91Ó°ÊÓ

In vector calculus, the derivative of a vector function involves taking the derivative of each component separately. Imagine a vector function like an instruction telling us how to move in space at any time, depending on the parameter, usually denoted as \( t \). This parameter can represent time or any other variable. To find how each part of this instruction changes as \( t \) changes, we differentiate.

For the vector function \( \mathbf{r}(t) = e^{2t} \mathbf{i} + e^{-t} \mathbf{j} \), we have two parts:

• \( e^{2t} \mathbf{i} \) for the \( x \)-component.
• \( e^{-t} \mathbf{j} \) for the \( y \)-component.

The derivative of \( e^{2t} \) is \( 2e^{2t} \), meaning the \( x \)-direction's rate of change is getting scaled by 2. Similarly, for \( e^{-t} \), the derivative is \( -e^{-t} \), indicating the speed and direction of change in the \( y \)-direction. This provides us with the derivative vector \( \mathbf{r}'(t) = 2e^{2t} \mathbf{i} - e^{-t} \mathbf{j} \), the new instruction on how to move.

Evaluating it at \( t = 0 \), we get specific values, \( \mathbf{r}'(0) = 2\mathbf{i} - \mathbf{j} \), showing the exact direction and speed at that particular moment.

Exponential functions are a fundamental aspect of calculus and appear frequently in the study of growth and decay processes. They are functions in the form \( e^{x} \), where \( e \) is a constant approximately equal to 2.71828. This constant, known as Euler's number, has unique mathematical properties.

Differentiating exponential functions is particularly straightforward and useful::

• The derivative of \( e^{x} \) is itself, \( e^{x} \). This means that the function grows at a rate proportional to its current value.
• For other forms like \( e^{2t} \), the derivative involves the chain rule, yielding \( 2e^{2t} \). This shows that as \( t \) grows, the function's growth rate is twice as fast.
• The function \( e^{-t} \) represents exponential decay. Its derivative, \( -e^{-t} \), indicates a decrease over time.

Recognizing and working with exponential functions are crucial in vector calculus, simplifying computations related to derivatives and integrals.

Taking the second derivative of a vector function opens another layer of understanding. The first derivative provides the rate of change, while the second derivative indicates how that rate itself changes.

For the vector \( \mathbf{r}(t) = e^{2t} \mathbf{i} + e^{-t} \mathbf{j} \):

• We've already computed the first derivative as \( \mathbf{r}'(t) = 2e^{2t} \mathbf{i} - e^{-t} \mathbf{j} \).
• We take the derivative of this new function to obtain the second derivative.
  • The term \( 2e^{2t} \mathbf{i} \) differentiates to \( 4e^{2t} \mathbf{i} \)
  • \( -e^{-t} \mathbf{j} \) differentiates to \( e^{-t} \mathbf{j} \)

This results in the second derivative, \( \mathbf{r}''(t) = 4e^{2t} \mathbf{i} + e^{-t} \mathbf{j} \), which offers insights into the concavity and acceleration of the motion described by the original vector function.

Evaluating at \( t = 0 \) simplifies the second derivative to \( \mathbf{r}''(0) = 4\mathbf{i} + \mathbf{j} \), supplying specific information on how movement patterns evolve at that time point.