91Ó°ÊÓ

In vector calculus, the velocity vector is a crucial concept when studying motion. It describes the direction and rate of change of an object's position over time. For a position function like \[ \mathbf{r}(t) = e^{-t} \mathbf{i} + t^{2} \mathbf{j} + \tan t \mathbf{k} \] the velocity vector, \(\mathbf{v}(t)\), is found by differentiating this position vector with respect to the parameter \(t\). This is because the derivative measures how a function changes as its variable changes, providing a snapshot of motion at any given instant. To find the velocity vector, we differentiate each component of the position vector:

• \( \frac{d}{dt}(e^{-t}) \mathbf{i} = -e^{-t} \mathbf{i}\)
• \( \frac{d}{dt}(t^2) \mathbf{j} = 2t \mathbf{j}\)
• \( \frac{d}{dt}(\tan t) \mathbf{k} = \sec^2 t \mathbf{k}\)

The resulting velocity vector is:\[ \mathbf{v}(t) = -e^{-t} \mathbf{i} + 2t \mathbf{j} + \sec^2 t \mathbf{k} \]This vector not only tells us the speed but also the direction at which the object is traveling along the x, y, and z axes.

Acceleration is defined as the rate at which an object’s velocity changes over time. In other words, it's a measure of how quickly the velocity vector itself is changing. To find the acceleration vector, \(\mathbf{a}(t)\), we differentiate the velocity vector with respect to \(t\). Continuing with our example:

• The derivative of \(-e^{-t} \mathbf{i}\) with respect to \(t\) is \(e^{-t} \mathbf{i}\)
• The derivative of \(2t \mathbf{j}\) is \(2 \mathbf{j}\)
• The derivative of \(\sec^2 t \mathbf{k}\) requires applying the chain rule, yielding \(2\sec^2 t \tan t \mathbf{k}\)

Thus, the acceleration vector is:\[ \mathbf{a}(t) = e^{-t} \mathbf{i} + 2 \mathbf{j} + 2\sec^2 t \tan t \mathbf{k} \]This vector describes not only how quickly the speed is changing but also how the direction of travel is shifting across the coordinate axes.

Speed is a scalar quantity, which means it has magnitude but no direction. It can be extracted from the velocity vector by calculating its magnitude. Using the velocity vector from our example:\[ \mathbf{v}(t) = -e^{-t} \mathbf{i} + 2t \mathbf{j} + \sec^2 t \mathbf{k} \]The formula for calculating speed is the square root of the sum of the squares of its components:\[ \text{Speed} = \sqrt{(-e^{-t})^2 + (2t)^2 + (\sec^2 t)^2} \]Upon simplifying:\[ \text{Speed} = \sqrt{e^{-2t} + 4t^2 + \sec^4 t} \]This value provides the magnitude of velocity, effectively describing how fast the object is moving regardless of its direction.

Differentiation is a fundamental tool in calculus used to determine how a function changes as its input changes. It is particularly useful in vector calculus when analyzing position functions to find velocity and acceleration vectors.

• Derivative of a constant: The derivative of a constant function is zero, as constants do not change.
• Power Rule: If \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\).
• Exponential Functions: The derivative of \(e^{-t}\) is \(-e^{-t}\).
• Trigonometric Functions: The derivative of \(\tan t\) is \(\sec^2 t\), derived from the quotient rule for trigonometric identities.

Differentiation reveals the rate of change in different directions, which is why it's essential for calculating both velocity and acceleration vectors.

The position function \(\mathbf{r}(t)\) describes the location of an object at any given time \(t\). It's a vector comprised of components along the \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) directions, often corresponding to three-dimensional coordinates.

• Exponential Component: \(e^{-t} \mathbf{i}\) might imply a decaying movement along the x-axis, as the exponential term decreases over time.
• Polynomial Component: \(t^2 \mathbf{j}\) indicates motion along the y-axis, with the speed increasing as time progresses due to the power of the polynomial.
• Trigonometric Component: \(\tan t \mathbf{k}\) represents the movement along the z-axis, influencing by the behavior of the tangent function.

Understanding the position function is the first step to predicting how an object moves through space over time, as it forms the basis for further derivatives that yield velocity and acceleration.