Question

Use a graph to find a number \(\delta \) such that if \(\left| {x - 2} \right| < \delta \), then \(\left| {\sqrt {{x^2} + 5} - 3} \right| < 0.3\)b

Short answer

The number is \(0.426\).

Step-by-step solution

Plot the graph of the function.

Draw the graph of the function\(f\left( x \right) = \sqrt {{x^2} + 5} \)by using the graphingcalculator as:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(\sqrt {{X^2} + 5} \)in the\({Y_1}\)tab.

2. Enter the “GRAPH” button in the graphing calculator.

The graph of the function \(f\left( x \right) = \sqrt {{x^2} + 5} \) is shown below:

Solve the given condition

Apply the absolute property, that is, if \(\left| x \right| < a\) then \( - a < x < a\).

Rewrite the given condition \(\left| {\sqrt {{x^2} + 5} - 3} \right| < 0.3\) as:

\(\begin{aligned} - 0.3 < \sqrt {{x^2} + 5} - 3 < 0.3\\ - 0.3 + 3 < \sqrt {{x^2} + 5} < 0.3 + 3\\2.7 < \sqrt {{x^2} + 5} < 3.3\end{aligned}\)

Solve the obtained condition for \({\bf{x}}\)

Take square on each side of the inequality and simplify.

\(\begin{aligned}{\left( {2.7} \right)^2} < {\left( {\sqrt {{x^2} + 5} } \right)^2} < {\left( {3.3} \right)^2}\\7.29 < {x^2} + 5 < 10.89\\2.29 < {x^2} < 5.89\\1.513 < x < 2.426\end{aligned}\)

Observe the graph

It is observed that the condition \(2.7 < f\left( x \right) < 3.3\) is satisfied when \(x\) lies between \(1.513\) and \(2.426\), that is, \(1.513 < x < 2.426\).

From the graph, it is observed that \(2 - {\delta _1} = 1.513\) and \(2 + {\delta _2} = 2.426\).

Obtain the value of \({\delta _1}\) and \({\delta _2}\)

Solve the obtained condition.

\(\begin{aligned} 2 - {\delta _1} & = 1.513\\{\delta _1} & = 2 - 1.513\\ & = 0.487\end{aligned}\)

\(\begin{aligned} 2 + {\delta _2} & = 2.426\\{\delta _2} &= 2.426 - 2\\ &= 0.426\end{aligned}\)

Obtain the value of \(\delta \)

The value of\(\delta \)is the minimum value of \({\delta _1}\), and\({\delta _2}\).

\(\begin{aligned}\delta & = {\rm{min}}\left\{ {0.487,0.426} \right\}\\ & = 0.426\end{aligned}\)

Therefore, the value of \(\delta \) is \(0.426\).