Q. 5 - Calculus

Question

Find punctured intervals on which the function $f\left(x\right)=\frac{1}{x^2-x}$is defined, centered around

$\left(\mathrm{a}\right) x=1.5 \left(\mathrm{b}\right) x=0.25 \left(\mathrm{c}\right) x=1$

Step-by-Step Solution

Answer

Part (a). The punctured intervals is$\left(1,1.5\right)\cup \left(1.5,2\right)$.

Part (b). The punctured intervals is $\left(0,0.25\right)\cup \left(0.25,0.5\right)$.

Part (c). The punctured intervals is $\left(0,1\right)\cup \left(1,2\right)$.

Step 1. Given information.

The given function is $f\left(x\right)=\frac{1}{x^2-x}$.

Part (a) Step 1. Calculation.

For $x=1.5$

The punctured interval  $\left(c-\delta ,c\right)\cup \left(c,c+\delta \right)$ is given by:
$\left(1.5-0.5,1.5\right)\cup \left(1.5,1.5+0.5\right)=\left(1,1.5\right)\cup \left(1.5,2\right)$.

Part (b) Step 1. Calculation.

For $x=0.25$

The punctured interval $\left(c-\delta ,c\right)\cup \left(c,c+\delta \right)$is given by:

$\left(0.25-0.25,0.25\right)\cup \left(0.25,0.25+0.25\right)=\left(0,0.25\right)\cup \left(0.25,0.5\right)$

Part (c) Step 1. Calculation.

For $x=1$

The punctured interval $\left(c-\delta ,c\right)\cup \left(c,c+\delta \right)$is given by:

$\left(1-1,1\right)\cup \left(1,1+1\right)=\left(0,1\right)\cup \left(1,2\right)$