Problem 55 - Calculus III

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Components

The vector-valued function \( \mathbf{r}(t) = \langle \sin(t), \ln(t), \sqrt{t) \rangle \) has three components: \( \sin(t) \), \( \ln(t) \), and \( \sqrt{t} \). To find the domain of \( \mathbf{r}(t) \), you need to determine the domains of each of these components.

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

The function \( \sin(t) \) is defined for all real numbers, \( t \in (-\infty, \infty) \). There are no restrictions on \( t \) for this component.

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

The natural logarithm \( \ln(t) \) is defined when \( t > 0 \). Therefore, \( t \) must be positive for this component.

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

The square root function \( \sqrt{t} \) is defined for \( t \geq 0 \). This means \( t \) must be zero or positive.

Step 5: Intersection of Domains

To find the overall domain of \( \mathbf{r}(t) \), take the intersection of the domains from steps 2, 3, and 4. The strictest condition is \( t > 0 \) from the \( \ln(t) \) component. Therefore, \( \mathbf{r}(t) \) is defined for \( t > 0 \).

Key Concepts

Domain of Functions

The domain of a function refers to the set of all possible input values for which the function is defined. In simpler terms, it tells us the values that we can plug into the function without running into any mathematical issues like division by zero or taking the logarithm of a negative number. For vector-valued functions, like \( \mathbf{r}(t) = \langle \sin(t), \ln(t), \sqrt{t) \rangle \), we determine the domain by looking at each component of the vector in turn.

• \( \sin(t) \) is defined for all real numbers, meaning you can put any number into it and it will give you an output.
• \( \ln(t) \) requires its input to be strictly greater than zero, since the logarithm of zero or a negative number is undefined.
• \( \sqrt{t} \) is only defined for zero or positive values, as trying to take the square root of a negative number would involve imaginary numbers.

When determining the domain of the entire vector function, you have to consider all these constraints together. The strictest constraint here comes from \( \ln(t) \), indicating that \( t \) must be greater than zero.

Natural Logarithm Domain

The natural logarithm function, represented as \( \ln(x) \), is a powerful mathematical tool but comes with specific conditions for its input. It is defined only when its argument is greater than zero.

• If \( x \) is positive, \( \ln(x) \) will produce a real number output.
• If \( x \) is zero or negative, \( \ln(x) \) becomes undefined, which means there is no real number result.

This is crucial to remember when evaluating \( \ln(t) \) as part of a vector-valued function. To ensure we remain in the realm of real numbers, \( t \) must always be greater than zero whenever natural logarithms are involved. Failure to ensure this would lead to undefined behavior in mathematical computations.

Square Root Domain

The square root function, represented as \( \sqrt{x} \), also has specific rules governing what inputs it can take. It is vital to follow these rules to avoid stepping into undefined or complex number territories.

• \( \sqrt{x} \) gives a real number output when \( x \geq 0 \).
• An input of zero will produce a simple result, \( \sqrt{0} = 0 \).
• If \( x \) is negative, \( \sqrt{x} \) steps into the domain of complex numbers, which is typically not considered for domain purposes unless specially mentioned.

In cases like the function \( \sqrt{t} \), \( t \) needs to be at least zero. This ensures the square root part of the vector remains within the realm of real numbers, making the function "well-behaved" and easy to work with in real-valued contexts.