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Magazine Chapter 13: Problem 13
(a) Find an equation in \(x\) and \(y\) whose graph contains the points on the curve \(C .\) /b ) Sketch the graph of \(C\) and indicate the orientation. $$ x=t^{2}, \quad y=2 \ln t ; \quad t>0 $$
Short Answer
Expert verified
The equation of the curve is \( y = \ln(x) \).
Step by step solution
01
Solve for Parameter t in Terms of x
Given the equation for x is \( x = t^2 \). To find \( t \) in terms of \( x \), solve for \( t \):\[ t = \sqrt{x} \].
02
Substitute t in y Equation
We have \( y = 2 \ln t \). Substitute \( t = \sqrt{x} \) into the y equation to express \( y \) in terms of \( x \):\[ y = 2 \ln(\sqrt{x}) \].
03
Simplify the Expression for y
Simplify the expression \( y = 2 \ln(\sqrt{x}) \) using the property of logarithms \( \ln(a^b) = b \ln(a) \): \[ y = \ln(x) \].
04
Write the Equation of the Curve
Now that we have \( y \) in terms of \( x \), the equation of the curve \( C \) is \[ y = \ln(x) \].
05
Sketch the Graph of the Curve
The curve \( y = \ln(x) \) is the natural logarithm function which has a domain of \( x > 0 \). This curve passes through the point \( (1,0) \) and increases slowly for increasing \( x \). Indicate the orientation by noting that as \( t \) increases from 0 to infinity, \( x = t^2 \) increases from 0 to infinity and \( y = 2 \ln t \) also increases.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Natural Logarithm Function
The natural logarithm function, denoted as \( \ln(x) \), is a fundamental mathematical function with a base of the number \( e \), approximately equal to 2.718. It is the inverse of the exponential function \( e^x \). The natural logarithm measures the time it takes for something to grow to a particular level under continuous growth conditions. This function is widely used in everyday applications, science, and engineering.
- Properties: The natural logarithm is defined for all positive real numbers \( x > 0 \).
- Base: Its base, \( e \), is an irrational number and a limit of \( (1 + \frac{1}{n})^n \) as \( n \to \infty \).
- Zero point: The function intersects the x-axis at \( (1, 0) \) because \( \ln(1) = 0 \).
Graph Sketching
Graph sketching involves translating an equation into a visual representation on a coordinate plane. For the function \( y = \ln(x) \), this means mapping the transformation of x-values into y-values that represent the natural logarithm.
To sketch this graph, follow these steps:
To sketch this graph, follow these steps:
- Identify the Domain: Since logarithm functions are only defined for positive values, start your graph where \( x > 0 \).
- Plot Key Points: Begin with the familiar points such as \( (1, 0) \), then calculate \( y = \ln(x) \) for selected values of \( x \) greater than 1, like \( (e, 1) \) since \( \ln(e) = 1 \).
- Draw the Curve: Connect the points smoothly, creating a curve that rises to the right but at a decreasing rate, illustrating the natural logarithm's characteristic slope that approaches zero.
Curve Orientation
Curve orientation refers to the direction that a curve travels as its parameter, often \( t \), increases. In this context, the curve described by \( x = t^2 \) and \( y = 2 \ln(t) \) involves a parameter \( t \), which indicates how the curve is traced through its space.
As \( t \) increases from 0 to infinity:
As \( t \) increases from 0 to infinity:
- The value of \( x = t^2 \) increases from 0 upwards.
- The value of \( y = 2 \ln(t) \) also increases, reflecting growth both in the x-direction (positive) and the y-direction (positive).
- The graph鈥檚 orientation points to the right and upward.
Logarithmic Transformation
A logarithmic transformation is a mathematical application that involves applying a logarithm to a function or dataset to linearize growth and ease data interpretation. Through transformation, complex multiplicative relationships become more comprehensible.
- Purpose: By converting exponential growth problems into linear plots, it's easier to visualize and analyze data.
- Transformation Example: In our exercise, substituting \( t = \sqrt{x} \) into \( y = 2 \ln(t) \) to get \( y = \ln(x) \) is an example of how logarithms can simplify expressions, enhancing understanding and analytic capabilities.
- Reduced Complexity: This transformation turns multiplicative relationships into additive ones, bringing simplicity to various mathematical models.
