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 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:
Need Graph Paper?
Sketch 
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
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 → -∞, 
→ 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?
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:
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:
Example:
Sketch 
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
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