Curve sketching
Differential calculus is needed in the curve sketching. Here is just a short summary.
A curve sketching usually includes the following examinations:
| Property | Condition | |
|---|---|---|
| Axis intersections | ||
| Interfaces x-axis (Zeros of a function) | $f(x)=0$ | |
| Calculate interface y-axis | $f(0)$ | |
| Symmetry | ||
| Axis symmetry to the y-axis | $f(-x)=f(x)$ | |
| Point symmetry to the origin | $f(-x)=-f(x)$ | |
| Monotony behavior | ||
| monotone increasing | $f'(x)\ge0$ | |
| monotone decreasing | $f'(x)\le0$ | |
| strictly monotone increasing | $f'(x)>0$ | |
| strictly monotone decreasing | $f'(x)<0$ | |
| Maxima and minima (Extrema) | ||
| Maximum point | $f'(x_E)=0$ und $f''(x_E)<0$ | |
| Minimum point | $f'(x_E)=0$ und $f''(x_E)>0$ | |
| Inflection points | ||
| Inflection point | $f''(x_W)=0$ und $f'''(x_W)\neq0$ | |