Problem 16 - 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. $$ g(x)=\frac{1}{x} \text { on }(-1,1) $$

Step-by-Step Solution

Answer

The graph of \(g(x) = \frac{1}{x}\) on the interval \((-1, 1)\) has a vertical asymptote at \(x = 0\) and is decreasing throughout the interval with no critical points. The absolute maximum value is -1 at \(x = -1\), and the absolute minimum value is 1 at \(x = 1\).

Step 1: Find the Domain and Asymptotes of g(x)

The domain of \(g(x) = \frac{1}{x}\) consists of all real numbers except \(x = 0\). Since we are given the interval \((-1, 1)\), the function is not defined at the endpoints or at \(x = 0\). To find any vertical asymptotes, we set the denominator to zero and solve for x. In this case, the vertical asymptote is at \(x = 0\).

Step 2: Find the Critical Points of g(x)

Now we'll find the critical points of the function. Critical points occur where the derivative is either equal to zero or doesn't exist. We find the derivative of \(g(x)\): \[ g'(x) = -\frac{1}{x^2} \] Because the derivative is always negative for \(x \in (-1, 1)\), there are no points where the derivative is zero. Therefore, there are no critical points in this interval.

Step 3: Determine the End Behavior of g(x)

Since there are no critical points of the function on the interval \((-1, 1)\), the function must be either increasing or decreasing throughout this interval. We can analyze the end behavior by taking the limits of the function as x approaches the interval's bounds: As \(x \rightarrow -1^+\): \(g(x) \rightarrow -1\) As \(x \rightarrow 1^-\): \(g(x) \rightarrow 1\) Since the derivative is negative for all values in the interval, the function is decreasing throughout the interval.

Step 4: Sketch the Graph of g(x)

Using the information from steps 1-3, we can sketch the graph of \(g(x) = \frac{1}{x}\) on the interval \((-1, 1)\). The graph will have a vertical asymptote at \(x = 0\), will be decreasing throughout the interval, and will not have any critical points. Be sure to show the end behavior as x approaches the interval's boundaries.

Step 5: Find the Absolute Maximum and Absolute Minimum

Since the function is decreasing throughout the interval, the absolute maximum value must occur at \(x = -1\), and the absolute minimum value must occur at \(x = 1\). Evaluating the function at these endpoints, we find: Absolute maximum: \(g(-1) = -1\) Absolute minimum: \(g(1) = 1\) In conclusion, the graph of the function \(g(x) = \frac{1}{x}\) on the interval \((-1, 1)\) has an absolute maximum value of -1 at \(x = -1\) and an absolute minimum value of 1 at \(x = 1\).

Key Concepts

Graph Sketching

Graph sketching involves drawing a rough outline of a function's graph using key features such as intercepts, critical points, and asymptotes. For the function \(g(x) = \frac{1}{x}\) over the interval \((-1, 1)\), we start by identifying important characteristics. The nature of the function has a direct impact on its graph. Here, we have an inverse relationship, which typically forms a hyperbolic shape.
To sketch the graph of \(g(x)\), understand that:

• There is a vertical asymptote at \(x = 0\), meaning the graph approaches but never touches this vertical line.
• The function is decreasing, which we know from the derivative \(g'(x) = -\frac{1}{x^2}\), indicating it slopes down as \(x\) increases.
• Concentrate on the behavior of \(g(x)\) near the boundaries \(-1\) and \(1\): the graph moves from approaching \(-1\) as \(x\) approaches \(-1^+\) to \(1\) as \(x\) approaches \(1^-\).

These clues guide us to a clear, albeit rough, graph that captures how \(g(x)\) behaves across its domain.

Absolute Maximum

An absolute maximum refers to the greatest value a function achieves over a certain interval. For \(g(x) = \frac{1}{x}\) in \((-1, 1)\), because the function is always decreasing, the largest value occurs at the left boundary. This behavior is typical when a function does not have critical points within the interval due to a consistent increasing or decreasing pattern.
When looking at this particular function:

• The absolute maximum occurs at \(x = -1\).
• Evaluating the function at this point gives \(g(-1) = -1\).

So, \(g(x)\) does not achieve higher than \(-1\) within this specified interval. Understanding maximum values helps anchor the function’s highest point, crucial for determining overall graph behavior.

Absolute Minimum

As with absolute maxima, absolute minima represent the lowest value of a function within a defined interval. For \(g(x) = \frac{1}{x}\) on \((-1, 1)\), since the function decreases steadily, the smallest value occurs at the opposite boundary compared to the maximum.
This means:

• The absolute minimum occurs at \(x = 1\).
• At this point, evaluating \(g(1) = 1\) reveals the smallest value the function takes in the interval.

Recognizing absolute minima is pivotal for comprehending the downward trajectory of a function across its specified domain.

Vertical Asymptotes

Vertical asymptotes are lines that a graph approaches but never crosses or touches. They usually occur where a function is undefined, and they often indicate a point of discontinuity. For \(g(x) = \frac{1}{x}\), the vertical asymptote provides significant insight into the function's behavior.
For this function, the vertical asymptote is:

• Located at \(x = 0\).
• Indicative of how \(g(x)\) behaves as it gets very close to \(x = 0\) from either side; the function values tend to \(\pm\infty\).

Understanding vertical asymptotes is crucial when graphing functions, as they mark areas of drastic change and guide expectations for the graph's behavior around those points.