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

Symmetry about the y-axis can be helpful.

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

 

Example:

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Sketch math image

 

Find the following first:

1. x-intercepts

2. y-intercepts

3. Limit as x approaches infinity

4. Domain

    Range

5. maxima and minima

6. Second derivative

7. Behaviour near discontinuity

 

Solution:

1. x-intercepts

math image

 

2. y-intercepts:

We cannot have x = 0, so there is NO y-intercept, and there must be an asymptote at x = 0.

 

3. As x → -∞, math image → 0, so y → -∞ (In fact, y → -x)

As x → -∞, y → ∞ (In fact, y → x).

 

4. Domain all real x, except 0

   Range all real y

 

5. maxima and minima?

math image

 

So we have a max or min at (2,3).

Now, as x → -∞, dy/dx → 1,

and as x → ∞, dy/dx → 1.

 

6. Second derivative:

math image

So concave up for all x, except 0. So (2,3) is a MIN.

 

7. Near discontinuity:

As x → 0-, (0 from the negative side), y → ∞

[Try x = -1, -0.5, -0.1, -0.01, -0.001 etc to see this].

 

As x → 0+, (0 from the positive side), y → ∞

[Try x = 1, 0.5, 0.1, 0.01, 0.001 etc to see this].

So we are ready to sketch the curve:

math image

 

Example:

Sketch math image

 

Sketch this one by finding the following first:

1. x-intercepts

2. y-intercepts:

3. Limit as x approaches infinity:

4. Domain

   Range

5. maxima and minima

6. Second derivative


Answer



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