Problem 206 - Calculus Volume 3

Question

Find the domains of the vector-valued functions. \(\quad \mathbf{r}(t)=\langle\sin (t), \ln (t), \sqrt{t}\rangle\)

Step-by-Step Solution

Answer

The domain of \( \mathbf{r}(t) \) is \( (0, \infty) \).

Step 1: Understand the Components

The vector-valued function \( \mathbf{r}(t) = \langle \sin (t), \ln (t), \sqrt{t} \rangle \) consists of three components: \( \sin (t) \), \( \ln (t) \), and \( \sqrt{t} \). Each component has its own domain requirements.

Step 2: Domain of \( \sin(t) \)

The function \( \sin(t) \) is defined for all real numbers. Therefore, its domain is \( (-\infty, \infty) \). This means that \( \sin(t) \) does not impose any restrictions on \( t \).

Step 3: Domain of \( \ln(t) \)

The function \( \ln(t) \) is defined for \( t > 0 \). Therefore, to meet the conditions for \( \ln(t) \), \( t \) must be greater than zero.

Step 4: Domain of \( \sqrt{t} \)

The function \( \sqrt{t} \) is defined for \( t \geq 0 \). This means \( t \) must be non-negative. It further limits the domain to \( [0, \infty) \).

Step 5: Find the Common Domain

To find the overall domain of \( \mathbf{r}(t) \), we need the intersection of the domains from each component. The strictest requirement is \( t > 0 \) from \( \ln(t) \). Therefore, the domain of \( \mathbf{r}(t) \) is \( (0, \infty) \).

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (usually real numbers) for which the function is defined. For each component of a vector-valued function, you must consider the domain of each individual function.

In our exercise, the function in question is a vector-valued function, involving three separate components:

• \( \sin(t) \)
• \( \ln(t) \)
• \( \sqrt{t} \)

Each of these functions has specific constraints. When determining the domain of the combined vector-valued function, it is crucial to look for the intersection of the domains of the individual components. Only input values that satisfy all components' domain criteria will be included in the final domain of the vector-valued function.

Understanding that each function may have a different restriction is key, and finding the common ground among them involves seeking the most restrictive conditions. In this case, the overall function imposes a domain of \((0, \infty)\), driven by the \(\ln(t)\) component, which only allows for positive input values. Remember, analyzing each piece thoroughly and combining those insights leads to an accurate understanding of the function's domain.

Trigonometric Functions

Trigonometric functions, like \(\sin(t)\), play an integral role in mathematics, especially in contexts involving periodic behavior. These functions are defined for all real numbers, which gives them a broad domain compared to other mathematical functions that we encounter.

The sine function, \(\sin(t)\), for instance, is periodic with a period of \(2\pi\), meaning it repeats its values in regular intervals. This feature makes it particularly useful in modeling phenomena like waves and circular motion.

• \(\sin(t)\) outputs values in the range [-1, 1]. Due to its wide domain, it doesn't impose any additional domain restrictions when part of a vector-valued function.
• Its behavior is predictable, and it can handle any real number input without any change in definition or need for restriction.
• This inherent property simplifies many analyses when \(\sin(t)\) is a component of larger functions because it does not limit possible inputs by itself.

This broad applicability is a boon when determining the domains of compound functions. Considering \(\sin(t)\)'s constant presence in periodic models highlights how trigonometric functions handle diverse situations.

Logarithmic Functions

Logarithmic functions, like \(\ln(t)\), have unique characteristics that influence their domains significantly. The natural logarithm \(\ln(t)\) is only defined for positive real numbers, hence its domain looks like \((0, \infty)\). This condition exists because you cannot take the logarithm of zero or a negative number, which would result in undefined real values.

Let's delve into some critical aspects:

• Each logarithmic function has a base; \(\ln(t)\) specifically uses the natural base \(e\), which is approximately \(2.718\).
• Logarithms transform multiplicative relationships into additive ones, and this property is particularly useful in solving equations and analyzing exponential growth or decay.
• Due to their inherent properties, logarithmic functions are not defined for zero or negative inputs, which directly impacts their domain.

Logarithmic functions often come up in diverse fields including science and engineering, as they solve many practical problems associated with exponential growth rates. This defined domain of \((0, \infty)\) profoundly impacts calculations, especially in the case of determining the domains of vector-valued functions where each component has its restrictions. For our vector function, the \(\ln(t)\) function becomes the most restrictive component, setting the limitations for the domain of \(\mathbf{r}(t)\) to \((0, \infty)\).