Use the given graph off to find a number \delta



Answered question


Use the given graph off to find a number δ such that if |x1|<δ then |f(x)1|<0.2

Answer & Explanation

Philip Williams

Philip Williams

Beginner2022-01-08Added 39 answers

|X|<a implies a<X<a 
Simplify by adding 1 to each
The graph shows that if 0.8<f(x)<1.2 x must verify 0.7<x<1.1 
On the other hand, x is required to confirm this inequality:
If we can find δ that verify 0.3δ<x1<δ0.1, we can make sure that 0.7<x<1.1 
so δ must verify two conditions: δ0.1 and 0.3δ 
which means that δ0.1 and 0.3δ 
We can choose δ=0.1, so to be sure two conditions are satisfied, but if you want, you can choose any value.

Toni Scott

Toni Scott

Beginner2022-01-09Added 32 answers

You can find it using the graph
1. Find the intervals where f(x) and x must be. In our case for f(x)=[0.8,1.2], for x=[0.7,1.1]
2. Find the middle of the first interval. We have 1 (0.8+1.22=1)
3. Find x0, wheree f(x0)Z=d≤=1, we have x0=1
4. Calculate the distance between x0 and the endpoints of the second interval, in our case: |x00.7|=|10.7|=0.3 and |x01.1|=|11.1|=0.1
5.δ is the minimum of these two values, thus it's 0.1

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