x-4|<\delta$ , then $|\sqrt{x}-2|<0.4$ for the function $f(x)=\sqrt{x}$ .

Studdy Solution

STEP 1

Assumptions1. The function is $f(x)=\sqrt{x}$ . We are given that $|\sqrt{x}-|<0.4$
3. We are asked to find $\delta$ such that $|x-4|<\delta$
4. The function $f(x)=\sqrt{x}$ is continuous and differentiable for $x>0$

STEP 2

First, let's rewrite the given inequality $|\sqrt{x}-2|<0.4$ in a more convenient form. We can do this by solving for $x$ .
$\sqrt{x}-2<0.4$ $\sqrt{x}<2.4$ $x<(2.4)^2$ $x<5.76$ And $\sqrt{x}-2>-0.4$ $\sqrt{x}>1.6$ $x>(1.6)^2$ $x>2.56$

STEP 3

Now we have two inequalities that give us the range of $x$ for which $|\sqrt{x}-2|<0.$ . We need to find a $\delta$ such that if $|x-|<\delta$ then $x$ is in this range.

STEP 4

We know that $|x-4|<\delta$ means that $x$ is within $\delta$ units of4. So, we need to find a $\delta$ such that $4-\delta>x>4+\delta$ is within the range $2.56<x<.76$ .

STEP 5

We can do this by finding the minimum distance from4 to the endpoints of the range $2.56$ and $5.76$ .
$\delta1 =4 -2.56 =1.44$ $\delta2 =5.76 -4 =1.76$

STEP 6

We choose the smaller of the two $\delta$ values as our $\delta$ because we want the smallest interval around4 that is entirely within the range $2.56<x<5.76$ .
$\delta = min(\delta1, \delta2)$

STEP 7

Calculate the value of $\delta$ .
$\delta = min(1.44,1.76) =1.44$ So, if $|x-4|<1.44$ then $|\sqrt{x}-2|<0.4$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms