Problem 20 - Calculus III

Question

Find the domain of \(\mathbf{r}(t)=2 e^{-t} \mathbf{i}+e^{-t} \mathbf{j}+\ln (t-1) \mathbf{k}\).

Step-by-Step Solution

Answer

The domain is \((1, \infty)\).

Step 1: Identify Restrictions on the Exponential Part

The terms involving the exponential function, such as \(2 e^{-t} \mathbf{i}\) and \(e^{-t} \mathbf{j}\), have no restrictions on \(t\) because the exponential function \(e^{-t}\) is defined for all real numbers \(t\). Therefore, these terms contribute no restrictions to the domain.

Step 2: Analyze the Logarithmic Term

The term \(\ln(t-1) \mathbf{k}\) involves the natural logarithm function. The logarithm \(\ln(x)\) is defined only for \(x > 0\), so \(t-1 > 0\) must hold. This implies \(t > 1\) is required to ensure the argument of the logarithm is positive.

Step 3: Combine the Conditions

Considering both parts of the vector function, the only restriction comes from the logarithmic term \(\ln(t-1)\). Thus, the vector function \( \mathbf{r}(t) \) is defined for \(t > 1\).

Step 4: State the Domain

The domain of the vector function \(\mathbf{r}(t)=2 e^{-t} \mathbf{i}+e^{-t} \mathbf{j}+\ln(t-1) \mathbf{k}\) is all \(t\) such that \(t > 1\). So, in interval notation, the domain is \((1, \infty)\).

Key Concepts

Exponential Functions

Exponential functions are incredibly important in mathematics due to their unique properties and applications. An exponential function is generally expressed as \( f(x) = a e^{bx} \), where \( e \) is Euler's number (approximately 2.718). A standout feature of exponential functions is their growth or decay characteristics.

Key aspects of exponential functions:

• **Growth and Decay:** When the exponent's coefficient \( b \) is positive, the function represents exponential growth. A negative \( b \) indicates exponential decay, as seen in \( e^{-t} \).
• **Domain:** Importantly, exponential functions like \( e^{-t} \) are defined for all real numbers, meaning there are no restrictions on \( t \) based on the exponential part alone.
• **Graph Behavior:** The graph of an exponential function is continuously smooth and never touches the x-axis—it just approaches it as \( x \) becomes very large or very small (depending on the sign of \( b \)).

Exponential functions are used all around us: in population growth models, radioactive decay, and even finance to calculate compound interest. Their ability to model real-world phenomena quickly and accurately makes them a valuable tool in various fields.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. A common form is \( y = \ln(x) \), where \ln denotes the natural logarithm, logarithm base \( e \). The logarithm answers the question, "to what exponent must we raise \( e \) to obtain \( x \)?"

Some essential points about logarithmic functions:

• **Domain:** Logical functions are only defined for positive arguments. Therefore, \( \ln(t-1) \) requires that \( t-1 > 0 \), which simplifies to \( t > 1 \).
• **Basic Properties:** Logarithms convert multiplication into addition, division into subtraction, and exponentiation into multiplication, making them useful for simplifying calculations and solving equations.
• **Graph:** The graph of a logarithmic function passes through the point (1,0) since \( \ln(1) = 0 \). It continuously increases but at a decreasing rate and has a vertical asymptote at \( x = 0 \).

In practice, logarithmic functions find applications in measuring the intensity of earthquakes (Richter scale), sound (decibels), and in computational algorithms where they help handle large numerical values efficiently.

Interval Notation

Interval notation is an efficient way of describing a set of numbers on the real number line. It is especially useful when stating the domain or range of functions.

Here's a simple guide to understanding interval notation:

• **Parentheses \(( )\) and Brackets \([ ]\):** Parentheses \(( )\) are used when a boundary is not included in the interval, indicating an open interval. Brackets \([ ]\) indicate a closed interval, including the endpoint.
• **Types of Intervals:**
  • *Open Interval:* \((a, b)\) means all numbers greater than \( a \) and less than \( b \).
  • *Closed Interval:* \([a, b]\) includes all numbers from \( a \) to \( b \), including \( a \) and \( b \) themselves.
  • *Half-Open Interval:* \((a, b]\) or \([a, b)\) includes one endpoint but not the other.
• **Infinity:** Because we can't reach infinity, \((\infty)\) and \( (-\infty)\) use parentheses to denote the interval extends indefinitely, but does not "include" infinity.

Interval notation provides a clear and concise way to write down the domain of functions, such as \((1, \infty)\) for \( \ln(t-1) \) where \( t > 1 \), making it a valuable tool in mathematics.