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A vector function is a function that takes in one or more variables, usually denoted as \( t \), and returns a vector. This means that instead of producing a single numerical output like a function in calculus, a vector function produces an output in the form of a vector, with each component typically representing different dimensions or directions in space. To visualize this, think of a vector function as a rule that assigns a specific vector to every value of \( t \). This can be thought of as a path or trajectory in a multi-dimensional space.
In our problem, the vector function is given as\[ \mathbf{r}(t) = 3e^{t} \mathbf{i} + 2e^{-3t} \mathbf{j} + 4e^{2t} \mathbf{k} \]
This function maps different values of \( t \) to a vector composed of three components (\( \mathbf{i}, \mathbf{j}, \mathbf{k} \)) in 3D space. Each component is a function of \( t \) that applies exponential transformations. The trajectory is governed by how these components change as \( t \) changes, making it crucial to understand and work with vector functions in physics, engineering, and computer graphics.

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of the function in relation to its input variable. For a vector function, differentiation is applied component-wise to each part of the vector.
In the given exercise, we need to differentiate:

• \( 3e^t \mathbf{i} \)
• \( 2e^{-3t} \mathbf{j} \)
• \( 4e^{2t} \mathbf{k} \)

Each of these components is differentiated separately with respect to \( t \). Derivatives of exponential functions are particularly important here as they determine how each component grows or shrinks. Key differentiation rules, such as the chain rule and constant multiplication rule, play a crucial role.
The derivative of the vector function is:\[ \mathbf{r}'(t) = 3e^t \mathbf{i} - 6e^{-3t} \mathbf{j} + 8e^{2t} \mathbf{k} \]
This derivative, or the tangent vector, tells us the direction and rate of change of the vector function at any particular value of \( t \). It is particularly useful in understanding motion and trajectories in space.

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. In the context of a vector function, they can represent rates of growth or decay depending on the sign and magnitude of the exponent.
For the vector function \( \mathbf{r}(t) \), the exponential expressions are:

• \( e^t \) indicates growth with \( t \).
• \( e^{-3t} \) signifies decay as \( t \) increases.
• \( e^{2t} \) shows growth, but at a faster rate compared to \( e^t \).

Exponential functions are characterized by their rapid growth or decay, often making them suitable to model real-world phenomena like population growth, radioactive decay, or anything involving continuous compounding. In the problem, when estimating \( \mathbf{r}(t) \) at \( t = \ln(2) \), we utilize the property that \( e^{\ln(a)} = a \).
For instance:

• \( e^{\ln(2)} = 2 \)
• \( e^{-3\ln(2)} = 2^{-3} = \frac{1}{8} \)
• \( e^{2\ln(2)} = 2^2 = 4 \)

These calculations simplify evaluating the vector function and its derivative, reflecting how vital exponential functions are in analyzing changes in systems modeled by vector functions.