91Ó°ÊÓ

Parametric equations are a way of defining a curve using parameters rather than traditional x and y terms. In our exercise, the curve is defined as \(x=2-3 \cos \theta\) and \(y=3+2 \sin \theta\). Each value of the parameter \(\theta\) corresponds to a point on the curve. This form of representation allows for the description of more complex curves that might be difficult or impossible to describe using a simple y = f(x) equation.

For example, the given parametric equations describe an ellipse, with \(\theta\) determining the position on the ellipse for any given point. Students should understand that by changing the parameter \(\theta\), you move along the curve, and the coordinates \(x\) and \(y\) change accordingly. This concept is crucial in finding the slope of the tangent line at a specific point on the curve.

The derivative represents the rate of change of a function with respect to a variable. When we deal with parametric equations, we encounter the need for partial derivatives, like \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\), which give us the rates of change of x and y with respect to the parameter \(\theta\).

In the solution to our exercise, we calculate these derivatives to understand how the x and y coordinates of the curve change as \(\theta\) changes. The overall slope of the tangent line at any point is then given by the derivative \(\frac{dy}{dx}\), which requires the chain rule: \(\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}\). This step is essential for understanding how steep or flat the tangent line is at a given point on the curve.

The slope of the tangent line to a curve at a particular point is a measure of how steep the line is at that exact spot. For curves defined by parametric equations, we need to first find \(\frac{dy}{dx}\), the derivative of y with respect to x.

In our exercise, after finding \(\frac{dy}{dx}\), we compute the slope for each given point by substituting the corresponding \(\theta\) values found from the parametric equations. For instance, when \(\theta = \pi\), the slope of the tangent line is zero, indicating a horizontal line. It's important for students to recognize that a zero slope means horizontal, while an undefined slope, which occurs when \(\frac{dy}{dx}\) cannot be evaluated (like a division by zero situation), indicates a vertical line.

The point-slope form is an equation of a line that is derived from a known point on the line and the slope of the line. It is written as \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \(\left(x_1, y_1\right)\) is the point. This form is particularly useful for writing the equation of a tangent line once you've found the slope at the point of tangency.

As seen in the solution for our exercise, after determining the slope at each point, we use the point-slope form to write the equation of the tangent lines. Students should note how, for a horizontal line, the slope \(m\) is zero, simplifying the equation to \(y = y_1\), and for a vertical line, where the slope is undefined, the equation simplifies to \(x = x_1\). These steps illustrate direct applications of the point-slope form to different situations.