Question

Find the horizontal asymptotes of the curve and use them, together with concavity and intervals of increase and decrease, to sketch the curve.

y = x/x^2 + 1

Answer 1

Step 1:

The rational function

Find the horizontal asymptote :

Since degree of numerator less than the degree of denominator, the horizontal asymptote .

Find the vertical asymptotes by solving zeros of denominator.

There is no vertical asymptotes.

Step 2:

Find the critical numbers by equate the first derivative to zero.

The function does not exist when .

The denominator does not have real roots.

Equate the numerator of to zero.

Substitute the values of in original function.

Critical points are and .

Consider the test intervals as   and .

Interval	Test Value	Sign of	Conclusion
			Decreasing
			Increasing
			Decreasing

 The function is increasing on the interval .

And the function is decreasing on the intervals  and .

Answer 2

Contd...

Step 3:

Find the inflection points by equate the second derivative to zero.

The function does not exist when .

The denominator does not have real roots.

Equate the numerator of to zero.

and

and

Substitute the values of in original function.

Inflection points are , , .

Answer 3

Contd...

Consider the test intervals as   and .

Interval	Test Value	Sign of	Conclusion
			Down
			Up
			Down
			Up

Graph:

Draw the coordinate plane.

Plot the critical points and inflection points of the curve.

.

Comments

Solution:

Horizontal asymptote .

Graph of

.