Question

Derive Sluse's rule for the special case \(f(x, y)=g(x)-y\) from Fermat's rule for determining the subtangent to \(y=\) \(g(x) .\) Derive Sluse's general rule from the modification of Fermat's rule discussed in Exercise \(7 .\)

Short answer

Question: Derive Sluse's rule for the special case \(f(x, y)=g(x)-y\) from Fermat's rule for determining the subtangent to \(y=g(x)\) and then derive the general rule of Sluse's method using the modification of Fermat's rule discussed in Exercise 7. Answer: For the special case \(f(x, y)=g(x)-y\), Sluse's rule is \(x - x_0 = \frac{-f(x_0, y_0)}{g'(x_0)}\). The general rule of Sluse's method, derived using the modification of Fermat's rule, is \(x - x_0 = \frac{-y_0f_x(x_0, y_0)}{f_y(x_0, y_0)}\).

Step-by-step solution

Recall Fermat's rule for subtangents

According to Fermat's rule, the subtangent of a curve \(y = g(x)\) at a point \(P(x_0, y_0)\) is given by the formula: \(ST = \frac{y_0}{g'(x_0)}\), where \(g'(x_0)\) is the derivative of \(g(x)\) evaluated at \(x_0\).

Find the tangent to the curve \(y = g(x)\)

The equation of the tangent line to a curve of the form \(y = g(x)\) at a point \(P(x_0, y_0)\) can be found using the point-slope form: Tangent: \(y - y_0 = g'(x_0)(x - x_0)\).

Derive Sluse's rule for the special case \(f(x, y) = g(x) - y\)

Now we can rewrite the equation for the tangent at point P as: \(y - g(x_0) = g'(x_0)(x - x_0)\). Since \(f(x, y) = g(x) - y\), the tangent equation becomes: \(-f(x_0, y_0) = g'(x_0)(x - x_0)\). We can rewrite the expression for the tangent as: \(x - x_0 = \frac{-f(x_0, y_0)}{g'(x_0)}\). This is Sluse's rule for the special case \(f(x, y) = g(x) - y\).

Recall the modification of Fermat's rule from Exercise 7

Now, we need to consider the modification of Fermat's rule from Exercise 7. It states that the subtangent of a curve defined by \(f(x, y) = 0\) at \(P(x_0, y_0)\) can be given by: \(ST = \frac{y_0f_x(x_0, y_0)}{f_y(x_0, y_0)}\).

Derive Sluse's general rule using the modification of Fermat's rule

Using the formula from Step 4, we can now find the tangent equation at point P: \(y - y_0 = \frac{y_0 f_x(x_0, y_0)}{f_y(x_0, y_0)}(x - x_0)\). Rearranging this equation, we obtain: \(x - x_0 = \frac{-y_0f_x(x_0, y_0)}{f_y(x_0, y_0)}\). Thus, we have derived Sluse's general rule: \(x - x_0 = \frac{-y_0f_x(x_0, y_0)}{f_y(x_0, y_0)}\).

Key concepts

Fermat's Rule for Subtangents

In the fascinating realm of calculus, one historical gem is Fermat's Rule for Subtangents, a concept that provides insights into the nature of curves long before the formal definition of derivatives was established. This rule gives us a method to determine the length of the subtangent, which is the segment of the tangent line to a curve that lies between the point of tangency and the x-axis.

Imagine you're looking at a curve on a graph, and you drop a vertical line from the curve down to the x-axis. The length along the x-axis from this drop-down point to where the tangent intersects is called the subtangent. According to Fermat, the subtangent at a point on the curve where y = g(x) is given by the simple proportion \(ST = \frac{y_0}{g'(x_0)}\), where \(g'(x_0)\) is the slope of the tangent at that point, and \(x_0, y_0)\) are the coordinates of the point of tangency.

Understanding this historical rule helps students grasp more modern concepts in differential calculus and provides an excellent example of how geometry and algebra can come together to solve problems involving rates of change and motion.

Derivative in Calculus

The concept of a derivative sits at the heart of calculus, acting as a fundamental tool to explore the rates at which things change. When we take the derivative of a function, we're measuring how sensitive the output (dependent variable) is to a change in the input (independent variable). In simple terms, it tells us the slope of the tangent line to the function's graph at any given point.

The technical definition of the derivative of a function f(x) at a point \(x_0\) is the limit of the average rate of change of the function as the interval over which the change is measured shrinks to zero. Mathematically, this is expressed as \(f'(x_0) = \lim_{h \to 0} \frac{f(x_0+h) - f(x_0)}{h}\).

Derivatives are not only foundational in calculations involving motion and growth but are equally vital in various fields, including engineering, economics, and beyond. They are a precursor to more advanced studies, such as differential equations and optimization problems.

Tangent to a Curve

Delving into the realm of calculus, the term 'tangent' often stirs up the image of a line that barely touches a curve at a single point. This concept is crucial for visualizing and comprehending the behavior of functions. A tangent to a curve at a point P is a straight line that just 'grazes' the curve at P, representing the direction of the curve at that instant.

The slope of this elusive tangent line at the point \(x_0, y_0)\) on the curve y = g(x) is given by the derivative of the function at that point, \(g'(x_0)\), as we've explored with derivatives. Equipped with the slope and a point on the tangent, one can employ the point-slope form to craft the equation of the tangent line: \(y - y_0 = g'(x_0)(x - x_0)\).

This simple yet powerful line enables us to approximate the values of the function near the point of tangency and sets the stage for solving more involved problems such as curve sketching and optimization.

Differential Equations

Among the array of mathematical tools, differential equations are like a Swiss Army knife, versatile in modeling situations where change is continuously occurring. These equations relate functions with their derivatives, providing a description of the relationship between a varying quantity and its rates of change.

There are various types of differential equations, from simple, separable equations to more complex, non-linear systems. But the idea is the same: to find a function that satisfies the relationship outlined by the differential equation. For example, if you have an equation involving \(\frac{dy}{dx}\), where y is some quantity that depends on x, solving the differential equation means finding a functional form for y.

In the context of our earlier discussion on subtangents and tangents, differential equations often emerge as underlying models that predict the movement and growth of phenomena, encompassing everything from celestial mechanics to the flow of heat. Students exploring differential equations will learn not just to solve them but also to apply them in real-world situations that require understanding complex systems.