91Ó°ÊÓ

Calculus lies at the heart of understanding changes and movement in mathematics, especially when it comes to functions. A fundamental component of calculus is analyzing how objects vary over time or within a space. In the case of the vector-valued function \( \mathbf{r}(\theta)=2\cos^{3}\theta\mathbf{i}+3\sin^{3}\theta\mathbf{j} \), we're particularly interested in how the function behaves as the parameter \(\theta\) changes, thus examining its smoothness.

Smoothness of a curve is an important concept in calculus since it relates to continuity and differentiability. A function is considered smooth over an interval if it is continuously differentiable over that interval, meaning we can find a derivative for every point within the interval without any interruptions in the function's behavior. This concept ties directly into the understanding of motion and change because smoothness ensures that there are no abrupt changes or corners along the path of the curve.

When dealing with vector-valued functions, like \( \mathbf{r}(\theta) \), derivatives tell us about the rate at which the function is changing in each of its components at any given point. The derivative of a vector-valued function is obtained by differentiating each component of the function separately.

For our function \( \mathbf{r}(\theta)=2\cos^{3}\theta\mathbf{i}+3\sin^{3}\theta\mathbf{j} \), we compute the derivatives of its components, resulting in \( -6\cos^{2}\theta * \sin\theta \mathbf{i} \) and \( 9\sin^{2}\theta * \cos\theta \mathbf{j} \). These derivatives represent the instantaneous rate of change of the function's \(i\) and \(j\) components with respect to \(\theta\). The existence of these derivatives across all values of \(\theta\) is a strong indicator that our curve is smooth over the real-number domain.

Parametric equations describe a set of relationships where one or more equations depend on a separate variable, often a parameter like \(\theta\) in our example function \( \mathbf{r}(\theta) \). The smoothness of a curve represented by parametric equations can be deduced by looking at the derivatives of these equations.

The notion of smoothness in the context of parametric equations reflects the absence of sharp turns or cusps in the curve. In other words, the curve should flow in a continuous manner without any breaks. When we differentiate the components of \( \mathbf{r}(\theta) \) and find that there are no points where the derivative is undefined or discontinuous, we can conclude the curve does not have abrupt direction changes or infinite slopes, characterising it as smooth over the range of \(\theta\).

Given that the derivatives of both components of \( \mathbf{r}(\theta) \) are well-defined for all real numbers, we can conclude that the curve is smooth across all open intervals in its domain. Thus, parametrically defined curves like this vector-valued function give us elegant ways to describe complex shapes and motions, and calculus helps us to ensure these descriptions represent continuous and smooth movements.