Question

Explain how to compute the derivative of \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\).

Short answer

Answer: The derivative of the vector-valued function \(\mathbf{r}(t)\) is obtained by finding the derivatives of its components and forming a new vector. The derivative is given by \(\mathbf{r}'(t) = \langle f'(t), g'(t), h'(t)\rangle\), where \(f'(t)\), \(g'(t)\), and \(h'(t)\) are the derivatives of the components \(f(t)\), \(g(t)\), and \(h(t)\) with respect to \(t\).

Step-by-step solution

Identify the components of the vector-valued function

The given vector-valued function is \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\). The components of this function are \(f(t)\), \(g(t)\), and \(h(t)\).

Compute the derivatives of the components

To calculate the derivative of \(\mathbf{r}(t)\), we need to find the derivatives of its components with respect to \(t\). Let's denote these derivatives as \(f'(t)\), \(g'(t)\), and \(h'(t)\): 1. Calculate \(f'(t)\), the derivative of \(f(t)\) with respect to \(t\). 2. Calculate \(g'(t)\), the derivative of \(g(t)\) with respect to \(t\). 3. Calculate \(h'(t)\), the derivative of \(h(t)\) with respect to \(t\).

Form the derivative vector of \(\mathbf{r}(t)\)

Now that we have the derivatives for all the components, we can form the derivative vector of our function. The derivative of the vector-valued function \(\mathbf{r}(t)\) is also a vector whose components are the individual derivatives of the original components: \(\mathbf{r}'(t) = \langle f'(t), g'(t), h'(t)\rangle\) This new vector function, \(\mathbf{r}'(t)\), is the derivative of the original function \(\mathbf{r}(t)\).

Key concepts

Derivative of Vector-Valued Function

A vector-valued function is a function that outputs a vector rather than a single scalar value. For example, the function \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\) outputs a vector with three components. Each component represents a function of \(t\). The derivative of a vector-valued function is a critical concept in vector calculus. It involves taking the derivative of each component function individually. This way, we can understand how the entire vector function changes as the variable \(t\) changes. This operation is straightforward: the derivative of \(\mathbf{r}(t)\) is simply \(\mathbf{r}'(t) = \langle f'(t), g'(t), h'(t)\rangle\), where each \(f'(t)\), \(g'(t)\), and \(h'(t)\) represents the derivative of its corresponding function with respect to \(t\).
This new vector \(\mathbf{r}'(t)\) tells us how the direction and magnitude of \(\mathbf{r}(t)\) change, offering insights into the motion or behavior described by the function.

Vector Function Components

Vector functions are made up of components, each being a separate function of the same variable, \(t\). In the case of \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\), the components are \(f(t)\), \(g(t)\), and \(h(t)\). These components might represent various dimensions, such as spatial coordinates or other values that together define a system. Understanding the components is crucial because the derivative process affects each component individually.
Key Points:

• Components are treated as separate entities when performing operations like differentiation.
• The behavior of the entire vector function is determined by the interplay between its components.

When computing derivatives or evaluating behavior, examining each component reveals valuable insights about the whole function.

Differentiation

Differentiation is the process of finding the derivative, which measures how a function changes as its input changes. For a vector-valued function, differentiation occurs component-wise. Suppose we have \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\). To differentiate \(\mathbf{r}(t)\), we compute the derivative of each component:

• \(f'(t)\) is the derivative of \(f(t)\).
• \(g'(t)\) is the derivative of \(g(t)\).
• \(h'(t)\) is the derivative of \(h(t)\).

These derivatives are combined into a new vector \(\mathbf{r}'(t) = \langle f'(t), g'(t), h'(t)\rangle\).
Understanding differentiation within vector calculus enables the analysis of rates of change, velocities, and other dynamic properties of functions critical in fields like physics and engineering. Each derivative provides insight into how the system evolves over time.