Problem 22 - Single Variable Calculus

Question

In Exercises \(7-24\), sketch the graph of the function and find its absolute maximum and absolute minimum values, if any. $$ g(x)=\ln x \text { on }(0, e) $$

Step-by-Step Solution

Answer

The graph of the function \(g(x) = \ln(x)\) on the interval (0, e) has no critical points within the interval. As x approaches 0 from the right, g(x) approaches negative infinity, which means there is no absolute minimum value. The absolute maximum value is reached at x = e, where \(g(e) = \ln(e) = 1\).

Step 1: Sketch the graph

First, we need to have an idea of what the graph of g(x) = ln(x) looks like. The function represents the natural logarithm, which means it is undefined for x ≤ 0 and approaches negative infinity as x approaches 0 from the right. The graph is monotonic increasing and passes through the point (1,0). Therefore, the graph of g(x) will start from negative infinity on the y-axis, pass through the point (1,0), and continue to rise as x increases.

Step 2: Find the derivatives

To find critical points, we first need to find the first derivative of the function g(x): \(g'(x) = \frac{1}{x}\)

Step 3: Determine critical points

To find critical points, we identify where the first derivative is either equal to 0 or undefined: \(g'(x) = \frac{1}{x} = 0\) Since the fraction 1/x can never be equal to zero, there are no critical points where the derivative is zero. However, we still need to check for points where the derivative is undefined: g'(x) is undefined when x = 0. However, this point is not in the given interval (0, e). Therefore, there are no critical points within the interval (0, e).

Step 4: Analyze interval endpoints

Since there are no critical points within the interval (0, e), we only need to analyze the interval endpoints to find the absolute maximum and minimum. As x approaches 0 from the right, g(x) approaches negative infinity, which means there is no absolute minimum value. When x = e, g(x) = ln(e) = 1. Since the graph of g(x) is continuously increasing within the interval (0, e), the value g(e) = 1 represents the absolute maximum value.

Step 5: State the results

The graph of the function g(x) = ln(x) on the interval (0, e) has no absolute minimum value. The absolute maximum value is g(e) = 1.

Key Concepts

Natural Logarithm

The natural logarithm is denoted as "ln" and represents logarithms with a base of "e," where "e" is approximately equal to 2.71828. It is widely used in mathematics to solve equations involving growth and decay, compound interest, and much more. Unlike the common logarithm that uses base 10, the natural logarithm uses the irrational number "e."
For the function \( g(x) = \ln(x) \), we note a few key characteristics:

• The natural logarithm is only defined for positive values of \( x \). Hence, the domain of \( \ln(x) \) is \( x > 0 \).
• The function is continuous and differentiable over its domain.
• It is monotonic increasing, meaning the function increases as \( x \) increases.
• The graph of \( \ln(x) \) passes through the point (1, 0), since \( \ln(1) = 0 \).
• As \( x \) approaches zero from the right, \( \ln(x) \) approaches negative infinity.

Understanding these features helps in sketching the graph and analyzing the behavior of the function on a given interval.

Derivatives

Derivatives allow us to determine the rate at which a function is changing at any given point. For the function \( g(x) = \ln(x) \), the first derivative helps us understand the slope of the tangent lines to the graph.

• The first derivative of \( g(x) = \ln(x) \) is \( g'(x) = \frac{1}{x} \).
• This derivative tells us how steeply \( g(x) \) is increasing or decreasing at any point \( x > 0 \).
• Since \( \frac{1}{x} \) is positive for all \( x \) in the domain of the natural logarithm, \( g(x) \) is constantly increasing.
• The derivative \( \frac{1}{x} \) becomes larger as \( x \) gets smaller within the interval and approaches 0 from the right.

By studying the derivative, we gain insight into the function's behavior which is crucial for identifying critical points and extrema.

Critical Points

Critical points occur where a function's derivative is either zero or undefined. Identifying critical points helps us find local extremum points which could be potential maximum or minimum points of the function.

• For the function \( g(x) = \ln(x) \), its first derivative is \( g'(x) = \frac{1}{x} \).
• Setting \( g'(x) = 0 \) would imply \( \frac{1}{x} = 0 \), which is not possible because the fraction can never equal zero for any real number \( x \).
• We also check where the derivative is undefined, which is at \( x = 0 \). However, this point is outside the interval (0, \( e \)).

Thus, within the interval (0, \( e \)), \( g(x) = \ln(x) \) has no critical points.

Absolute Maximum and Minimum

Finding the absolute maximum and minimum values of a function over an interval involves evaluating the function at critical points (if any) and the boundaries of the interval.

• Since there are no critical points for \( g(x) = \ln(x) \) within the interval (0, \( e \)), we look at the interval's endpoints.
• As \( x \) approaches 0 from the right, \( g(x) = \ln(x) \) approaches negative infinity. Thus, no absolute minimum value exists.
• At \( x = e \), the function has the value \( g(e) = \ln(e) = 1 \), which is the absolute maximum value.

Therefore, on the open interval (0, \( e \)), the function achieves an absolute maximum of 1 and has no absolute minimum.

Graph Sketching

Sketching the graph of a function like \( g(x) = \ln(x) \) requires understanding its behavior and key features:

• Plot the known point \( (1, 0) \) since \( \ln(1) = 0 \).
• The graph starts from negative infinity as \( x \) approaches zero from the right.
• It continuously rises through the interval towards the maximum point at \( x = e \), where the function value is 1.
• Draw a smooth curve that passes through the point \( (1, 0) \) up to the point \( (e, 1) \).
• There are no turning points, as the function steadily increases across the interval.

These details provide a full picture of the logarithmic function's graph between 0 and \( e \). Being able to combine analytical work (derivatives, critical points) with graphical intuition is powerful for understanding real-world applications.