Question

Modify Fermat's tangent method to be able to apply it to curves given by equations of the form \(f(x, y)=c\). Begin by noting that if \((x+e, \bar{y})\) is a point on the tangent line near to \((x, y)\), then \(\bar{y}=\frac{t+\varepsilon}{t} y\). Then adequate \(f(x, y)\) to \(f(x+\) \(e, \frac{t+\varepsilon}{t} y\) ). Apply this method to determine the subtangent to the curve \(x^{3}+y^{3}=p x y\)

Short answer

Question: Determine the subtangent, \(t\), to the curve \(x^{3}+y^{3}=pxy\) using the modified Fermat's tangent method. Answer: The subtangent, \(t\), to the curve \(x^{3}+y^{3}=pxy\) is given by the equation \(t = \frac{3x(x+1)}{y-x}\).

Step-by-step solution

Introduction of Fermat's tangent method

Fermat's tangent method allows us to find the tangent line of a curve given by an equation of the form \(f(x, y) = c\). It is based on the observation that if \((x+e, \bar{y})\) is a point on the tangent line near to \((x, y)\), then \(\bar{y}=\frac{t+\varepsilon}{t} y\), where \(t\) is the subtangent, and \(\varepsilon\) is a small positive number.

Modify Fermat's tangent method

Based on the given conditions, we want to modify Fermat's tangent method for equations of the form \(f(x, y) = c\). To do so, we will replace \(x\) with \(x+e\) and \(\bar{y}\) with \(\frac{t+\varepsilon}{t}y\) in the function \(f(x, y)\). This gives us: $$f(x+e, \frac{t+\varepsilon}{t} y)$$

Apply the modified method to the given equation

We will now apply the modified Fermat's tangent method to the given equation: $$x^3 + y^3 = pxy$$ replacing \(x\) with \(x+e\) and \(\bar{y}\) with \(\frac{t+\varepsilon}{t}y\): $$(x+e)^3+\left(\frac{t+\varepsilon}{t}y\right)^3=p(x+e)\frac{t+\varepsilon}{t}y$$

Expand equation

Now we need to expand the equation to eliminate the \(\varepsilon\) term: $$(x^3 +3x^2e+3xe^2+e^3)+\left(\frac{(t+\varepsilon)^3}{t^3}y^3\right)=p(x+e)\frac{t+\varepsilon}{t}y$$

Consider the limit as \(\varepsilon\) approaches 0

As \(\varepsilon\) approaches 0, we get: $$\lim_{\varepsilon\to 0} (x^3+3x^2e + 3xe^2 + e^3 + \frac{(t+\varepsilon)^3}{t^3}y^3) = px(t+\varepsilon)y$$ By taking the limit, we arrive at the following equation: $$x^3+3x^2e+3xe^2+e^3+\frac{t^3}{t^3}y^3 = pxt(y + \varepsilon)$$

Use elimination to solve for subtangent

We can eliminate \(\varepsilon\) from the equation by subtracting the original equation from the expanded equation with the limit: $$[x^3 + 3x^2e + 3xe^2 + e^3 + y^3] - [x^3 + y^3] = pxty - [pxy]$$ Simplifying, we get: $$3x^2e + 3xe^2 + e^3 = pt(y - x)$$ Divide both sides by \(e^2\) since we assume \(e\neq0\): $$3x^2 + 3x + e = \frac{pt(y-x)}{e^2}$$

Determine the subtangent

For the given curve, \(x^{3}+y^{3}=pxy\), let the subtangent be denoted by \(t\). Since we want to determine \(t\), we can set \(e\to 0\) in the simplified equation from Step 6: $$3x^2+3x=pt(y-x)$$ Now we have the equation for the subtangent, \(t\), to the curve \(x^{3}+y^{3}=pxy\): $$t = \frac{3x(x+1)}{y-x}$$

Key concepts

Subtangent

The concept of the subtangent is quite fascinating when analyzing curves. It is a geometric entity that helps understand the nature of a tangent line to a curve at a specific point. Imagine standing on a curve, like being on the surface of a hill.
The tangent line is like a flat board making just one point of contact with your hill. Now, the subtangent is a horizontal line extending from this point on your curve to where this tangent line meets the x-axis.

• Think of it as a way to measure how horizontally stretched out this first contact point on a curve is.
• It shows the change along the x-axis while the height (y-axis) remains tangent at a specific point.

Many times in mathematics, especially involving curves like those described by equations, this subtangent offers deeper insight.
For example, in the analysis of curves like the one from the original problem, determining the subtangent can help assess the curve's behavior and comprehend its geometric properties. Essentially, finding the subtangent means computing a specific part of the tangent, which aids in examining how the curve varies!

Curve Equations

Curve equations represent the backbone of understanding geometric shapes analytically. They can define curves ranging from simple lines and circles to more complex shapes like ellipses and hyperbolas.
In mathematics, they are typically presented as an equation involving both x and y variables.
The equation given in the original problem, for example, is of the form \(x^3 + y^3 = pxy\). This particular curve is quite intricate, as it involves a polynomial expression of both x and y to a third degree.

• They allow us to predict and calculate points lying on that curve.
• Understanding them helps find intersections, tangents, or even areas under these curves.

For every equation given, such as \(x^3 + y^3 = pxy\), there's an infinite set of points that solve it, forming the curve.
This particular equation defines a relationship where both x and y affect each other to balance the equation equal to some constant form. This balance adds to the curve's distinct shape and properties, making it a fundamental tool in calculus and geometry.

Differential Calculus

Differential calculus is a branch of mathematics focused on how things change. It's like having a magnifying glass to zoom in on tiny changes in values, understanding their incremental differences.
At its core, it involves differentiation - which is the process of computing derivatives.
A derivative measures how a function changes as its input changes. It asks the question, "If I move a little on the x-axis, how much does the y-value change?"

• It’s about slopes, rates of change, and tangents.
• In problems like the original one, you're finding tangents and dealing with subtangents using calculus tools.

Differential calculus helps formalize ideas like those Fermat hinted at with his tangent method. By understanding derivatives, we gain insights into curve behavior, identifying slope, steepness, and specific tangent lines.
It plays an integral role not only in mathematics but also in science and engineering fields, where such detailed analysis is necessary to predict and analyze natural phenomena and technological advancements.