Final answer:
The direction field for the differential equation y' = tan(1/2πy) is characterized by vertical lines at points where 1/2πy equals an odd multiple of π/2. Between these vertical lines, the slopes correspond to values of the tangent function. The correct answer is C) Vertical lines.
Step-by-step explanation:
The question asks about the characteristics of a direction field for the differential equation y' = tan(1/2πy). A direction field is a graphical representation that shows the slope of solutions to a differential equation at various points in the plane. By examining the differential equation, we notice that the slope of the tangent to the solution curves is given by the tangent of a constant multiple of y, which is a periodic function.
Unlike parabolic curves, which open up or down, or spiral patterns, which wind around a central point, or circular patterns, which maintain a constant radius from a center, the tangent function repeatedly increases to infinity and decreases to negative infinity as its argument approaches odd multiples of π/2 due to its periodicity.
Therefore, the direction field for y' = tan(1/2πy) would consist of vertical lines at the points where the tangent function is undefined (where 1/2πy equals an odd multiple of π/2). Between these vertical lines, the direction field will show slopes that correspond to the values of the tangent function.
In conclusion, the correct answer to the question about what a direction field for y' = tan(1/2πy) shows is C) Vertical lines.