Problem 33 - 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 the initial point of the path corresponding to \(\mathbf{r}(0) ?\)

Step-by-Step Solution

Answer

The initial point is \( (50, 0, 0) \).

Step 1: Substitute t in the vector function

To find the initial point of the path, we need to evaluate the function \( \mathbf{r}(t) \) at \( t=0 \). Substitute \( t=0 \) into the vector function \( \mathbf{r}(t)=\left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k} \).

Step 2: Evaluate Exponential and Trigonometric Components

Compute the different components: 1. Evaluate the exponential function \( e^0 = 1 \).2. Evaluate the trigonometric functions: - \( \cos(0) = 1 \) - \( \sin(0) = 0 \)

Step 3: Calculate the i-component

Substitute the evaluated values into the i-component: \[ 50 e^{0} \cos(0) = 50 \times 1 \times 1 = 50 \]

Step 4: Calculate the j-component

Substitute the evaluated values into the j-component: \[ 50 e^{0} \sin(0) = 50 \times 1 \times 0 = 0 \]

Step 5: Calculate the k-component

Substitute the evaluated values into the k-component:\[ 5 - 5e^{0} = 5 - 5 \times 1 = 5 - 5 = 0 \]

Step 6: Write the initial point

Combine the components identified in the previous steps to get the initial point of the path:\( \mathbf{r}(0) = 50\mathbf{i} + 0\mathbf{j} + 0\mathbf{k} \). Thus, the initial point is \( (50, 0, 0) \).

Key Concepts

Understanding the Initial Point of a Vector-Valued Function

The initial point of a path described by a vector-valued function is akin to the starting position of an object moving along a specific trajectory in space. It is determined by evaluating the function at the initial parameter value, generally referred to as \( t = 0 \). Substituting this value into the function reveals the coordinates in each direction.

• The "i-component" reflects the position along the x-axis.
• The "j-component" shows the position along the y-axis.
• The "k-component" provides the position along the z-axis.

In the given vector function, the components are \( (50e^{-t}\cos t)\mathbf{i} + (50e^{-t}\sin t)\mathbf{j} + (5-5e^{-t})\mathbf{k} \). For the initial point, substituting \( t = 0 \) reveals \( (50, 0, 0) \), indicating the path starts from this coordinate in 3D space.

Decoding the Exponential Function

The exponential function, represented as \( e^{-t} \), plays a significant role in vector calculus. It describes how values change exponentially over time, and it's crucial for capturing decay or growth trends.When evaluating at \( t = 0 \), the exponential function simplifies to 1, since any non-zero number raised to the power of zero is 1

• This simplification helps in calculating initial conditions quickly.
• It reduces the complexity of mathematical expressions, making them easier to solve.

In the context of the exercise, setting the exponential term to 1 simplifies the analysis of the vector components, allowing us to swiftly find the initial trajectory of the path. The exponential function essentially indicates that initially, there is no decay or growth impact on the function.

Exploring Trigonometric Functions within Vectors

Trigonometric functions, \( \cos(t) \) and \( \sin(t) \), are used to describe periodic behavior, often reflecting circular or rotational movements. These functions help model phenomena that repeat over time or space, such as the winding of a spring or the orbit of a planet.For the angle \( t = 0 \):

• \( \cos(0) = 1 \) implies full horizontal alignment with the angle.
• \( \sin(0) = 0 \) indicates no change along the vertical direction.

In vector-valued functions, the inclusion of trigonometric components can capture oscillatory motion. This is important for understanding how paths develop not just in magnitude (size), but direction as well. In our exercise, these functions clarify the initial orientation of the path in the xy-plane.

Introducing Vector Calculus in Path Analysis

Vector calculus is a mathematical technique used to analyze vector fields, particularly in understanding curves and surfaces in three dimensions. By managing multiple dimensions simultaneously, it helps us map and predict trajectories of moving objects. Main elements of vector calculus:

• It combines the study of vectors, which possess both magnitude and direction.
• Calculus tools extend these concepts, allowing differentiation and integration of vector fields.
• Vector-valued functions can effectively describe the path or orbit of points in space.

In the exercise, vector calculus helps in evaluating the path's initial state and subsequent evolution. Each component of the vector influences spatial orientation. Understanding such multi-component systems enables students to analyze real-world movements and phenomena, like fluid dynamics and electromagnetism.