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Chapter 9: Problem 8

Draw a sketch of the graph of the curve having the given equation. \(y=x \ln x\)

Short Answer

Expert verified
The curve \(y = x \, \ln x\) starts just above x-axis at x > 0, passes through (1, 0), and rises without bound.

Step by step solution

01

Understand the equation

The given equation is \(y = x \, \ln x\). This represents a curve where \(x\) is multiplied by the natural logarithm of \(x\).
02

Identify the domain

The natural logarithm function \(\ln x\) is only defined for \(x > 0\). Therefore, the domain of the function \(y = x \, \ln x\) is \((0, \infty)\).
03

Calculate key points

Find the values of \(y\) at select \(x\) values: - When \(x = 1\), \(y = 1 \, \ln 1 = 0\).- When \(x = e (\approx 2.718)\), \(y = e \, \ln e = e\).
04

Determine the behavior at extremes

Analyze the behavior as \(x\) approaches 0 and infinity: - As \(x \to 0^+\), \(y = x \, \ln x \to 0\) (since \(\ln x \to -\infty\) but \(x \to 0\)). - As \(x \to \infty\), \(y = x \, \ln x\) increases without bound as both \(x\) and \(\ln x\) increase.
05

Sketch the graph

Using the points and behaviors identified, draw the graph starting from just above the x-axis at x = 0, passing through the point (1, 0), and continuing to rise as x increases.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Logarithm
The natural logarithm, denoted as \(\ln x\), is a logarithmic function with the base \(e\) (where \(e\) is approximately 2.718). It is a fundamental function in mathematics that tells us how many times we need to multiply \(e\) to get a number \(x\).

Some important properties of the natural logarithm are:
  • \(\ln 1 = 0\) because \(e^0 = 1\).
  • \(\ln e = 1\) because \(e^1 = e\).
  • \(\ln(ab) = \ln(a) + \ln(b)\)
  • \(\ln(a^b) = b \ln(a)\)
In the equation \(y = x \ln x\), \(\ln x\) modifies the behavior of the function significantly. For \(x = 1\), \(\ln 1 = 0\), making \(y = 0\). As \(x \) increases, \(\ln x\) increases, enhancing the value of \(y\). Understanding these properties helps in graphing such equations.
Domain and Range
The domain of a function refers to all possible input values (or \('x'\) values) for which the function is defined.

In the given equation \(y = x \ln x\), the term \(\ln x\) is only defined when \(x > 0\). This is because you can't take the logarithm of zero or a negative number. Therefore, the domain of \(y = x \ln x\) is \((0, \infty)\).

The range of a function is all possible output values (or \('y'\) values) that the function can take.
For small values of \(x\) (closer to 0), \(y = x \ln x\) approaches 0. For large values of \(x\), since \(\ln x\) increases but more slowly than \(x\), \(y\) will continue to grow. Thus, the range of \(y = x \ln x\) is \((-\infty, \infty)\).
Curve Sketching
Curve sketching is the process of drawing a diagram that represents the behavior of a function. For the equation \(y = x \ln x\), follow these steps to sketch the curve:

**Identify Key Points:** Find specific points by substituting values of \(x\) into the equation. For instance:
  • When \(x = 1\), \(y = 1 \ln 1 = 0\).
  • When \(x = e (\approx 2.718)\), \(y = e \ln e = e\).
**Analyze Behavior at Extremes:**
  • As \(x \rightarrow 0^+\), \(y = x \ln x \rightarrow 0\). This happens because \(\ln x\) approaches \(-\infty\) but \(x\) approaches 0.
  • As \(x \rightarrow \infty\), \(y = x \ln x\) increases without bound since both \(x\) and \(\ln x\) increase.
**Sketch the Graph:**
  • Start from just above the x-axis at \(x = 0\).
  • Pass through the point \((1, 0)\).
  • Continue to rise as \(x\) increases.
By following these steps and understanding the function's behavior, you can accurately sketch the graph of \(y = x \ln x\).

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Most popular questions from this chapter

In a telegraph cable, the measure of the speed of the signal is proportional to \(x^{2} \ln (1 / x)\), where \(x\) is the ratio of the measure of the radius of the core of the cable to the measure of the thickness of the cable's winding. Find the value of \(x\) for which the speed of the signal is greatest.

\(y=e^{-3 x^{2}}\)

Use differentials to find an approximate value of the given logarithm and express the answer to three decimal places. $$ \log _{10} 997 $$

A loan of \(\$ 100\) is repaid in one payment at the end of a year. If the interest rate is \(8 \%\) compounded continuously, determine (a) the total amount repaid and (b) the effective rate of interest.

A simple electric circuit containing no condensers, a resistance of \(R\) ohms, and an inductance of \(L\) henrys has the electromotive force cut off when the current is \(I_{0}\) amperes. The current dies down so that at \(t\) sec the current is \(i\) amperes and $$ i=I_{0} e^{-(R / L) t} $$ Show that the rate of change of the current is proportional to the current.
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