Analyzing Slope in Implicit Equations

Given the differential equation $x^2 + y^2 - 2 = 0$, determine the locations in the graph where the slope ($\frac{dy}{dx}$) is zero, positive, and negative.

This problem requires the analysis of the slope on the graph of an implicit equation. Such tasks often involve implicit differentiation, a crucial concept in calculus when dealing with equations where both variables are interdependent, and direct algebraic solutions for one variable in terms of the other are not readily available. Implicit differentiation allows us to find the derivative of dependent variables with respect to an independent variable when they are linked together by an equation and not easily separable.

To approach this problem, consider the geometric and analytical implications of the given equation, which forms a circle. The points of interest for this task are determined where the derivative, dy/dx, changes sign or is zero. This involves understanding where the slope is horizontal (dy/dx = 0), pointing upwards (positive dy/dx), and downwards (negative dy/dx). Finding these points requires applying implicit differentiation to obtain dy/dx and then analyzing the derivative to find where these conditions occur.

Moreover, this task touches upon the idea of critical points and their classification. In this case, looking for zero slopes corresponds to identifying horizontal tangents. Checking where the slope is positive or negative can give insights into the direction of change of y with respect to x along the curve. Such skills are vital for solving problems related to curve sketching and understanding the nature of functions represented implicitly, preparing students for more complex analyses in fields involving multivariable calculus and dynamical systems.