Question

Multiple Choice A slope field for the differential equation \(d y / d x=42-y\) will show (A) a line with slope \(-1\) and \(y\) -intercept 42 . (B) a vertical asymptote at \(x=42\) . (C) a horizontal asymptote at \(y=42\) (D) a family of parabolas opening downward. (E) a family of parabolas opening to the left.

Short answer

The correct answer is (C) a horizontal asymptote at \(y = 42\).

Step-by-step solution

Understand the differential equation and its slope field.

The given differential equation is a first order linear differential equation with form \(dy/dx = a - y\), where \(a\) is a constant and \(y\) is the dependent variable. The slope field of such an equation is a representation with directions of the slopes at different points, which corresponds to the y values. The equilibrium or steady state is achieved when \(dy/dx = 0\), hence giving \(y = a\). For the provided equation, the equilibrium point is \(y = 42\). The slope field will have lines with different slopes according to the equation \(dy/dx = 42 - y\), and will have equilibrium at \(y=42\) where the slopes will be zero.

Identify the correct choice

By understanding the characteristics of the slope field for the given differential equation, it can be inferred that at \(y = 42\), the slope will be horizontal. This is equivalent to a horizontal asymptote in graphical terms. Looking at the multiple choices given, option (C) presents a horizontal asymptote at \(y = 42\), aligning with the characteristics of the slope field for the provided differential equation.