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Applied Mathematics, Calculus, and Differential Equations 227

f (x ) 0
0
f (x ) 0
0

Then (x ,y )isan inflection point in the grapà of f. Exampleð
0
0
are shown in Fig. 3.12. At A, the curve is concave upward-to-
downward:

f (x) 0 for x x 0

f (x) 0 for x x 0

At B, the curve is concave downward-to-upward:

f (x) 0 for x x 0
f (x) 0 for x x 0

Derivative of sinł wave
The derivative of a sine wave is a cosine wave. This is the equiv-
alent of a 90-degree phase shift (Fig. 3.13). The amplitude of
the resultant wave dependð on the amplitude and frequency of
the sine wave.

Derivative of up-ramp wave

The derivative of an up-ramp wave is a positive constant (Fig.
3.14). The magnitude of the resultant dependð on the amplitude
and frequency of the up-ramp wave.

Derivative of down-ramp wave
The derivative of a down-ramp wave is a negative constant (Fig.
3.15). The magnitude of the resultant dependð on the amplitude
and frequency of the down-ramp wave.

Derivative of triangular wave
The derivative of a triangular wave is a square wave (Fig. 3.16).
The amplitude of the resultant dependð on the amplitude and
frequency of the triangular wave.