Question

Perpendicular Slope Fields If the slope fields for the differ- ential equations \(d y / d x=\sec x\) and \(d y / d x=g(x)\) are perpendicu- lar (as in Exercise \(55 ),\) find \(g(x)\) .

Short answer

The function \(g(x)\) that describes the slope of the line perpendicular to the given line is \(g(x) = -\cos x\)

Step-by-step solution

Express the given Slope

We are provided with one slope \(dy/dx=\sec x\). We also know that for two lines to be perpendicular, the product of their slopes is -1.

Determine the perpendicular Slope

If \(\sec x\) is the slope of one line, then the slope of the line which is perpendicular to it is therefore \(-1/\sec x\), because the product of the slopes of two perpendicular lines is -1.

Simplify the expression

Note that \(\sec x\) is equivalent to \(1/\cos x \). Consequently, \(-1/\sec x\) becomes \(-1/(1/\cos x)\) which simplifies to \(-\cos x\). Hence, the function \(g(x)\) that describes the slope of this line is \(-\cos x\)