Q. 15 - Calculus

Question

If $x\in (1.5, 2.5)$, what is the largest interval $I=\left(4-\epsilon ,4+\epsilon \right)$ for which we can guarantee that $x^2\in I$ ?

Step-by-Step Solution

Answer

The interval is $\left(1.75,6.25\right)$.

Step 1. Given information.

The given function is $x^2$.

Step 2. Calculation.

For $x=1.5$

$$\begin{gathered} f\left(x\right)=x^2 \\ f(1.5)={(1.5)}^{2}f(1.5)=2.25 \end{gathered}$$

So the interval, where $\epsilon =2.25$ is:

$$\begin{gathered} I=\left(4-\epsilon ,4+\epsilon \right) \\ I=(4-2.25,4+2.25) \\ I=\left(1.75,6.25\right) \end{gathered}$$

Step 3. Conclusion.

Therefore, the interval is $\left(1.75,6.25\right)$.