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Magazine Chapter 17: Problem 106
Are the statements true or false? Give reasons for your answer.
A parameterization of the graph of \(y=\ln x\) for \(x>0\) is given by \(x=e^{t},
y=t\) for \(-\infty
Short Answer
Expert verified
True, the parameterization represents the graph accurately.
Step by step solution
01
Understand the Parameterization
The parameterization given is \(x = e^t\) and \(y = t\) for \(-\infty < t < \infty\). A parameterization is a way of expressing the coordinates \((x, y)\) of points on a curve as functions of a variable, in this case, \(t\).
02
Substitute Parameter into Equation
Substituting \(x = e^t\) into the equation \(y = \ln{x}\), we get \(y = \ln{e^t}\). Since \(\ln{e^t} = t\), the expression becomes \(y = t\), which matches the parameterized \(y\).
03
Check the Domain
For the parameterization, \(x = e^t\) implies \(x > 0\) for all \(t\) since the exponential function \(e^t\) is always positive. This matches the requirement \(x > 0\) for the original function \(y = \ln{x}\).
04
Confirm the Range of Y
The parameter \(t\) varies from \(-\infty\) to \(\infty\), which means \(y = t\) can take any real number, covering the entire range of \(y\) for the given graph \(y = \ln{x}\).
05
Conclusion
Since both the functional form \( (y = \ln x) \) and the domain \( (x > 0) \) and range for \(y\) are correctly represented by the parameterization \(x = e^t, y = t\), the statement is true.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parameterization
In multivariable calculus, parameterization is a technique used to express a curve or surface within a certain space using equations that depend on one or more parameters, which are often labeled as variables such as \( t \) or \( u \). This provides a way to describe the geometric properties of an object, like a curve, without directly dealing with the implicit or explicit form of the equation.
- Curve Description: When a curve is parameterized, each point on the curve is described by a pair or set of functions. In the given exercise, the curve described by the graph of \( y = \ln x \) is parameterized as \( x = e^t, y = t \), where \( t \) is the parameter.
- Parameters: A parameter \( t \) acts as an additional variable that tunes the conditions of \( (x, y) \) in terms of other functions, like \( x = e^t \), which simplifies describing nonlinear curves or functions.
- Advantages: Parameterizations provide a convenient way to handle integrals, derivatives, and other calculations along a curve, which is particularly useful when analyzing complex shapes or paths in higher dimensions.
Exponential Functions
Exponential functions are a class of functions characterized by an equation of the form \( f(x) = a \cdot e^{bx} \), where \( e \) is the base of the natural logarithms, approximated as 2.71828, and \( a \) and \( b \) are constants. In the context of the parameterization in the exercise, \( x = e^t \) is identified as an essential exponential function.
- Exponential Growth: The function \( x = e^t \) exhibits exponential growth because for any real number \( t \), \( e^t \) is positive and increases rapidly, especially as \( t \) moves towards infinity.
- Domain and Range: The domain of \( e^t \) is all real numbers \( t \), and its range is all positive real numbers, which fits the condition \( x > 0 \) for the given function \( y = \ln x \).
- Inverse Function: The exponential function \( e^t \) is the inverse of the natural logarithm \( \ln x \). This relationship is fundamental in calculus, as it allows converting between these two mathematical operations easily.
Natural Logarithm
The natural logarithm, denoted as \( \ln x \), is the logarithm to the base \( e \). This function is the inverse of the exponential function \( e^x \). It occupies a special place in calculus owing to its fundamental properties and appearance in complex calculations.
- Inverse Relationship: An important property of the natural logarithm is that it's the inverse of the exponential function, which means \( \ln(e^t) = t \). This confirms that the parameterization given in the problem \( x = e^t, y = t \) accurately represents the curve \( y = \ln x \).
- Properties: Natural logarithms are defined for \( x > 0 \), with the output range being all real numbers \(-\infty < y < \infty\). This aligns with the parameter \( t \), which does the same in the parameterized form.
- Applications: The natural logarithm is useful for simplifying complex algebraic expressions and solving exponential growth and decay problems. It's widely used in economics, physics, and engineering.
