Question

Suppose \(x\) and \(y\) are related by the equation \(F(x, y)=0 .\) Interpret the solution of this equation as the set of points \((x, y)\) that lie on the intersection of the surface \(z=F(x, y)\) with the \(x y\) -plane \((z=0)\). a. Make a sketch of a surface and its intersection with the \(x y\) -plane. Give a geometric interpretation of the result that \(\frac{d y}{d x}=-\frac{F_{x}}{F_{y}}\). b. Explain geometrically what happens at points where $$F_{y}=0.$$

Short answer

Answer: The result \(dy/dx = -F_x / F_y\) geometrically represents the slope of the tangent to the curve formed by the intersection of the surface \(z = F(x, y)\) with the \(xy\)-plane. When \(F_y = 0\), this indicates that the surface does not change with respect to \(y\), and thus the tangent to the curve could be either vertical or the curve could be locally constant along the \(y\) direction. In this case, the result \(dy/dx = -F_x / F_y\) becomes undefined.

Step-by-step solution

Sketch the surface and its intersection with the \(xy\)-plane

To get started, let's consider a generic surface \(z = F(x, y)\). We are interested in the points where this surface intersects the \(xy\)-plane, or when \(z = 0\). To find these points, we can set \(F(x, y) = 0\) and solve for \((x, y)\). The intersection of the surface with the \(xy\)-plane will be a curve formed by these points.

Compute the partial derivatives \(F_x\) and \(F_y\)

To find the partial derivatives of \(F(x, y)\), compute the derivatives of \(F\) with respect to \(x\) and \(y\): (i) \(F_x = \frac{\partial F}{\partial x}\) represents how the function \(F(x, y)\) changes with respect to \(x\). (ii) \(F_y = \frac{\partial F}{\partial y}\) represents how the function \(F(x, y)\) changes with respect to \(y\).

Understand the geometric interpretation of \(\frac{dy}{dx} = -\frac{F_x}{F_y}\)

This result can be interpreted geometrically as describing the slope of the tangent to the curve formed by the intersection of the surface \(z = F(x, y)\) with the \(xy\)-plane. The negative sign of the fraction indicates that an increase in \(x\) would lead to a decrease in \(y\), or vice versa.

Analyze the case when \(F_y = 0\)

When \(F_y = 0\), this indicates that the surface does not change with respect to \(y\). Geometrically, this can be interpreted as the tangent to the curve formed by the intersection of the surface with the \(xy\)-plane being either vertical or the curve being locally constant along the \(y\) direction. In this case, the result of \(\frac{dy}{dx} = -\frac{F_x}{F_y}\) becomes undefined as it involves division by zero. This suggests that the curve might have a vertical tangent at these points or the curve might be constant in the \(y\) direction.

Key concepts

Implicit Differentiation

When dealing with equations involving two or more variables combined in a function like \(F(x, y) = 0\), finding how one variable changes concerning another can be tricky. This is where implicit differentiation comes into play. It's a tool we use to find derivatives of functions that aren't explicitly solved for one variable. For instance, if you cannot solve for \(y\) as a function of \(x\), you can still find the derivative \(\frac{dy}{dx}\) by treating \(y\) as a function of \(x\) and using chain rules.
Specifically, when you differentiate \(F(x, y) = 0\) concerning \(x\), you apply standard rules to \(x\) directly while applying the chain rule to \(y\). This gives a derivative like \(\frac{dy}{dx} = -\frac{F_x}{F_y}\), suggesting how \(y\) changes as \(x\) changes. It's like solving a puzzle where the solution appears when using this framework, allowing us to analyze the rate of change along the curve defined by the intersection of our surface with the \(xy\)-plane.

Tangent Lines

Tangent lines are straight lines that gracefully touch a curve at exactly one point without crossing it. Think of them as instantaneous directions where the curve is headed. In calculus, these lines are highly valued as they provide immediate information about the slope of the curve at a given point.
By finding the derivative \(\frac{dy}{dx}\), which in our implicit function scenario is \(-\frac{F_x}{F_y}\), we understand the slope of the tangent line to the curve that results from the surface intersecting the \(xy\)-plane. The sign and magnitude of the slope tell us not just how steep or flat the curve is at that point, but also which direction it's moving in.

• A positive slope means the curve ascends as \(x\) values increase.
• A negative slope indicates a downward trajectory.
• When \(F_y\) becomes zero, the tangent becomes vertical, creating a peculiar situation where the tangent line could theoretically stretch infinitely upwards or downwards, making the slope, in practical terms, undefined.

Critical Points

In mathematical analysis, critical points are where the curve's interesting behavior resides, particularly where it might reach a peak, bottom out, or change direction. In the context of our implicit function \(F(x, y)\), critical points are special because they occur where \(F_y = 0\).
Understanding what happens when \(F_y = 0\) can unlock insights into the surface's behavior. At these points:

• The derivative \(\frac{dy}{dx} = -\frac{F_x}{F_y}\) becomes undefined, flagging these points for further investigation.
• These critical situations can mean a sharp turn in the curve or even a vertical tangent line, indicating abrupt directional changes.
• These are the curves' decision-makers, pinpointing where surface characteristics flip or where the surface might be perfectly level in one direction but not the other.

By identifying and analyzing critical points, we gain clues about the overall topology of the surface and the nature of its intersecting curve with the \(xy\)-plane.