Problem 304 - Calculus Volume 1

Question

Draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior. \(\quad y=x \ln (x), x>0\)

Step-by-Step Solution

Answer

The function has a local minimum at \(x=\frac{1}{e}\), no inflection points, and asymptotically grows to infinity.

Step 1: Understanding the Function

The function given is \(y = x \ln(x)\), defined for \(x > 0\). The function involves the natural logarithm, which implies certain behavior to explore, such as its definition only for positive \(x\). Furthermore, because it is expressed as a product of \(x\) and \(\ln(x)\), it may have interesting properties worth examining, like its growth and shape.

Step 2: Find the Derivative

To find the critical points of the function, we first determine its derivative: \( \frac{dy}{dx} = x \cdot \frac{d}{dx}(\ln x) + \ln x \cdot \frac{d}{dx}(x) = x \cdot \frac{1}{x} + \ln x \cdot 1 = 1 + \ln x \).Setting the derivative to zero gives the critical points: \[ 1 + \ln x = 0 \Rightarrow \ln x = -1 \Rightarrow x = e^{-1} = \frac{1}{e} \].

Step 3: Test for Local Maxima or Minima

To determine whether \(x = \frac{1}{e}\) is a local maximum or minimum, consider the second derivative: \[ \frac{d^2y}{dx^2} = \frac{d}{dx}(1 + \ln x) = \frac{d}{dx}(\ln x) = \frac{1}{x} \].At \(x = \frac{1}{e}\), \[ \frac{d^2y}{dx^2} = e \gt 0 \],indicating that \(x = \frac{1}{e}\) is a local minimum.

Step 4: Identify Inflection Points

Inflection points occur where the concavity of the graph changes, i.e., where the second derivative changes sign. Since \( \frac{d^2y}{dx^2} = \frac{1}{x} \), the second derivative is always positive for \(x > 0\), indicating the graph is always concave up, and therefore, there are no inflection points.

Step 5: Determine Asymptotic Behavior

To explore asymptotic behavior, consider behavior as \(x\) approaches infinity and zero:- As \(x \to 0^+\), \(y = x \ln x \to 0 \cdot (-\infty) \to 0\) in a limiting sense, due to the \(x\) factor dominating the logarithm.- As \(x \to \infty\), \(y = x \ln x\) grows towards infinity due to the nature of both factors increasing.

Step 6: Sketch the Graph

With the gathered information:- There's a local minimum at \(x = \frac{1}{e}\).- No inflection points since the graph is always concave up.- The function tends towards zero as \(x\) approaches zero from the right, and increases infinitely as \(x\) approaches infinity.Sketch the graph starting slightly above zero, touching a minimum at \(x = \frac{1}{e}\), and then steadily rising.

Key Concepts

Graphing functions

Graphing a function is about visually representing its behavior by drawing its curve on a set of axes. The function provided is \( y = x \ln(x) \) for \( x > 0 \). Here are some steps you should consider when graphing:

• Identify Key Features: Look for points where the function changes direction, such as maxima and minima, as well as places it cannot go, shown by asymptotes.
• Domain and Range: Set the domain where the function exists, here it's \( x > 0 \), because the natural logarithm is undefined at zero or negative values. The range can be found using the function's behavior and critical points.
• Behavior at Extremes: Assess how the function behaves as \( x \) approaches critical values or infinity. For \( y = x \ln(x) \), as \( x \to 0^+ \), the function approaches zero, and as \( x \to \infty \), it tends towards infinity.

Breaking the function down in this way helps to visualize its curve accurately and understand how it behaves at different points.

Critical points

Critical points of a function occur where the first derivative is zero or undefined, pinpointing where the slope of the tangent is flat. Let's dissect the process using \( y = x \ln(x) \):

• First Derivative: Compute \( \frac{dy}{dx} = 1 + \ln x \).
• Set Derivative to Zero: Find \( x \) where \( 1 + \ln x = 0 \) to identify critical points, leading us to \( x = \frac{1}{e} \).
• Understanding Slope Changes: Critical points mark potential locations for a local maximum, minimum, or saddle point, which is determined by further analysis.

This calculated critical point suggests \( x = \frac{1}{e} \) as a significant point of interest on the function's curve, indicating where the function shifts its growth pattern.

Derivative

The derivative of a function represents its rate of change or the slope of the tangent line at any point along its curve. When exploring derivatives, consider:

• Definition: The derivative \( \frac{dy}{dx} \) for \( y = x \ln x \) is \( 1 + \ln x \), showing how the function's value changes with respect to \( x \).
• Purpose: Use derivatives to find critical points where the change halts and analyze the function's increasing or decreasing behavior.
• Slope Information: A positive derivative means the function is ascending, and a negative derivative means descending. Where it's zero, there's potential for critical behavior, like a peak or valley.

Understanding derivatives is fundamental to grasping calculus concepts, allowing you to delve deeper into how functions behave dynamically.

Asymptotic behavior

Asymptotic behavior describes how a function behaves as it continues far along in one direction, represented by lines that the curve approaches but never actually touches. For \( y = x \ln(x) \), this involves:

• Near the Origin: As \( x \to 0^+ \), \( y \to 0 \) indicating no asymptote exists, but it suggests a starting position for the graph.
• Towards Infinity: As \( x \to \infty \), \( y = x \ln x \to \infty \) shows the function climbing without bound. This indicates no horizontal or vertical asymptotes.
• Resulting Trajectory: Recognize that while no asymptotes exist, the behavior informs the gradual transitioning of function values.

This assessment helps in characterizing the long-term behavior of the function, informing how the graph might look as \( x \) becomes very small or very large.

Inflection points

Inflection points indicate where a function's concavity changes—the curve shifts from concave up to concave down, or vice versa. Here's how to examine inflection points for \( y = x \ln(x) \):

• Second Derivative Test: The second derivative \( \frac{d^2y}{dx^2} = \frac{1}{x} \) is positive for all \( x > 0 \).
• No Sign Change: Since the second derivative is always positive, the function is always concave up with no changes in curvature.
• Concavity Information: The absence of inflection points signals a continuous concave upward shape for the function.

Understanding inflection points, or the lack thereof, helps in sketching the overall shape and revealing the function's persistent curvature tendencies.