Problem 2 - Calculus Volume 3

Question

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

Step-by-Step Solution

Answer

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

Step 1: Evaluate at π/4 (Part a)

To find \( \mathbf{r}\left(\frac{\pi}{4}\right) \), substitute \( t = \frac{\pi}{4} \) into the vector function: \( \mathbf{r}(t) = 3 \sec t \mathbf{i} + 2 \tan t \mathbf{j} \). First, calculate \( \sec \frac{\pi}{4} = \frac{1}{\cos \frac{\pi}{4}} = \sqrt{2} \). Next, calculate \( \tan \frac{\pi}{4} = 1 \). Thus, \( \mathbf{r}\left(\frac{\pi}{4}\right) = 3\sqrt{2} \mathbf{i} + 2 \mathbf{j} \).

Step 2: Evaluate at π (Part b)

To find \( \mathbf{r}(\pi) \), substitute \( t = \pi \) into the vector function. Calculate \( \sec \pi = \frac{1}{\cos \pi} = -1 \) and \( \tan \pi = 0 \). Thus, \( \mathbf{r}(\pi) = 3(-1) \mathbf{i} + 2(0) \mathbf{j} = -3 \mathbf{i} \).

Step 3: Evaluate at π/2 (Part c)

Substitute \( t = \frac{\pi}{2} \) into the vector function. Calculate \( \sec \frac{\pi}{2} = \frac{1}{\cos \frac{\pi}{2}} \), which is undefined, and \( \tan \frac{\pi}{2} \), which is also undefined. As both components are undefined, \( \mathbf{r}\left(\frac{\pi}{2}\right) \) does not exist.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in topics involving angles and periodic phenomena. They use the ratios of sides in a right triangle to describe angles and are widely used in calculus and physics. The primary trigonometric functions include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)), along with their reciprocals: cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)). For an angle \(\theta\), these functions are defined as:

• Sine: \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\)
• Cosine: \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\)
• Tangent: \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\)
• Secant: \(\sec \theta = \frac{1}{\cos \theta}\)

These functions are periodic and have specific values for common angles. For example, \(\sec\left(\frac{\pi}{4}\right)=\sqrt{2}\), revealing how trigonometric functions often involve square roots and simple fractions. When evaluating these functions at specific angles, you should also recognize where functions become undefined, such as when dividing by zero, which happens for \(\sec\left(\frac{\pi}{2}\right)\).
Understanding these properties helps in evaluating vector-valued functions that include trigonometric components.

Vector-Valued Functions

Vector-valued functions extend the concept of functions to vectors, where each input, often parameterized by a variable like \(t\), produces a vector. This is essential in calculus as it allows the representation of curves and motion in multi-dimensional spaces.
A vector-valued function is generally expressed as:\[\mathbf{r}(t) = f(t)\mathbf{i} + g(t)\mathbf{j} + h(t)\mathbf{k}\]where \(f(t), g(t),\) and \(h(t)\) are scalar functions, and \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) are unit vectors in the Cartesian coordinate system.
Specific functions, such as \(\mathbf{r}(t) = 3 \sec t \mathbf{i} + 2 \tan t \mathbf{j}\), show how vector components can be composed of trigonometric functions, which dictate the vector's position in a plane.
Calculating the output for a given \(t\) means substituting \(t\) into each component function and evaluating them individually. This allows us to describe dynamic systems, like oscillations or rotations, comprehensively.

Limits of Functions

Limits in calculus help us understand the behavior of functions as they approach a certain point. This becomes crucial when dealing with functions that might not be well-defined at specific inputs, but their limiting behavior offers vital insights.
In terms of function \(f(t)\), the limit as \(t\) approaches a value \(a\) is denoted as \(\lim_{t \to a} f(t)\). This concept not only allows the analysis of function near undefined points but also helps in dealing with infinities.
When examining vector-valued functions like \(\mathbf{r}(t)\), the limit must be considered for each component:

• If \(\lim_{t \to a} f(t)\) or \(g(t)\) is undefined, the whole vector \(\mathbf{r}(t)\) might not exist at \(t=a\). For instance, at \(t = \frac{\pi}{2}\), both \(3 \sec t\) and \(2 \tan t\) are undefined, illustrating when vector functions are incomplete.
  

Understanding limits is pivotal in calculus to predict the behavior and paths of objects described by vector-valued functions.