91Ó°ÊÓ

In mathematics, the concept of a limit allows us to observe the behavior of a function as it approaches a certain point. When dealing with vector-valued functions, the limit involves evaluating each component of the vector independently. For the given vector-valued function \[\mathbf{r}(t)=\left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k}\], we can regard it as having three separate component functions: - \(x(t) = 50 e^{-t} \cos t\)- \(y(t) = 50 e^{-t} \sin t\)- \(z(t) = 5 - 5 e^{-t}\)Each component functions' limit must be found individually as \(t\) approaches infinity. The exponential terms \(e^{-t}\) decay to zero, simplifying the evaluation of each part. - For \(x(t)\) and \(y(t)\), as explained in the steps, \(e^{-t}\) makes these approach zero since \(\cos t\) and \(\sin t\) are bounded.- The function \(z(t)\) simplifies to \(5\) as \(e^{-t}\) approaches zero.Thus, the limit is effectively a vector \((0) \mathbf{i} + (0) \mathbf{j} + (5) \mathbf{k}\). Understanding limits in vector-valued functions is crucial when studying dynamics where directions and magnitudes change as a parameter moves toward infinity.

Exponential functions are critical in understanding growth and decay processes. They are defined as functions of the form \(f(t) = a e^{bt}\), where \(e\) is the base of the natural logarithm, and \(a\) and \(b\) are constants. In the context of our vector-valued function, the exponential term \(e^{-t}\) describes decay since the exponent is negative. As \(t\) grows, \(e^{-t}\) rapidly declines towards zero, affecting the behavior of both \(x(t) = 50 e^{-t} \cos t\) and \(y(t) = 50 e^{-t} \sin t\). are essential when examining phenomena like radioactive decay, cooling processes, or population dynamics. These functions serve as a foundation for modeling a wide range of real-world scenarios involving time-dependent changes.

Parametric equations involve expressing a set of equations as functions of one or more parameters. In our exercise, the vector function \(\mathbf{r}(t)\) can be seen as a set of parametric equations for components \(x(t)\), \(y(t)\), and \(z(t)\). The parameter \(t\) often represents time, allowing us to observe how a particle moves through 3D space. Each component of the vector function describes movement along a coordinate axis.

• \(x(t)\) and \(y(t)\) components illustrate motion in the \(xy\)-plane, affected by the functions \(\cos t\) and \(\sin t\).
• \(z(t)\) alters the height, showing vertical movement independent of modulation in \(xy\)-plane.

This approach facilitates an understanding of geometric shapes and motion paths, with applications in physics and engineering. Evaluating the limit for these parametric components allows students to predict end behavior as time progresses.

Understanding three-dimensional vectors is essential in visualizing objects and their trajectories within space. A vector is defined by both direction and magnitude, represented in space by components along the x, y, and z axes. The vector \(\mathbf{r}(t)=\left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k}\) conveys motion expressed in these three axes.

• The \(\mathbf{i}\) component (50 \(e^{-t} \cos t\)) affects movement along the x-axis.
• The \(\mathbf{j}\) component (50 \(e^{-t} \sin t\)) determines movement along the y-axis.
• The \(\mathbf{k}\) component (5 - 5 \(e^{-t}\)) adjusts the vector vertically along the z-axis.

In dynamics, these vectors help depict an object's position over time. By mastering vector operations and limits, one can derive richer insights into the mechanics of motion, forces, and other vector-dependent phenomena in physics and engineering.