91Ó°ÊÓ

When working with curves in 3-dimensional space, parametric equations play a crucial role. Unlike traditional equations, which express variables in terms of each other, parametric equations define each coordinate—x, y, and z—as functions of an independent parameter, often denoted as \( t \). This allows us to describe complex curves and paths.
In our given vector-valued function, we have:

• \( x(t) = 50 e^{-t} \cos t \)
• \( y(t) = 50 e^{-t} \sin t \)
• \( z(t) = 5 - 5e^{-t} \)

These parametric equations give us a clear way to understand the position of a point in 3D space as \( t \) changes. Each function draws a path by assigning coordinates based on \( t \), allowing us to elegantly describe curves that would be complicated using a single equation.

Visualizing vector-valued functions often requires us to think in three dimensions. The function \( \mathbf{r}(t) \) describes a path in 3D space, using the i, j, and k unit vectors to define movement along the x, y, and z axes respectively.
This helps us understand the physical representation of these equations:

• The \( i \) component (\( x(t) \)) influences movement along the x-axis.
• The \( j \) component (\( y(t) \)) dictates movement along the y-axis.
• The \( k \) component (\( z(t) \)) defines movement along the z-axis.

Each component combines to create a trajectory in 3D, giving us the ability to model behaviors like spirals or arcs in physical space. Visualizing these paths can be challenging, so using technology like graphing software becomes essential. It lets us see these complex 3D curves, helping us grasp their geometric nuances.

Exponential decay is a fundamental concept in mathematics that describes the process of reducing faster at the beginning and leveling off as time progresses due to a constant decay rate. In our function, exponential decay is expressed through terms like \( e^{-t} \).
In the vector-valued function, the terms \( 50 e^{-t} \cos t \) and \( 50 e^{-t} \sin t \) represent diminishing amplitudes as \( t \) increases:

• As \( t \to \infty \), \( e^{-t} \to 0 \), reducing the significance of the x and y components, causing the point to "decay" toward the center of the spiral (0, 0, 5).
• The \( 5 - 5e^{-t} \) term approaches 5 in the z-direction, showing a vertical stabilization.

This combination results in a spiral that not only winds down due to decaying x and y coordinates but also asymptotically approaches a fixed z-level, creating a distinctive "corkscrew" effect towards the point (0, 0, 5). Understanding exponential decay helps predict how this function behaves over time, providing insights into natural processes and phenomena.