Problem 34 - Calculus III

Question

Consider the curve described by the vector-valued function \(\mathbf{r}(t)=\left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k}\). What is \(\lim _{t \rightarrow \infty} \mathbf{r}(t) ?\)

Step-by-Step Solution

Answer

\( \lim_{t \to \infty} \mathbf{r}(t) = 5\mathbf{k} \)

Step 1: Understanding the Problem

We need to find the limit of the vector function \( \mathbf{r}(t) \) as \( t \) approaches infinity. The vector \( \mathbf{r}(t) \) has the form \( \mathbf{r}(t)= \left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k} \). This requires evaluating the limit for each component separately.

Step 2: Evaluating the x-component

The x-component of \( \mathbf{r}(t) \) is \( 50 e^{-t} \cos t \). As \( t \to \infty \), the term \( e^{-t} \) approaches 0 because the exponential decay dominates. Thus, \( \lim_{t \to \infty} 50 e^{-t} \cos t = 0 \).

Step 3: Evaluating the y-component

The y-component of \( \mathbf{r}(t) \) is \( 50 e^{-t} \sin t \). Similarly to the x-component, as \( t \to \infty \), \( e^{-t} \) approaches 0. Hence, \( \lim_{t \to \infty} 50 e^{-t} \sin t = 0 \).

Step 4: Evaluating the z-component

The z-component of \( \mathbf{r}(t) \) is \( 5-5 e^{-t} \). As \( t \to \infty \), \( e^{-t} \) approaches 0, so \( 5 e^{-t} \) goes to 0. Therefore, \( \lim_{t \to \infty} (5 - 5 e^{-t}) = 5 \).

Step 5: Combining the Limits

The limits for the x, y, and z components as \( t \to \infty \) are 0, 0, and 5 respectively. Consequently, the limit of the vector \( \mathbf{r}(t) \) as \( t \to \infty \) is \( \mathbf{r}(\infty) = 0\mathbf{i} + 0\mathbf{j} + 5\mathbf{k} \) or simply \( 5\mathbf{k} \).

Key Concepts

Vector-Valued Functions

Vector-valued functions are functions that have vectors as their outputs. These vectors typically exist in three-dimensional space and are written in the form \( \mathbf{r}(t) = x(t) \mathbf{i} + y(t) \mathbf{j} + z(t) \mathbf{k} \), where \( x(t) \), \( y(t) \), and \( z(t) \) are the component functions in terms of \( t \). Each component function is responsible for the behavior of the vector in the corresponding direction.

In the given exercise, the vector-valued function is \( \mathbf{r}(t)=\left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k} \). This expression describes a path or curve in space as the parameter \( t \) changes. This type of function is fundamental for describing the motion of objects.

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

Each component indicates how much the vector \( \mathbf{r}(t) \) extends in the directions of the unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \). Such functions are often used to model physical phenomena involving three-dimensional movement.

Limits in Calculus

The concept of limits is vital in calculus, providing a way to analyze the behavior of functions as the input approaches a particular point or becomes very large. In the context of vector-valued functions, we examine each of the vector's components separately to find the limit.

In our scenario, as \( t \rightarrow \infty \), we investigate the individual limits of each component function:

• For the x-component, \( \lim_{t \to \infty} 50 e^{-t} \cos t = 0 \)
• For the y-component, \( \lim_{t \to \infty} 50 e^{-t} \sin t = 0 \)
• For the z-component, \( \lim_{t \to \infty} (5 - 5 e^{-t}) = 5 \)

These calculations benefit from the property of exponential functions such as \( e^{-t} \), where the exponential decay means \( e^{-t} \) approaches zero as \( t \rightarrow \infty \). Understanding limits not only facilitates the analysis of vector-valued functions but proves indispensable in broader calculus operations, such as finding tangents or solving differential equations.

Exponential Decay

Exponential decay describes a process where quantities reduce at rates proportional to their current value. The mathematical form involves expressions such as \( e^{-t} \), which decrease rapidly as \( t \) increases.

In the given problem, exponential decay is observed in the x and y components of the vector-valued function. Both include the factor \( e^{-t} \), which leads to rapid decrease towards zero.

• For the x-component: \( 50 e^{-t} \cos t \)
• For the y-component: \( 50 e^{-t} \sin t \)

As \( t \) approaches infinity, \( e^{-t} \) trends towards zero, causing both components to become negligible. However, in the z-component, the term \(-5 e^{-t}\) decreases to zero, leaving a constant term behind, which defines the behavior of the vector's z-component at infinity. Exponential decay is crucial not only in calculus but in various fields, including physics for modeling processes such as radioactive decay and cooling, and in finance for depreciation.