Problem 91 - 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. $$ f(x)=x^{3}-2 x+\cos (x) $$

Step-by-Step Solution

Answer

The derivative of the function is \( f'(x) = 3x^2 - 2 - \sin(x) \). Plotting reveals local extrema at points where the derivative changes sign.

Step 1: Find the derivative of the function

To find the derivative of the function given, \( f(x) = x^3 - 2x + \cos(x) \), we apply the rules of differentiation.The derivative of \( x^3 \) is \( 3x^2 \).The derivative of \( -2x \) is \( -2 \).The derivative of \( \cos(x) \) is \( -\sin(x) \).Combining these, the derivative is:\[ f'(x) = 3x^2 - 2 - \sin(x) \]

Step 2: Plot the derivative function

Using graphing software or a graphing calculator, plot the derivative \( f'(x) = 3x^2 - 2 - \sin(x) \). Look for points where the derivative changes sign; these points are candidates for local extrema of the original function \( f(x) \).

Step 3: Identify candidates for local extrema

From the plot of \( f'(x) \), identify the points where the graph crosses the x-axis. These points correspond to where the derivative is zero, indicating potential local maxima or minima.

Step 4: Plot the original function

Now, plot the original function \( f(x) = x^3 - 2x + \cos(x) \) on the same axes where \( f'(x) \) was plotted.Compare the xmaxima and minima of \( f(x) \) to the x-intercepts of \( f'(x) \) to determine the behavior of the function at each candidate for local extrema.

Step 5: Analyze the behavior of the function

Observe the graph of \( f(x) \) and \( f'(x) \):- If \( f'(x) \) changes from positive to negative, \( f(x) \) has a local maximum.- If \( f'(x) \) changes from negative to positive, \( f(x) \) has a local minimum.Note the x-values where these changes occur to determine the behavior of \( f(x) \) at each critical point.

Key Concepts

Local Extrema

When you're studying functions, local extrema refer to the highest or lowest points in a small region around a specific point on the graph. These are crucial for understanding the behavior of the function. Particularly, they are not necessarily the highest or lowest points on the entire graph, but they are higher or lower compared to their immediate surroundings. To identify local extrema, we generally find where the derivative of a function is zero or undefined.

Here's how you can spot them:

• First, calculate the derivative of the function. This will help us to understand where the slope of the tangent is zero, indicating a possible local extremum.
• Next, look at where the sign of the derivative changes. If it passes from positive to negative, it's a local maximum; from negative to positive, it's a local minimum.

Remember that using graphing tools can help visually confirm those areas by making it easier to see where the derivative crosses the x-axis or is undefined. By examining both the original function and its derivative plotted together, you get a better insight into the nature of these points.

Differentiation Rules

Differentiation is a core calculus concept, and mastering its rules is essential for finding the derivative of a function. The primary differentiation rule used in calculating the derivative of our function \( f(x) = x^3 - 2x + \cos(x) \) involves breaking down each piece of the function and applying specific rules to each.

Here's the process:

• For powers of \( x \), like \( x^3 \), you apply the power rule: the derivative of \( x^n \) is \( nx^{n-1} \). So, the derivative of \( x^3 \) is \( 3x^2 \).
• Linear terms, such as \( -2x \), have straightforward derivatives: the coefficient itself, \( -2 \).
• For trigonometric functions like \( \cos(x) \), the derivative is \( -\sin(x) \).

Combine these derivatives to get the overall derivative: \( f'(x) = 3x^2 - 2 - \sin(x) \). Practicing these steps until they become second nature will greatly simplify solving problems involving derivatives.

Graphing Derivatives

Once you've calculated the derivative, graphing is a powerful way to understand the function better. When you graph both the original function and its derivative, you can see how they relate and how the behavior of one informs the other.

Here's why graphing derivatives is so insightful:

• Graphing \( f'(x) \) shows you where the slope of \( f(x) \) is zero. These x-intercepts are critical because they suggest where the local extrema of \( f(x) \) might be.
• Between these intercepts, observe if the derivative is above or below the x-axis. Above indicates the original function is increasing, and below indicates it is decreasing.

When you overlay the graph of \( f(x) \) with \( f'(x) \), you can directly see how changes in \( f'(x) \) coincide with key features of \( f(x) \), enhancing your understanding of the function's overall shape and behavior.