91Ó°ÊÓ

Implicit differentiation is a powerful tool in calculus that helps us find the rates of change or derivatives of functions that are not easily separable. It is especially useful when dealing with equations where variables are intertwined and cannot be isolated on one side of the equation.

In our exercise, we're working with the curve defined by the equation \(x^2 + xy + y^2 = 7\). Here, \(y\) is not expressed explicitly as a function of \(x\), making traditional differentiation difficult. Instead, we apply implicit differentiation.

Implicit differentiation involves differentiating both sides of the equation with respect to \(x\), while treating \(y\) as an implicit function of \(x\). This means you also have to apply the chain rule when differentiating terms involving \(y\). Hence, each time you differentiate a term with \(y\), include a \(\frac{dy}{dx}\), since \(y\) is essentially a function of \(x\).

Through implicit differentiation of our equation, you derive the expression \(2x + y + x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0\), which can be rearranged to solve for \(\frac{dy}{dx}\), giving us the slope of the tangent line at any point \((x, y)\) on the curve.

Intersection points between a curve and an axis occur where the curve crosses the axis. These points are crucial because they often simplify the calculation of slopes and can reveal symmetrical properties of the curve.

In this problem, the goal is to find where the curve intersects the \(x\)-axis, which means setting \(y = 0\) because the definition of the \(x\)-axis is where \(y\) equals zero.

When you substitute \(y = 0\) into the equation \(x^2 + xy + y^2 = 7\), it simplifies to \(x^2 = 7\). This indicates the curve intersects the \(x\)-axis at \(x = \sqrt{7}\) and \(x = -\sqrt{7}\). Therefore, the intersection points are \((\sqrt{7}, 0)\) and \((-\sqrt{7}, 0)\).

Finding these points is a straightforward yet essential step, as it lays the groundwork for determining the parallel nature of tangents, by allowing these specific \(x\) values to be used in further calculations.

The tangent slope at a point on a curve is the slope of the line that just "touches" the curve at that point, providing the best linear approximation of the curve there. It is determined by finding the derivative of the curve's equation at that specific point.

With implicit differentiation, we've found the derivative \(\frac{dy}{dx} = \frac{-(2x + y)}{x + 2y}\). This formula helps us determine the slope of the tangent line, \(m\), at any given point on the curve by plugging in the coordinates of the point into the derivative.

For the intersection points \((\sqrt{7}, 0)\) and \((-\sqrt{7}, 0)\), substituting these values into the derivative gives us the same slope \(-2\) for both.

Since both tangent slopes are \(-2\), it confirms that the tangents at these points are parallel. Parallel lines share the same slope, and in this exercise, the parallel tangents at the intersection points reaffirm that property with \(m = -2\) being the common slope.