6. More on Curve Sketching Using Differentiation

by M. Bourne

This section deals with curves which are NOT polynomials. They have discontinuities or other unusual behaviour. It is important to understand these types of graphs, since they arise out of real-life situations. Also, we need to be able to interpret error messages or other unexpected behaviour when we are using computers to draw them.

We use all the techniques applied in Section 5 Curve Sketching and also examine the behaviour of the function as

  • x → −∞
  • x → +∞
  • x `→` left side of the discontinuity
  • x ` →` right side of the discontinuity


We can use symmetry about the y-axis to help us sketch the curve (it will be a mirror image about the y-axis).

Domain and range

The domain (all possible x-values) and range (all resulting y-values) is important when graphing certain types of questions (e.g. those involving square root).

Our method

Find the following first:

1. x-intercepts

2. y-intercepts

3. Limit as x approaches infinity

4. Domain and Range

5. Maxima and minima

6. Second derivative

7. Behaviour near discontinuity

Example 1

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Sketch `y=x+4/x^2`

Example 2

Sketch `y=(9x)/(x^2+9)`