Problem 22 - Calculus Early Transcendentals

Question

More graphing Make a complete graph of the following functions. If an interval is not specified, graph the function on its domain. Use a graphing utility to check your work. $$f(x)=\ln \left(x^{2}+1\right)$$

Step-by-Step Solution

Answer

Question: Sketch the graph of the function $$f(x) = \ln (x^2 + 1)$$, and identify its domain and critical points. Answer: The graph of the function $$f(x) = \ln (x^2 + 1)$$ is continuous and increasing on its domain, which is \((-\infty, \infty)\). It has a single critical point at \((0, 0)\), which is a minimum point on the graph.

Step 1: Find the domain of the function

The domain of the natural logarithm is \((0, \infty)\). Since we are given a function inside the logarithm, we need to ensure the input is never 0 or negative. Thus, we need to find the domain of the function $$x^{2}+1$$. Note that squaring any real number will yield a non-negative result. Therefore, the domain of the function $$x^{2}+1$$ is \((-\infty, \infty)\). As a result, the domain of $$f(x)=\ln \left(x^{2}+1\right)$$ is also \((-\infty, \infty)\).

Step 2: Find critical points

Critical points are potential maximum or minimum points on the graph. To find them, take the first derivative of the function and set it equal to 0. So we have: $$f'(x) = \frac{d}{dx}\left[\ln(x^2+1)\right]$$ Using the Chain Rule, we get $$f'(x) = \frac{1}{x^2+1} \cdot \frac{d}{dx}(x^2+1)$$ Now find the derivative of \(x^2+1\) and substitute it into the equation: $$f'(x) = \frac{1}{x^2+1} \cdot 2x$$ $$f'(x) = \frac{2x}{x^2+1}$$ Now set this equal to 0 and solve for \(x\): $$\frac{2x}{x^2+1} = 0$$ The only way the fraction can equal 0 is if the numerator equals 0, so \(2x = 0\). Thus, the only critical point is at \(x = 0\) and \(f(0) = \ln(1) = 0\).

Step 3: Sketch the graph

Now that we have the domain and critical points, we can assemble the information and sketch the overall graph. 1. The domain is \((-\infty, \infty)\), so the graph extends in both directions. 2. The critical point is at \((0, 0)\), which is a candidate for a minimum or maximum. Since the natural logarithm is an increasing function, \(f(x) = \ln(x^2 + 1)\) will also be increasing on its domain. Use these facts to sketch the graph, and then use a graphing utility to verify your work. The graph should show that the function is continuous and increasing on its domain, with a low point at the origin.

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial for graphing logarithmic functions. The domain refers to the set of all possible input values (x-values) for which the function is defined. In the case of the natural logarithm function, denoted as \( \text{ln}(x) \), the domain is all positive real numbers, \( (0, \text{\infty}) \).

However, for functions within a logarithm, such as \( f(x) = \text{ln}(x^{2}+1) \), we must ensure the input to the logarithm is always positive. After evaluating the inside function \( x^{2}+1 \), it's clear that for all real numbers, the square of any number plus one will always be positive. Hence, the domain of \( f(x) \) is \( (-\text{\infty}, \text{\infty}) \), meaning that we can plot this function for any value of x on the real number line. Identifying the domain is the first step before proceeding to find features such as critical points or behavior of the function.

First Derivative Test

The first derivative test is a powerful tool used to determine where a function's graph has relative minima or maxima. By taking the derivative of a function, we can find the rate at which the function's value changes with respect to x. Where the first derivative (\( f'(x) \)) equals zero, or does not exist, we might find critical points, which are potential locations for these relative extrema.

The signed values of the first derivative on the intervals around these critical points indicate whether the function is increasing or decreasing. For a critical point to be a relative minimum, the first derivative should change from negative to positive; for a relative maximum, it should change from positive to negative. This assessment is fundamental for sketching the graph of a function, as it highlights important features of its curvature and overall shape.

Critical Points

Critical points of a function are where its first derivative is zero or undefined; these points are where the function's graph may change direction, creating peaks or troughs. Identifying critical points is essential in the analysis of functions to determine their local extrema and concavity. In the example given, \( f'(x) \) is set to zero to find the critical points of the logarithmic function \( f(x) = \text{ln}(x^{2}+1) \). The equation \( \frac{2x}{x^{2}+1} = 0 \) yields the only critical point at \( x = 0 \). This information is used to judge the potential for a relative minimum or maximum at that point; for the given function, the critical point at the origin signifies the lowest point on the graph. In combination with other aspects such as the domain and concavity, critical points guide us in graphing the function correctly.

Chain Rule

The chain rule is a fundamental differentiation technique in calculus, used when dealing with compositions of functions. It allows us to differentiate a function that is nested within another function by taking the derivative of the outer function and multiplying it by the derivative of the inner function. In our logarithmic function example, \( f(x) = \text{ln}(x^{2}+1) \), applying the chain rule helps us find the first derivative, \( f'(x) \).

The process involves differentiating the logarithmic part, \( \text{ln}(u) \), where \( u = x^{2}+1 \), to get \( \frac{1}{u} \), and then multiplying by the derivative of the inner function, \( u' = 2x \). This gives us \( f'(x)= \frac{2x}{x^{2}+1} \), the derivative we use to determine critical points and the function's rate of change. Understanding the chain rule is thus pivotal for analyzing how the function behaves and changes at various points along its graph.