Problem 15 - Applied Calculus

Question

Describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing). Then use the derivative and algebra to explain the shape of the graph. $$ f(x)=x+2 \sin x $$

Step-by-Step Solution

Answer

The graph of \( f(x) = x + 2\sin x \) has wavy fluctuations due to \( 2 \sin x \), with critical points at \( x = \frac{2\pi}{3} + 2k\pi \) and \( x = \frac{4\pi}{3} + 2k\pi \), where it changes monotonicity.

Step 1: Identify the Function

The function given is \( f(x) = x + 2 \sin x \). This is a combination of a linear function, \( x \), and a trigonometric function, \( 2 \sin x \). This combination suggests non-linear behavior, influenced by the sine wave's periodic oscillation.

Step 2: Determine the Derivative

To find critical points and understand monotonicity, calculate the derivative: \( f'(x) = 1 + 2 \cos x \). This derivative will help identify where the function is increasing or decreasing.

Step 3: Find Critical Points

Critical points occur where the derivative equals zero: \( 1 + 2 \cos x = 0 \). Solving for \( x \), we get \( \cos x = -\frac{1}{2} \). The general solutions for \( x \) in this case are \( x = \frac{2\pi}{3} + 2k\pi \) and \( x = \frac{4\pi}{3} + 2k\pi \) for \( k \in \mathbb{Z} \).

Step 4: Test Intervals for Monotonicity

Check intervals between critical points using values within those ranges in the derivative \( f'(x) = 1 + 2 \cos x \). For example, test values like \( x = 0, \pi, \frac{3\pi}{2} \) to determine if the derivative is positive or negative, which indicates if the function is increasing or decreasing.

Step 5: Analyze Sine's Influence

Due to the \( 2 \sin x \) component, the function's direction will oscillate following the sine wave's peaks and troughs. Periodically, \( \sin x \) oscillates between -1 and 1, multiplying by 2 gives oscillations between -2 and 2, which adds to the linear component.

Step 6: Summarize Graph Features

The function \( f(x) = x + 2 \sin x \) has critical points where \( \cos x = -\frac{1}{2} \). Between these points, the derivative tells us where the function increases or decreases. The sine component adds periodic oscillations superimposed on the linear trend, creating a wavey line.

Key Concepts

Critical Points

Critical points are essential for understanding the behavior of a function. They occur where the derivative of the function equals zero or is undefined. In this particular problem, we are dealing with the function \( f(x) = x + 2 \sin x \). To find the critical points, we first take the derivative: \( f'(x) = 1 + 2 \cos x \).

Setting this derivative to zero gives us the equation \( 1 + 2 \cos x = 0 \). Solving for \( \cos x \), we find \( \cos x = -\frac{1}{2} \). This happens at specific points on the unit circle. Specifically, \( x = \frac{2\pi}{3} + 2k\pi \) and \( x = \frac{4\pi}{3} + 2k\pi \) for \( k \in \mathbb{Z} \).

These are the critical points of the function. Here, \( f'(x) \) changes sign, indicating a potential switch between increasing and decreasing behavior. Knowing these points allows us to further analyze the graph's interesting features.

Monotonicity

Monotonicity refers to the intervals where the function is either consistently increasing or decreasing. Monotonic behavior can be identified by examining the sign of the derivative.

For the function \( f(x) = x + 2 \sin x \), we earlier calculated the derivative: \( f'(x) = 1 + 2 \cos x \). By testing values of \( x \) in different intervals between the critical points, we determine the sign of \( f'(x) \).

Some key points to consider are:

• If \( f'(x) > 0 \), the function is increasing.
• If \( f'(x) < 0 \), the function is decreasing.

For instance, at \( x = 0 \), \( f'(x) = 1 + 2 \cdot 1 = 3 \), which is positive, meaning the function is increasing around this point. Testing at \( x = \pi \), \( f'(x) = 1 + 2(-1) = -1 \), shows the function is decreasing there.

Testing various intervals, as shown in the steps, helps us map out where the function's direction changes due to these derivatives, providing insights into the overall shape of the graph.

Trigonometric Function

Trigonometric functions add an interesting layer of complexity to function behavior. In \( f(x) = x + 2 \sin x \), the \( 2 \sin x \) component introduces periodic oscillations on top of the linear trend of \( x \).

The sine function, \( \sin x \), oscillates between -1 and 1. So \( 2\sin x \) oscillates between -2 and 2. This results in a waveform that regularly affects the slope and shape of \( f(x) \).

When superimposed on a linear function, these oscillations add peaks and valleys, creating a wavy overall appearance. Analyzing such a graph involves understanding both the periodic nature of \( \sin x \) and the linear trend of \( x \).

The trigonometric component ensures that the graph of \( f(x) \) isn't just a straight line but rather one that undulates, showing how periodic behaviors interact with linear elements in complex functions. Understanding this helps when exploring more advanced trigonometric applications in calculus.