Math Application differential calculus Curve sketching

# Curve sketching

Differential calculus is needed in the curve sketching. Here is just a short summary.

A curve sketching usually includes the following examinations:

PropertyCondition
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 $f'''(x_W)\neq0$