Problem 17 - Calculus Volume 3

Question

Find the domain of the vector-valued functions. Domain: \(\mathbf{r}(t)=\left\langle\csc (t), \frac{1}{\sqrt{t-3}}, \ln (t-2)\right\rangle\)

Step-by-Step Solution

Answer

Domain: \( (3, \infty) \setminus \{ n\pi \mid n \in \mathbb{Z} \} \).

Step 1: Understand the Constraints of Cosecant Function

The cosecant function, represented as \( \csc(t) \), is defined as the reciprocal of the sine function. Therefore, \( \csc(t) \) is undefined when \( \sin(t) = 0 \). Since sine is zero at integer multiples of \( \pi \) (i.e., \( t = n\pi \), where \( n \) is an integer), \( \csc(t) \) is undefined at these points. Thus, \( t eq n\pi \).

Step 2: Analyze the Square Root Function Constraint

For the term \( \frac{1}{\sqrt{t-3}} \), the term \( \sqrt{t-3} \) implies that \( t - 3 \) must be positive (i.e., \( t > 3 \)) to avoid division by zero or taking the square root of a negative number. This further means that \( t \) must be greater than 3.

Step 3: Consider the Natural Log Function

The natural logarithm, denoted by \( \ln(t-2) \), is defined only for positive numbers. Therefore, \( t-2 > 0 \), which simplifies to \( t > 2 \). However, since we know from Step 2 that \( t > 3 \), this constraint is already satisfied as it is stronger.

Step 4: Combine All Constraints for the Domain

Combine all restrictions from the previous steps. We have: \( t > 3 \) due to the square root and natural logarithm constraints, and \( t eq n\pi \) due to the cosecant constraint. Thus, the domain of the function \( \mathbf{r}(t) \) is: \( (3, \infty) \), excluding \( n\pi \), i.e., \( t \in (3, \infty) \setminus \{ n\pi \mid n \in \mathbb{Z} \} \).

Key Concepts

Cosecant Function

The cosecant function is a fascinating mathematical construct closely related to the sine function. It is denoted as \( \csc(t) \) and is defined as the reciprocal of sine:

• \( \csc(t) = \frac{1}{\sin(t)} \)

Because it is a reciprocal, the cosecant function is only defined when sine is non-zero. This means that \( \csc(t) \) is undefined when \( \sin(t) = 0 \). Sine equals zero at integer multiples of \( \pi \) (e.g., \( t = n\pi \), where \( n \) is an integer). Hence, the cosecant function has vertical asymptotes at these points.
Visually, the graph of the cosecant function approaches infinity when \( t \) nears the points where the sine function is zero.
Understanding these gaps is crucial because they directly affect the domain of expressions involving the cosecant function.

Natural Logarithm

The natural logarithm, symbolized as \( \ln(x) \), is a special type of logarithmic function with base \( e \), where \( e \approx 2.71828 \). It is particularly useful in various scientific and engineering contexts. Essential properties of the natural logarithm include:

• It is defined only for positive values. Thus, the expression \( \ln(t-2) \) requires \( t-2 > 0 \), meaning \( t > 2 \).
• The function grows logarithmically, which means it increases but at a decreasing rate.

The natural logarithm serves as the inverse of the exponential function, providing solutions to equations involving exponential growth or decay.
In our domain analysis, the constraint \( t > 2 \) from the natural logarithm is important, although other constraints can make it redundant when \( t > 3 \).
This shows the integral role the natural logarithm can play in defining the domain of more complex functions.

Square Root Function

The square root function is represented mathematically as \( \sqrt{x} \) and is a fundamental function in algebra and calculus. It extracts the principal (non-negative) square root of a given number. Key points to understand include:

• The function is defined only for non-negative inputs, which means \( \sqrt{x} \) is only valid for \( x \geq 0 \).
• In the case of \( \frac{1}{\sqrt{t-3}} \), not only must \( t-3 \) be non-negative, but to prevent division by zero, \( t-3 > 0 \).

This leads to a crucial constraint: \( t > 3 \). The requirement that \( t \) be strictly greater than 3 ensures the function remains in its defined domain and avoids undefined expressions like division by zero.
These properties help shape the domain of more complex vector-valued functions, demonstrating how the square root function's constraints guide calculations and analysis in various mathematical contexts.