Functionf(x,y) = p sin(qy)
Domainx and y are elements of the set of all Real numbers
DerivativeWith respect to x: 0
With respect to y: p cos(qy)q
Critical points are formed where the two derivatives equal 0. These critical points may be minimums, maximums, or saddle points. In the first graph, for y, 0 = cos(y) which happens at n*Pi/2, for all n. For x, 0 = 0 for all x values. From the graph, it shows that the lines for y = n*Pi/2 are the ridges that run high and low.
IntegralWith respect to x: p sin(qy)x
With respect to y: (-cos(qy)p)/q
When working with this function, there are many different items to look at. The p scales the values up and down. If it is left out, the graph goes from -1 to 1. The graph goes from -p to p. This is shown in the graph of 2 sin(y). The q value scales the angle. If it is left out, the values remain the same as the angle. This means that the graph completes one cycle in 2*Pi radians. The graph of sin (2y) shows this when the angles are doubled, the graph completes one cycle in only Pi radians.
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