Problem 27 - Calculus III

Question

The position vector of a particle is \(\mathbf{r}(t)=5 \sec (2 t) \mathbf{i}-4 \tan (t) \mathbf{j}+7 t^{2} \mathbf{k}\) a. Graph the position function and display a view of the graph that illustrates the asymptotic behavior of the function. b. Find the velocity as \(t\) approaches but is not equal to \(\pi / 4\) (if it exists).

Step-by-Step Solution

Answer

The velocity is undefined at \( t \to \frac{\pi}{4} \) due to asymptotes.

Step 1: Understanding the Position Vector

The given position vector is \( \mathbf{r}(t) = 5 \sec(2t) \mathbf{i} - 4 \tan(t) \mathbf{j} + 7t^2 \mathbf{k} \). This represents the particle's position over time in 3D space.

Step 2: Graphing the Position Function

To graph the position function, recognize that \( \sec(2t) \) and \( \tan(t) \) are trigonometric functions with asymptotes. The function will have vertical asymptotes where these functions are undefined. Use a graphing tool to sketch the function over a chosen interval. The asymptotic behavior is visible at points where \( \tan(t) \) and \( \sec(2t) \) approach infinity.

Step 3: Calculating the Derivative for Velocity

The velocity vector \( \mathbf{v}(t) \) is the derivative of the position vector \( \mathbf{r}(t) \) with respect to time \( t \). We find: \( \mathbf{v}(t) = \frac{d}{dt}[5 \sec(2t)] \mathbf{i} + \frac{d}{dt}[-4 \tan(t)] \mathbf{j} + \frac{d}{dt}[7t^2] \mathbf{k} \).

Step 4: Derivatives of Trigonometric Functions

Find the derivatives: \[ \frac{d}{dt}[5 \sec(2t)] = 10 \sec(2t) \tan(2t), \quad \frac{d}{dt}[-4 \tan(t)] = -4 \sec^2(t), \quad \frac{d}{dt}[7t^2] = 14t \].

Step 5: Express Velocity Vector

Combine the derivatives to express \( \mathbf{v}(t) \): \[ \mathbf{v}(t) = (10 \sec(2t) \tan(2t)) \mathbf{i} - (4 \sec^2(t)) \mathbf{j} + (14t) \mathbf{k} \].

Step 6: Finding Velocity at \( t \to \pi/4 \)

As \( t \to \frac{\pi}{4} \), evaluate each component of \( \mathbf{v}(t) \). The term with \( \sec(t) \) will tend to infinity since \( \sec(\pi/4) \) and \( \tan(\pi/2) \) are undefined at multiples of \( \pi/2 \). Thus, the velocity does not exist as it approaches infinity.

Key Concepts

Velocity Vector

The velocity vector is a fundamental concept in physics and mathematics that describes both the speed and direction in which an object moves. When we have a position vector given by a function of time, \(\mathbf{r}(t)\), like \(\mathbf{r}(t) = 5 \sec(2t) \mathbf{i} - 4 \tan(t) \mathbf{j} + 7t^2 \mathbf{k}\), the corresponding velocity vector, \(\mathbf{v}(t)\), is the derivative of the position vector with respect to time.To find this derivative, calculate the rate of change of each component of the position vector separately. This involves differentiating trigonometric functions, which can introduce unique challenges.- Understanding a velocity vector helps us describe motion fully.- It's crucial for predicting future motion.- This vector gives a complete picture of movement in three-dimensional space.

Derivative of Trigonometric Functions

The derivative of trigonometric functions is a fundamental topic that involves finding how these functions change over time. For the position vector given in our exercise, we are concerned with trigonometric functions such as \(\sec(2t)\) and \(\tan(t)\).When differentiating these functions, we utilize specific rules:- The derivative of \(\sec(u)\) gives us \(\sec(u)\tan(u)\frac{du}{dt}\), which, for \(\sec(2t)\), becomes \(10 \sec(2t)\tan(2t)\), as the chain rule requires we multiply everything by the derivative of \(2t,\) which is \(2\).- The derivative of \(\tan(t)\) is \(\sec^2(t).\)Understanding the derivatives of these functions means you can anticipate how they will behave over time, which is crucial for predicting motion, especially in physics.

Asymptotic Behavior

Asymptotic behavior refers to how a function behaves as it approaches certain critical points, often where it becomes undefined or approaches infinity. In the context of the exercise, this is particularly important when considering the \(\sec(2t)\) and \(\tan(t)\) components, which have vertical asymptotes when they reach specific values.- For \(\sec(2t),\) it is undefined where \(\cos(2t) = 0.\)- For \(\tan(t),\) it's undefined where \(\cos(t) = 0.\)These asymptotes result in the functions tending towards infinity. In physical applications, such behavior is significant as it can indicate points of non-existence in data or scenarios where calculations yield infinite results. Therefore, understanding asymptotic behavior is crucial for graphing and interpreting these functions.

3D Graphing

Graphing in three dimensions can appear daunting but is a skill that allows us to visually interpret complex functions, such as the given position vector \(\mathbf{r}(t)\). This vector describes a particle's motion in 3D space, which is best visualized through a 3D graph.When graphing:- **Identify each component**: Use \(\mathbf{i}, \mathbf{j},\) and \(\mathbf{k}\) to represent the x, y, and z axes, respectively.- **Understand asymptotic behavior**: Recognize areas where the functions become undefined, usually indicated by vertical asymptotes.3D graphing helps increase comprehension of how each component interacts over time, providing insights into the effects of motion in space. It becomes simpler to predict and verify graph behavior using such visual tools.