Problem 92 - Calculus Single Variable

Question

In each of Exercises \(91-94\), calculate and plot the derivative \(f^{\prime}\) of the given function \(f\). Use this plot to identify candidates for the local extrema of \(f\). Add the plot of \(f\) to the window containing the graph of \(f^{\prime} .\) From this second plot, determine the behavior of \(f\) at each candidate for a local extremum. $$ f(x)=x^{2}-2 x \ln \left(1+x^{2}\right)+x-4 $$

Step-by-Step Solution

Answer

Calculate the derivative, plot both functions, and check for local extrema using critical points.

Step 1: Identify the Function or Relation

We need to graph: \(91-94\).

Step 2: Determine Key Features

We identify important features such as intercepts, slope, vertex, asymptotes, or other characteristic points.

Step 3: Plot Key Points and Draw the Graph

Using the key features identified, we plot several points and connect them to form the graph.

Step 4: Describe the Graph

Calculate the derivative, plot both functions, and check for local extrema using critical points.

Key Concepts

Derivative

A derivative in calculus is a tool that tells us how a function changes at any given point. Think of it as the slope of the function at a particular spot. If you imagine yourself on a hill, the derivative tells you if you're climbing up, going down, or if it's just flat. To find the derivative of a function like \[ f(x) = x^2 - 2x \ln(1 + x^2) + x - 4 \] we apply differentiation rules. For each part of the function, we find how it contributes to the overall slope. Lines, such as the linear part \( x \) and constants like \(-4\), have straightforward derivatives: \(1\) and \(0\) respectively. The quadratic \(x^2\) gives us \(2x\), and for complex terms like the logarithmic one, we use more specific rules like the product rule. This involves differentiating each piece separately and then combining them accordingly. Using these steps gives us the precise derivative that helps in analyzing the function's behavior. The final derivative, \[ f'(x) = 2x + \left(-\frac{4x}{1 + x^2}\right) + 1 \], shows where and how sharply our function rises or falls as \( x \) changes.

Local Extrema

Local extrema are the high and low points in a small region of a graph. If you imagine climbing a mountain range, local maximum points are like mountain peaks, while local minimum points are like valleys.When we want to find local extrema of our function \( f(x) \), we look for points where the derivative is zero or undefined. That's because these are the places where the function's slope changes – from positive to negative or vice versa.For the given function, we find the derivative and solve for points where \( f'(x) = 0 \). These zero points are candidates for local maxima or minima. By further analysis, such as checking the second derivative or assessing the behavior of \( f' \) before and after these points, we can confirm whether these points are indeed peaks or valleys. This helps us map out the shape and trends of the function effectively.

Function Plotting

Plotting a function and its derivative provides a visual way to understand their behaviors. Seeing both graphs on the same plot enables us to directly observe the relationship between a function and its rate of change.The original function \( f(x) \) offers insight into overall trends like growth or decline. By plotting \[ f(x) = x^2 - 2x \ln(1 + x^2) + x - 4 \] alongside its derivative \[ f'(x) = 2x + \left(-\frac{4x}{1 + x^2}\right) + 1 \], we can visually locate potential local extrema where \( f' \) crosses the x-axis. At these crossings, the function changes its ascent or descent, showing critical points for analysis. Plotting allows for a straightforward assessment of how these points correspond to potential maxima or minima, and what behavior the function exhibits around these critical areas.

Differentiation Rules

Differentiation rules are the fundamental guidelines we follow to determine a derivative. Having a set of these rules ensures that no matter how complex the function, we can break it down piece by piece.In our function \( f(x) \), the rules like the power rule (\( \frac{d}{dx}[x^n] = nx^{n-1} \)), product rule (\( \frac{d}{dx}[uv] = u'v + uv' \)), and chain rule are employed. The power rule allows us to easily find derivatives of terms like \(x^2\). For elements like \(-2x\ln(1 + x^2)\), we use the product rule, differentiating each part separately and then combining them.The chain rule, in cases where we have compound functions like the logarithmic term, lets us handle internal functions by multiplying the derivative of the outer function by the derivative of the inside.Through the proper application of these rules, we derive a clear and accurate picture of how our function \( f(x) \) behaves at any given point.