Problem 44 - Calculus 3

Question

Compute the derivatives of the vector-valued functions. $$\mathbf{r}(t)=e^{t} \mathbf{i}+2 e^{t} \mathbf{j}+\mathbf{k}$$

Step-by-Step Solution

Answer

The derivative is \( e^t \mathbf{i} + 2e^t \mathbf{j} \).

Step 1: Identify the Components

The vector-valued function \( \mathbf{r}(t) \) is given by \( e^{t} \mathbf{i} + 2e^{t} \mathbf{j} + \mathbf{k} \). Identify the individual components as functions of \( t \): \( x(t) = e^t \), \( y(t) = 2e^t \), and \( z(t) = 1 \).

Step 2: Differentiate Each Component

Differentiate each component of \( \mathbf{r}(t) \) with respect to \( t \):1. \( \frac{dx}{dt} = \frac{d}{dt}(e^t) = e^t \).2. \( \frac{dy}{dt} = \frac{d}{dt}(2e^t) = 2e^t \).3. \( \frac{dz}{dt} = \frac{d}{dt}(1) = 0 \).

Step 3: Combine the Derivatives

Combine the derivatives of the individual components into a single vector:\[ \mathbf{r}'(t) = e^t \mathbf{i} + 2e^t \mathbf{j} + 0 \cdot \mathbf{k} = e^t \mathbf{i} + 2e^t \mathbf{j} \].

Key Concepts

Vector-Valued Functions

A vector-valued function associates a vector to each value of a given variable, often denoted as \( t \). These functions have multiple components, each of which is a scalar function of \( t \). For instance, a vector-valued function \( \mathbf{r}(t) \) might be written in the form \( x(t)\mathbf{i} + y(t)\mathbf{j} + z(t)\mathbf{k} \).

• Components: These are like the individual parts of the vector function. Instead of just one output, the function gives a vector, with each part following its own function of \( t \).
• Example: In the function \( \mathbf{r}(t) = e^{t} \mathbf{i} + 2e^{t} \mathbf{j} + \mathbf{k} \), the components are \( e^t \), \( 2e^t \), and \( 1 \).

Understanding vector-valued functions is important as they are used to describe motions in space, complex systems in engineering, and fields in physics.

Derivatives

The derivative of a vector-valued function is very similar to that of scalar functions, but we find the derivative of each component individually. The result is a new vector comprised of the derivatives of the original components.

• Symbol: Just like scalar functions, the derivative of a vector \( \mathbf{r}(t) \) is denoted as \( \mathbf{r}'(t) \).
• Example: If \( \mathbf{r}(t) = x(t)\mathbf{i} + y(t)\mathbf{j} + z(t)\mathbf{k} \), then \( \mathbf{r}'(t) = x'(t)\mathbf{i} + y'(t)\mathbf{j} + z'(t)\mathbf{k} \).

Calculating derivatives helps in understanding rates of change and motion along paths, making them essential in calculus and physics.

Differentiation

Differentiation is the process of finding a derivative, which involves calculating the rate at which a function changes. For vector-valued functions, differentiation is applied component by component.When you differentiate \( \mathbf{r}(t) = e^{t} \mathbf{i} + 2e^{t} \mathbf{j} + \mathbf{k} \), you treat each component as a separate function of \( t \):

• For \( x(t) = e^t \): The derivative is \( e^t \).
• For \( y(t) = 2e^t \): The derivative is \( 2e^t \).
• For \( z(t) = 1 \): The derivative is 0, since constants have a derivative of zero.

This step-by-step differentiation shows how each part of the vector is treated separately yet combined back into a single vector derivative.

Components of Vectors

Vectors in three-dimensional space have three components, often associated with the standard basis vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \), which align with the x-, y-, and z-axes, respectively.

• Identifying Components: Each scalar expression multiplied by the unit vector (\( \mathbf{i} \), \( \mathbf{j} \), \( \mathbf{k} \)) is a component. For example, in \( \mathbf{r}(t) = e^{t}\mathbf{i} + 2e^{t}\mathbf{j} + \mathbf{k} \), \( e^t\mathbf{i} \) is a component.
• Analysis and Application: Understanding these components helps in breaking down complex motions into simpler parts. It allows easier manipulation and analysis of changes in systems.

The components of a vector reflect how much influence each axis has in defining the direction and magnitude in space.