Chapter Contents

• Applications of Differentiation
• 1. Tangents and Normals
• 2. Newton's Method
• 3. Curvilinear Motion
• 4. Related Rates
• 5. Curve Sketching Using Differentiation
• 6. More Curve Sketching Using Differentiation
• 7. Applied Maximum and Minimum Problems
• 8. Radius of Curvature
• Applications of Differentiation Problem Solver

• Comments, Questions?

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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

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

Answer

Example 2

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

Answer