Question

1 –54 Use the guidelines of this section to sketch the curve.

27. \(y = \frac{{\sqrt {1 - {x^2}} }}{x}\)

Short answer

Graph of the given curve is:

Step-by-step solution

Steps for sketching a curve

There are following terms needed to examine forSketching a Graphfor any given Function:

1. Find the Domain of the Function.
2. Calculate the Intercepts.
3. Check forSymmetricity.
4. Find theAsymptotes.
5. Intervals ofIncrease and Decrease.
6. Evaluate theMaxima and Minimaof the function.
7. ExamineConcavityand the Point of Inflection.
8. Sketch the Graph.

(A) Determine Domain

The given function is \(y = \frac{{\sqrt {1 - {x^2}} }}{x}\).

The domain of the function will be defined when \(\left\{ {x|x \ne 0} \right\}\).

So, the domain of the given function is \(\left( { - 1,0} \right) \cup \left( {0,1} \right)\).

(B) Determine Intercepts

Find \(y\)-intercepts by substituting 0 for \(x\) into \(y = \frac{{\sqrt {1 - {x^2}} }}{x}\).

\(\begin{array}{c}y = \frac{{\sqrt {1 - {0^2}} }}{0}\\ = {\rm{not defined}}\end{array}\)

So, there is no \(y\)-intercept.

Find \(x\)-intercepts by substituting 0 for \(y\) into \(y = \frac{{\sqrt {1 - {x^2}} }}{x}\).

\(\begin{array}{c}0 = \frac{{\sqrt {1 - {x^2}} }}{x}\\ \pm 1 = x\end{array}\)

So, \(x\)-intercepts are \(\left( { \pm 1,0} \right)\).

(C) Determine symmetry

For the given function \(y = \frac{{\sqrt {1 - {x^2}} }}{x}\), \(f\left( { - x} \right) = - x\), so the function is symmetric about the origin.

(D) Find Asymptotes 

As the infinite point is not in the domain so horizontal asymptotes, do not exist.

Determine \(\mathop {\lim }\limits_{x \to {0^ + }} y\) and \(\mathop {\lim }\limits_{x \to {0^ - }} y\) for Vertical asymptotes.

\(\begin{array}{c}\mathop {\lim }\limits_{x \to {0^ + }} y = \mathop {\lim }\limits_{x \to {0^ + }} \left( {\frac{{\sqrt {1 - {x^2}} }}{x}} \right)\\ = \infty \end{array}\)

\(\begin{array}{c}\mathop {\lim }\limits_{x \to {0^ - }} y = \mathop {\lim }\limits_{x \to {0^ - }} \left( {\frac{{\sqrt {1 - {x^2}} }}{x}} \right)\\ = - \infty \end{array}\)

So, the Vertical asymptote is \(x = 0\).

(E) Find Intervals of Increase or Decrease 

Find the first derivative of the given function with respect to \(x\).

\(\begin{array}{c}y' = \frac{d}{{dx}}\left( {\frac{{\sqrt {1 - {x^2}} }}{x}} \right)\\ = \frac{{\frac{{ - {x^2}}}{{\sqrt {1 - {x^2}} }} - \sqrt {1 - {x^2}} }}{{{x^2}}}\\ = - \frac{1}{{{x^2}\sqrt {1 - {x^2}} }}\end{array}\)

For \(y' = 0\), there is no \(x\), as \(y' = 0\) does not exist.

Draw a table for the interval of increasing and decreasing.

Interval	\(y'\)	Behavior of \(y\)
\(\left( { - 1,0} \right)\)	-	Decreasing
\(\left( {0,1} \right)\)	-	Decreasing

(F) Find Local Minimum and Maximum values 

From the obtained table and the condition of local maxima and minima, it can be said that there are no maximum or minimum values as there is no sign of change.

(G) Determine Concavity and Points of Inflection 

Find \(y''\).

\(\begin{array}{c}y'' = \frac{d}{{dx}}\left( { - \frac{1}{{{x^2}\sqrt {1 - {x^2}} }}} \right)\\ = \frac{{2 - 3{x^2}}}{{{x^3}{{\left( {1 - {x^2}} \right)}^{\frac{3}{2}}}}}\end{array}\)

For \(y'' = 0\), \(x = \pm \sqrt {\frac{2}{3}} \).

Draw a table for Concavity for different intervals.

Interval	Sign of \(y''\)	Behavior of \(y\)
\(\left( { - 1, - \sqrt {\frac{2}{3}} } \right)\)	+	Concave upward
\(x = - \sqrt {\frac{2}{3}} \)	0	Inflection
\(\left( { - \sqrt {\frac{2}{3}} ,0} \right)\)	-	Concave downward
\(\left( {0,\sqrt {\frac{2}{3}} } \right)\)	+	Concave upward
\(x = \sqrt {\frac{2}{3}} \)	0	Inflection
\(\left( {\sqrt {\frac{2}{3}} ,1} \right)\)	-	Concave downward

Find \(y\left( { \pm \sqrt {\frac{2}{3}} } \right)\)as from the table we will get the inflection point.

\(\begin{array}{c}y\left( { \pm \sqrt {\frac{2}{3}} } \right) = \frac{{\sqrt {1 - {{\left( { \pm \sqrt {\frac{2}{3}} } \right)}^2}} }}{{ \pm \sqrt {\frac{2}{3}} }}\\ = \pm \frac{1}{{\sqrt 2 }}\end{array}\)

So, the inflection point is \(\left( { \pm \sqrt {\frac{2}{3}} , \pm \frac{1}{{\sqrt 2 }}} \right)\).

(H) Draw Graph

Using all the obtained information from steps 2 to 8, draw the graph of the given function.