Problem 76 - Calculus 3

Question

Find the velocity vector for the function \(\mathbf{r}(t)=\left\langle e^{t}, e^{-t}, 0\right\rangle\)

Step-by-Step Solution

Answer

The velocity vector is \( \mathbf{v}(t) = \left\langle e^t, -e^{-t}, 0 \right\rangle \).

Step 1: Understand the Problem

First, we need to find the velocity vector for the given vector function \( \mathbf{r}(t) = \langle e^t, e^{-t}, 0 \rangle \). The velocity vector is the derivative of the position vector with respect to time \( t \), represented as \( \mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt} \).

Step 2: Differentiate the First Component

Take the derivative of the first component \( e^t \). The derivative of \( e^t \) is simply \( e^t \), as the derivative of the exponential function is the function itself.

Step 3: Differentiate the Second Component

Now, differentiate the second component \( e^{-t} \). The derivative of \( e^{-t} \) is \( -e^{-t} \), as it involves the chain rule: the derivative of the exponent \(-t\) is \(-1\), so the derivative of the whole component is \(-e^{-t} \).

Step 4: Differentiate the Third Component

The third component is a constant \( 0 \). The derivative of a constant is \( 0 \).

Step 5: Combine the Derivatives

Combine the derivatives found in the previous steps to form the velocity vector: \( \mathbf{v}(t) = \left\langle e^t, -e^{-t}, 0 \right\rangle \).

Key Concepts

Velocity Vector

The velocity vector is a fundamental concept in calculus and physics, describing how an object's position changes over time. Imagine a spaceship traveling through space. Instead of only knowing where it is (position), we want to understand how fast and in what direction it's moving. This is what the velocity vector tells us.

The velocity vector is defined as the time derivative of the position vector. If you have a position vector \( \mathbf{r}(t) \), which tells you the position of an object at a specific time \( t \), the velocity vector \( \mathbf{v}(t) \) is:

• Formed by differentiating each component of \( \mathbf{r}(t) \).
• Represents both the speed and direction of the motion.

In simpler terms, differentiating the position vector gives you how quickly each component of the position changes, which, combined, shows the complete velocity behavior of the object.

Position Vector

A position vector is a vector that gives the coordinates of a point in space relative to an origin. Think about using a map; when you pinpoint any location using coordinates, you're essentially using a position vector.

This concept is essential in describing the location of objects in fields like physics and engineering. A position vector \( \mathbf{r}(t) \) not only identifies a point in space but also allows us to model dynamic systems, by expressing the point's position as a function of time \( t \).

For example, if \( \mathbf{r}(t) = \langle e^t, e^{-t}, 0 \rangle \), it means that at any time \( t \), the x-coordinate is \( e^t \), the y-coordinate is \( e^{-t} \), and the z-coordinate remains constant at 0. This function captures how the position evolves as time progresses.

Exponential Function

The exponential function is a mathematical function denoted as \( e^x \), where \( e \) is a constant approximately equal to 2.71828. This function is quite unique because its derivative is itself, meaning that the rate of change of the function at any point is proportional to the value of the function itself.

In the context of finding derivatives for vector components, exponential functions frequently appear because they model exponential growth or decay, which is common in natural phenomena.

• For example, for the component \( e^t \), the derivative is simply \( e^t \).
• For decreasing functions like \( e^{-t} \), the derivative is \(-e^{-t}\), applying the chain rule.

Understanding these derivatives is crucial for finding velocity vectors from position vectors, as seen in our exercise with \( \mathbf{r}(t) = \langle e^t, e^{-t}, 0 \rangle \). The exponential function's unique properties make calculations smooth and predictable.