Problem 57 - Calculus with Concepts in Calculus

Question

Plot the graph of \(f\). Using the zoom feature of the calculator, approximate to within \(0.1\) all values of \(c\) such that \(f(c)\) is a relative extreme value, and identify each as a relative maximum value or a relative minimum value. $$ f(x)=3 \sin x-x \text { for }-2 \pi \leq x \leq 2 \pi $$

Step-by-Step Solution

Answer

Relative maxima at \( x \approx -5.4, 1.8 \); relative minima at \( x \approx -1.8, 5.4 \).

Step 1: Understand the Problem

We need to find and plot the graph of the function \( f(x) = 3 \sin x - x \) over the interval \( -2\pi \leq x \leq 2\pi \). Then, using a graphing calculator, we must identify and approximate all points of relative extrema to within 0.1 accuracy on the x-axis.

Step 2: Graph the Function

Using a graphing calculator or graphing software, plot the function \( f(x) = 3 \sin x - x \) over the interval \( -2\pi \leq x \leq 2\pi \). Make sure the plot is clear and visible to identify peaks and valleys, which indicate relative extrema.

Step 3: Locate Relative Extrema

By using the zoom feature and the graph, visually inspect for points where the graph reaches a local maximum or minimum. These are found where the slope of the tangent is zero, indicating the potential for a relative maximum or minimum.

Step 4: Use Calculator Zoom Feature

Use the zoom feature on the graphing calculator to identify each of these extrema more precisely. Adjust the zoom level until you can approximate the x-values \( c \) where \( f(c) \) is at a relative maximum or minimum within 0.1 accuracy.

Step 5: Identify and Classify Extrema

Check for relative maxima (where the graph peaks) and minima (where the graph bottoms out). Approximate the x-values \( c \) for each of these, noting whether each is a maximum or minimum. Record the x-values \( c \) and identify each point on the graph.

Key Concepts

Understanding Relative Extrema

When studying calculus functions, a core concept to grasp is that of relative extrema. Relative extrema refer to the points on a graph where a function takes on either a maximum or minimum value within a small interval. For example, imagine you're hiking along a path that goes up and down. The points where you reach the peak of a hill or the lowest point in a valley are like the relative maxima and minima on a graph.
The calculations behind finding these points involve looking at the derivative of the function, since the slope of the tangent line at these points is zero. Think of the slope of the tangent line as a tool that shows you whether you are ascending or descending the hill.

For the sine function and its variations, like our function here, these concepts become visually evident through graph analysis—especially with a graphing tool that allows for zooming in on these specific features.

Characteristics of the Sine Function

The sine function, denoted as \( \sin x \), is a fundamental concept in trigonometry. It produces a smooth, oscillating wave that repeats every \( 2\pi \), known as its period. This function moves between a maximum value of 1 and a minimum value of -1.
Key characteristics of the sine function include:

• Periodic Nature: Repeats consistently every \( 2\pi \) interval.
• Symmetrical Graph: Exhibits symmetry about the origin in its graph.
• Amplitude: The height from the centerline to the peak, which is 1 for \( \sin x \).

In the given function \( f(x) = 3 \sin x - x \), the sine wave is adjusted by a coefficient, stretching its amplitude to 3. Additionally, subtracting \( x \) shifts and alters the pattern, creating a graph with varied peaks and valleys—ideal for investigating relative extrema in calculus.

Using a Graphing Calculator Effectively

Graphing calculators are excellent tools for visualizing mathematical functions and identifying features like relative extrema. To effectively use a graphing calculator, start by entering the function, such as \( f(x) = 3 \sin x - x \), into the calculator's graphing mode.
Here are some steps to maximize this tool's potential:

• Clear Plotting: Ensure that you've set the correct interval, such as \( -2\pi \leq x \leq 2\pi \), to visualize the entire necessary graph.
• Zoom Feature: Utilize the zoom function to get a closer look at the peaks and valleys. This helps in approximating x-values for relative maxima and minima more accurately.
• Trace Function: Use the calculator’s ‘trace’ or equivalent function to navigate along the graph. This can help pinpoint exact x-values of interest.

Remember: practice makes perfect. Using these features correctly will greatly aid you in recognizing where a graph hits its extreme values, and with precision. Mastering this skill can significantly enhance analysis and streamline future calculations.