Sketch A Curve
The techniques of sketching a curve consist of
These techniques are used to determine the curve’s trend. They may not be applicable in all cases, but, depending on the particular problem, some can always be used.
Example 1. Determine the relative maxima, relative minima, and points of inflection of the function:
Sketch the graph. Use the GraphFunc utility online to confirm the results.
1. The domain of f(x) is determined for all.
2. Derivatives:and .
· x = 0 => y = 0
· => y = 9/4
· => y = -9/4
· x = 1 => y = -5/4
· x = -1 => y = -5/4
4. Construct a table:
Based on the table’s results, we conclude
· and are relative minima becausechanges sign from negative to positive as x pass through these points.
· x = 0 is relative maxima becausechanges sign from positive to negative as x pass through this point.
· are the points of inflection because(second derivative) changes sign as x pass through these points.
· The graph’s trend is shown on the constructed table. The direction of the arrow is down or up in each interval is depended on the sign of is negative or positive, respectively. The Figure 1 illustrates the graph of f(x).
Figure 1 – the graph of