# 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

### Symmetry

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

### Need Graph Paper?

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

### Example 2

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

### Search IntMath, blog and Forum

### Online Algebra Solver

This algebra solver can solve a wide range of math problems.

Go to: Online algebra solver

### Calculus Lessons on DVD

Math videos by MathTutorDVD.com

Easy to understand calculus lessons on DVD. See samples before you commit.

More info: Calculus videos

### The IntMath Newsletter

Sign up for the free **IntMath Newsletter**. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!