Problem 12 - Calculus 3

Question

Find the limit of the following vector-valued functions at the indicated value of \(t .\) $$ \lim _{t \rightarrow \pi / 6}\left\langle\cos ^{2} t, \sin ^{2} t, 1\right\rangle $$

Step-by-Step Solution

Answer

The limit is \( \left\langle \frac{3}{4}, \frac{1}{4}, 1 \right\rangle \).

Step 1: Understanding the Limit of a Vector-Valued Function

When finding the limit of a vector-valued function, we need to calculate the limit of each of the components individually as \( t \) approaches a particular value. In this exercise, we will find the limits of \( \cos^2 t \), \( \sin^2 t \), and the constant 1 as \( t \to \frac{\pi}{6} \).

Step 2: Calculating the First Component Limit

Calculate \( \lim_{t \to \pi/6} \cos^2 t \). Start by finding \( \cos(\pi/6) \), which is \( \frac{\sqrt{3}}{2} \). Then square this result to get \( \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \).

Step 3: Calculating the Second Component Limit

Calculate \( \lim_{t \to \pi/6} \sin^2 t \). First, find \( \sin(\pi/6) \), which is \( \frac{1}{2} \). Then square this result to get \( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \).

Step 4: Calculating the Constant Component

The third component is the constant 1, and its limit as \( t \rightarrow \pi/6 \) is simply 1.

Step 5: Combining the Results

Combine the limits of all components. The limit of the vector-valued function as \( t \rightarrow \pi/6 \) is \( \left\langle \frac{3}{4}, \frac{1}{4}, 1 \right\rangle \).

Key Concepts

Component-Wise Limits

When dealing with vector-valued functions, an important principle is the component-wise method of finding limits. This involves breaking down a vector into its individual components and separately evaluating the limit for each. In our problem, we have the vector function \( \left\langle \cos^2 t, \sin^2 t, 1 \right\rangle \), which has three components. To find the limit as \( t \) approaches a specific value, in this case \( \pi/6 \), we evaluate:

• The first component \( \cos^2 t \)
• The second component \( \sin^2 t \)
• The constant component 1

For each component:- We independently calculate the limit- Then combine the results to form the limit of the entire vector

This approach simplifies complex problems by allowing us to focus on one piece of the vector at a time. It helps clarify that the vector’s behavior is simply a collection of the behaviors of its components.

Cosine and Sine Functions

The fundamental trigonometric functions, cosine and sine, play a pivotal role in calculus and beyond. In this exercise, understanding these functions' values at specific points is crucial.
The trigonometric identity and harmony between cosine and sine are particularly useful when exploring their squared values.

• Cosine Function \(\cos(t)\): At \( t = \pi/6 \), the cosine value is \( \frac{\sqrt{3}}{2} \). For the given vector, we're interested in \( \cos^2(t) \), so we square \( \frac{\sqrt{3}}{2} \) to get \( \frac{3}{4} \).
• Sine Function \(\sin(t)\): At \( t = \pi/6 \), the sine value is \( \frac{1}{2} \). Thus, \( \sin^2(t) \) results in \( \frac{1}{4} \).

These calculations emphasize the periodic nature of trigonometric functions and demonstrate the seamless transition from function values to squared outcomes. This principle is often used in problems involving oscillations and circular motion.

Vector Calculus

Vector calculus extends the realm of calculus into multiple dimensions, which is essential for understanding physical phenomena like motion and forces. With vector-valued functions, each component may describe different dimensions of a problem: direction, magnitude, or both.
In this context, evaluating vectors often involves:

• Understanding how changes in one component relate to changes in others
• Introducing mathematical operations like limits, derivatives, and integrals to vector functions

For our exercise, this translates to evaluating limits in a multi-dimensional setting. Where traditional calculus might focus on a single variable or function, vector calculus requires us to consider how vectors evolve over time or space.
The scalar limit of each vector component informs the entire vector’s limit, revealing interactions between different dimensions. This comprehensive view is pivotal for fields like physics and engineering, where vectors explicate complex systems encountered in reality.