Problem 2 - Calculus III

Question

Given \(\mathbf{r}(t)=3 \sec t \mathbf{i}+2 \tan t \mathbf{j}\), find the following values (if possible). a. \(\mathbf{r}\left(\frac{\pi}{4}\right)\) b. \(\mathbf{r}(\pi)\) c. r \(\left(\frac{\pi}{2}\right)\)

Step-by-Step Solution

Answer

a. \(3\sqrt{2} \mathbf{i} + 2 \mathbf{j}\); b. \(-3 \mathbf{i}\); c. undefined.

Step 1: Evaluate the vector for t=π/4

First, plug in the value \( t = \frac{\pi}{4} \) into the vector function \( \mathbf{r}(t) = 3 \sec t \mathbf{i} + 2 \tan t \mathbf{j} \). Recall that \( \sec t \) is \( \frac{1}{\cos t} \) and \( \tan t \) is \( \frac{\sin t}{\cos t} \). Thus, we have:\[ \sec \left(\frac{\pi}{4}\right) = \frac{1}{\cos \left(\frac{\pi}{4}\right)} = \sqrt{2} \quad \text{and} \quad \tan \left(\frac{\pi}{4}\right) = 1. \]Thus, \( \mathbf{r} \left( \frac{\pi}{4} \right) = 3\sqrt{2} \mathbf{i} + 2 \mathbf{j}. \)

Step 2: Evaluate the vector for t=π

Next, substitute \( t = \pi \) into \( \mathbf{r}(t) = 3 \sec t \mathbf{i} + 2 \tan t \mathbf{j} \). Using the identities \( \sec(\pi) = -1 \) and \( \tan(\pi) = 0 \), you find:\[ \mathbf{r}(\pi) = 3(-1) \mathbf{i} + 2(0) \mathbf{j} = -3 \mathbf{i}. \]

Step 3: Determine if r(t) is defined for t=π/2

Finally, check if the function is defined for \( t = \frac{\pi}{2} \). At \( t = \frac{\pi}{2} \), \( \sec \left(\frac{\pi}{2}\right) = \frac{1}{0} \) and \( \tan \left(\frac{\pi}{2}\right) = \frac{1}{0} \), both of which lead to undefined expressions. Therefore, \( \mathbf{r} \left(\frac{\pi}{2}\right) \) is undefined.

Key Concepts

Trigonometric Functions

Trigonometric functions, like \(\sec t\) and \(\tan t\), play a crucial role in calculus, especially when dealing with vector functions. These functions are defined based on the relationships within a right triangle. \(\sec t\), or secant, is the reciprocal of \(\cos t\), which can be understood as \(\sec t = \frac{1}{\cos t}\). Likewise, \(\tan t\), or tangent, is defined as \(\tan t = \frac{\sin t}{\cos t}\).

In our exercise, evaluating these trigonometric functions correctly was key to solving for the vector \(\mathbf{r}(t)=3 \sec t \mathbf{i}+2 \tan t \mathbf{j}\). Here are some points to remember:

• At \(t = \frac{\pi}{4}\), the secant and tangent functions simplify nicely due to the specific values of sine and cosine, making calculations straightforward.
• Trigonometric identities such as \(\sec(\pi) = -1\) and \(\tan(\pi) = 0\) greatly simplify computations, showing how knowing basic trigonometric values can be quite handy.
• Recognize the potential for undefined expressions when the cosine of an angle equals zero, as this affects both secant and tangent.

Understanding these foundational trigonometric concepts equips you to evaluate vector functions effectively and accurately.

Vector Function Evaluation

Evaluating a vector function involves substituting specific values for the variable \(t\) within the function and simplifying the expression. In this exercise, we worked with the vector function \(\mathbf{r}(t) = 3 \sec t \mathbf{i} + 2 \tan t \mathbf{j}\). Here's how to evaluate it at specific points:

1. **Substitution and Simplification:** For each desired value of \(t\), replace \(t\) in the function with the specific value.2. **Use Trigonometric Values:** Insert the known trigonometric values (such as \(\sec(\frac{\pi}{4}) = \sqrt{2}\) or \(\tan(\pi) = 0\)) to simplify the vector.3. **Combine Vector Components:** Combine all parts of the vector into simplified form, resulting in an expression with coefficients for \(\mathbf{i}\) and \(\mathbf{j}\).Remember:

• Always pay close attention to trigonometric identities, as they are fundamental to the simplification process.
• The context of each value \(t\) can change outcomes significantly, so be mindful of angles that result in undefined outcomes.

Mastering these steps allows for efficient evaluation of vector functions, providing critical insights into their behaviors at various points.

Undefined Expressions in Calculus

Undefined expressions occur in calculus when you perform operations that are not mathematically valid. These often arise when denominators equal zero, as seen in this exercise with \(\sec\left(\frac{\pi}{2}\right)\) and \(\tan\left(\frac{\pi}{2}\right)\).

In vector calculus, it is crucial to recognize when expressions become undefined:

• If a trigonometric function approaches infinity (e.g., \(\sec(\frac{\pi}{2}) = \frac{1}{0}\)), it indicates that the value is undefined.
• When an expression, such as tangent or secant, involves division by zero, the function is not defined at that point.
• Be careful during evaluation; undefined expressions can disrupt calculations, requiring careful examination of the function.

Dealing with undefined expressions is a vital skill in calculus as it necessitates understanding the conditions under which certain operations fail. This understanding prevents misinterpretations in evaluations, ensuring accurate calculations and analyses of vector functions.