Curve Sketching Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

hard
Sketch f(x)=x2x2โˆ’1f(x) = \dfrac{x^2}{x^2-1}: find domain, asymptotes, critical points, and concavity.

Solution

  1. 1
    Domain: xโ‰ ยฑ1x \neq \pm1. VAs at x=ยฑ1x=\pm1. HA: y=1y=1 (equal degrees).
  2. 2
    fโ€ฒ(x)=โˆ’2x(x2โˆ’1)2f'(x) = \frac{-2x}{(x^2-1)^2}. Critical point x=0x=0: local max at (0,0)(0,0).
  3. 3
    Concave down on (โˆ’1,1)(-1,1). No inflection in domain.

Answer

VA: x=ยฑ1x=\pm1; HA: y=1y=1; local max at (0,0)(0,0).
For rational functions, find asymptotes first. The quotient rule gives fโ€ฒf'.

About Curve Sketching

Using the first and second derivatives to determine a function's behavior: intervals of increase/decrease, local maxima/minima, concavity (up/down), and inflection points, then combining this information to sketch an accurate graph.

Learn more about Curve Sketching โ†’

More Curve Sketching Examples

Example 1 easy

Sketch [formula]: find critical points, monotonicity, concavity, and inflection points.

Example 3 easy

For [formula], find and classify all critical points.

Example 4 medium

Find the inflection points of [formula].

โ† All Curve Sketching Examples Curve Sketching Concept