Problem 35
Question
Graph the function. $$y=x \ln x$$
Step-by-Step Solution
Verified Answer
The function \(y=x \ln x\) has its root at x=1, undefined at x=0, decreases to negative infinity as x approaches 0 and increases slowly for \(x>1\). It crosses x-axis at \(x=1\) and keeps increasing for larger x-values.
1Step 1: Identify the key points and plot the root and y-intercept
The root of the function is where \(y=0\). For the function \(y=x \ln x\), setting \(y=0\), we get \(x=0, x=1\). So the function crosses the x-axis at \(x=0,1\). The y-intercept is the value of y when \(x=0\), but logarithm isn't defined for \(x=0\). So, there's no y-intercept.
2Step 2: Graph Behavior for Small and Large x-values
As x gets closer to 0 from the right (i.e., \(x > 0\)), \(y=x \ln x\) goes towards negative infinity. This is due to the natural logarithm's behavior near 0. As x goes larger, \(y = x \ln x\) grows, but slower than x. It is positive and increasing for \(x > 1\).
3Step 3: Drawing the graph
Combine all the information from the above two steps to graph the function. The graph goes downwards as it approaches zero from the right, crosses the x-axis at \( x = 1\), and continues to increase.
Key Concepts
Natural LogarithmFunction RootsEnd Behavior AnalysisAsymptotes in Functions
Natural Logarithm
The natural logarithm is a fundamental concept in mathematics, specifically dealing with exponential and logarithmic functions. It is denoted as \( \ln(x) \) and is defined as the inverse of the exponential function \( e^x \) where \( e \) is an irrational and transcendental number approximately equal to 2.71828. This means that if you have \( \ln(x) = y \) then \( e^y = x \).
The natural logarithm has unique properties that are particularly useful in calculus, such as the derivative of \( \ln(x) \) being \( 1/x \) and its integral resulting in \( x\ln(x) - x \) plus a constant of integration. It's also noteworthy that \( \ln(1) = 0 \) because \( e^0 = 1 \) and the natural logarithm of a product equals the sum of the logarithms \( \ln(xy) = \ln(x) + \ln(y) \). Understanding these properties is key to analyzing functions involving natural logarithms, like in the given exercise where the function \( y = x \ln x \) is evaluated.
The natural logarithm has unique properties that are particularly useful in calculus, such as the derivative of \( \ln(x) \) being \( 1/x \) and its integral resulting in \( x\ln(x) - x \) plus a constant of integration. It's also noteworthy that \( \ln(1) = 0 \) because \( e^0 = 1 \) and the natural logarithm of a product equals the sum of the logarithms \( \ln(xy) = \ln(x) + \ln(y) \). Understanding these properties is key to analyzing functions involving natural logarithms, like in the given exercise where the function \( y = x \ln x \) is evaluated.
Function Roots
Function roots, also known as zeros or x-intercepts, are the points where the function crosses or touches the x-axis. When given a function, finding its roots involves solving the equation \( f(x) = 0 \).
In the context of the provided exercise with the function \( y = x \ln x \), the root is found by setting \( y \) to zero and solving for \( x \). This yields \( x = 1 \) since \( \ln(1) = 0 \) and \( x\ln(x) \) would be 0. It’s important to remember that the logarithmic function is not defined for non-positive values of \( x \) and, therefore, the function does not have a root or an x-intercept at \( x = 0 \) despite the fact that \( x \ln x \) approaches 0 as \( x \) approaches 0 from the right.
In the context of the provided exercise with the function \( y = x \ln x \), the root is found by setting \( y \) to zero and solving for \( x \). This yields \( x = 1 \) since \( \ln(1) = 0 \) and \( x\ln(x) \) would be 0. It’s important to remember that the logarithmic function is not defined for non-positive values of \( x \) and, therefore, the function does not have a root or an x-intercept at \( x = 0 \) despite the fact that \( x \ln x \) approaches 0 as \( x \) approaches 0 from the right.
End Behavior Analysis
End behavior analysis refers to understanding how a function behaves as \( x \) approaches positive or negative infinity. The nature of a function’s end behavior can provide insight into its growth rate, direction, and overall shape far from the origin.
For the function \( y = x \ln x \) as \( x \) approaches 0 from the right, the function heads towards negative infinity due to the logarithmic term, which dramatically drops as it nears zero. As \( x \) increases beyond 1, the function tends to increase at a rate slower than \( x \) itself, indicated by the logarithmic function's tendency to grow slower as its input grows larger. This indicates a positive, yet decelerating growth as \( x \) tends towards infinity. By examining end behavior, one can predict the long-term trends of a function's graph.
For the function \( y = x \ln x \) as \( x \) approaches 0 from the right, the function heads towards negative infinity due to the logarithmic term, which dramatically drops as it nears zero. As \( x \) increases beyond 1, the function tends to increase at a rate slower than \( x \) itself, indicated by the logarithmic function's tendency to grow slower as its input grows larger. This indicates a positive, yet decelerating growth as \( x \) tends towards infinity. By examining end behavior, one can predict the long-term trends of a function's graph.
Asymptotes in Functions
Asymptotes in functions are lines that the graph of a function approaches but never actually reaches or crosses. These can be vertical, horizontal, or slant (oblique) and provide critical structural details about the function's graph.
In the case of the function \( y = x \ln x \), there is a vertical asymptote at \( x = 0 \). This is because \( \ln x \) goes to negative infinity as \( x \) approaches 0 from the right, but \( \ln x \) is not defined for \( x \leq 0 \), which means the function itself does not exist to the left of \( x = 0 \). This behavior creates a boundary that the graph approaches infinitely but never crosses, forming a vertical asymptote here. Recognizing asymptotes is essential for accurately sketching the shape and position of a function's graph.
In the case of the function \( y = x \ln x \), there is a vertical asymptote at \( x = 0 \). This is because \( \ln x \) goes to negative infinity as \( x \) approaches 0 from the right, but \( \ln x \) is not defined for \( x \leq 0 \), which means the function itself does not exist to the left of \( x = 0 \). This behavior creates a boundary that the graph approaches infinitely but never crosses, forming a vertical asymptote here. Recognizing asymptotes is essential for accurately sketching the shape and position of a function's graph.
Other exercises in this chapter
Problem 34
Use mathematical induction to prove that \(\sum_{i=1}^{n} i^{5}=\frac{n^{2}(n+1)^{2}\left(2 n^{2}+2 n-1\right)}{12}\) for all integers \(n \geq 1.\)
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Evaluate the definite integral. $$\int_{0}^{2} \frac{e^{x}}{1+e^{2 x}} d x$$
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The data come from a pneumotachograph, which measures air flow through the throat (in liters per second). The integral of the air flow equals the volume of air
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Find an equation of the tangent line at the given value of \(x\) $$y=\int_{2}^{x} \cos \left(\pi t^{3}\right) d t, x=2$$
View solution