Problem 17 - Calculus 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: \((3, \infty) \setminus \{n\pi \mid n \in \mathbb{Z}, n\pi > 3\}\).

Step 1: Determine domain for \( \csc(t) \)

The function \( \csc(t) = \frac{1}{\sin(t)} \) is defined for all \( t \) except where \( \sin(t) = 0 \). The sine function is zero at integer multiples of \( \pi \), so \( t eq n\pi \) where \( n \) is an integer.

Step 2: Find the domain for \( \frac{1}{\sqrt{t-3}} \)

The expression \( \frac{1}{\sqrt{t-3}} \) demands that the denominator and the square root are defined and non-zero. Hence, \( t-3 > 0 \). Solving the inequality \( t-3 > 0 \) gives \( t > 3 \).

Step 3: Determine the domain restriction from \( \ln(t-2) \)

The logarithmic function \( \ln(t-2) \) is defined when \( t-2 > 0 \). Solving this inequality, we find that \( t > 2 \).

Step 4: Combine domain restrictions

To find the overall domain of the vector function \( \mathbf{r}(t) \), intersect the individual domains: \( t eq n\pi \) (step 1), \( t > 3 \) (step 2), and \( t > 2 \) (step 3). The most restrictive condition is \( t > 3 \), so we take \( t > 3 \) and exclude integer multiples of \( \pi \): \( t \in (3, \infty) \setminus \{n\pi \mid n \in \mathbb{Z}, n\pi > 3\} \).

Key Concepts

Cosecant Function

The cosecant function, represented as \( \csc(t) \), is essentially the reciprocal of the sine function. In mathematical terms, this is expressed as \( \csc(t) = \frac{1}{\sin(t)} \). Understanding the cosecant function is crucial because it is only defined when the sine function, \( \sin(t) \), is not equal to zero. This is because division by zero is undefined in mathematics.

The sine function reaches zero at integer multiples of \( \pi \) (such as \( 0, \pi, 2\pi, \ldots \)). Consequently, the cosecant function will be undefined at these points, namely when \( t = n\pi \), where \( n \) is an integer. This gives us a critical restriction to account for when determining the domain of any expression involving \( \csc(t) \).

To properly define the domain involving \( \csc(t) \), it is essential to exclude all values that make \( \sin(t) = 0 \). This specific exclusion forms a foundational understanding when working with trigonometric functions and their reciprocals. Recognizing these restrictions helps avoid any mathematical computation errors, ensuring that functions remain properly defined across the intervals examined.

Inequalities

Inequalities are mathematical expressions used to compare two quantities. When dealing with inequalities in functions such as \( \frac{1}{\sqrt{t-3}} \), significant domain considerations come into play. These types of functions require conditions that involve not dividing by zero and ensuring that values under radicals are non-negative for real number outputs.

When examining \( \frac{1}{\sqrt{t-3}} \), it is crucial to ensure the expression under the square root is positive, because the square root of a negative number is not a real number. Solving the inequality \( t-3 > 0 \) leads us to \( t > 3 \). This indicates that our domain must only include values of \( t \) that are strictly greater than 3. This inequality is essential in maintaining a valid real number output for the function.

Understanding and applying inequalities like these help in properly determining the domain of functions, especially when involving square roots and reciprocals. Maintaining these conditions avoids undefined or complex numbers, ensuring the practical applicability of the functions in question.

Logarithmic Functions

Logarithmic functions, such as \( \ln(t-2) \), carry with them particular domain restrictions. These functions dictate that the argument of the logarithm, \( t-2 \) in this case, must be greater than zero. This is due to the definition of a logarithm, which only applies to positive real numbers.

Solving \( t-2 > 0 \) leads to the inequality \( t > 2 \), implying that the function \( \ln(t-2) \) is only defined for values of \( t \) greater than 2. However, when combined with other expressions, such as \( \frac{1}{\sqrt{t-3}} \), we note that the stricter condition \( t > 3 \) arises from the need for both components to have a valid real number domain.

Logarithmic functions require attentive consideration of their definitions to uphold real number standards. By thoughtfully examining these restrictions, particularly through inequalities, we ensure that all expressions retain their essential properties and integrity, allowing for precise and consistent calculations.