QuestionFind $\delta$ such that if $|x-1|<\delta$ , then $|f(x)-1|<0.2$ , given $f(1)=1$ , $f(1.1)=0.8$ , $f(1.2)=1.2$ .

Studdy Solution

STEP 1

Assumptions1. The function $f$ has a known point $(1,1)$ . . The function $f$ intersects the line $y=0.8$ at $x=1.1$ .
3. The function $f$ intersects the line $y=1.$ at $x=1.$ .
4. We need to find a number $\delta$ such that if $|x-1|<\delta$ then $|f(x)-1|<0.$ .

STEP 2

We are given that $f(x)$ intersects the lines $y=0.8$ and $y=1.2$ at $x=1.1$ and $x=1.2$ respectively. This means that $f(1.1) =0.8$ and $f(1.2) =1.2$ .

STEP 3

We are looking for a $\delta$ such that if $|x-1|<\delta$ then $|f(x)-1|<0.2$ . This means that the values of $f(x)$ should lie between $0.8$ and $1.2$ when $x$ is in the interval $(1-\delta,1+\delta)$ .

STEP 4

Since $f(1.1) =0.8$ and $f(1.2) =1.2$ , we can see that the values of $f(x)$ lie between $0.8$ and $1.2$ when $x$ is in the interval $(1.1,1.2)$ .

STEP 5

Therefore, we can choose $\delta$ such that $(1-\delta,1+\delta)$ is a subset of the interval $(1.1,1.2)$ .

STEP 6

The distance between $1$ and $1.1$ is $0.1$ and the distance between $1$ and $1.2$ is $0.2$ . Therefore, we choose $\delta$ to be the smaller of these two distances, which is $0.1$ .
So, $\delta =0.1$ .
With this choice of $\delta$ , if $|x-1|<\delta$ then $|f(x)-1|<0.2$ .

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