Problem 53 - Calculus Early Transcendentals: Pearson New International Edition

Question

Each of the following functions is periodic. Use a graphing calculator or a CAS to plot the graph of each of the following functions over one full period with the center of the interval located at the origin. Determine the coordinates of any of the global extrema and any inflection points. You should be able to give answers that are accurate to at least one decimal place. (a) \(f(x)=2 \sin x+\cos ^{2} x\) (b) \(f(x)=2 \sin x+\sin ^{2} x\) (c) \(f(x)=\cos 2 x-2 \cos x\) (d) \(f(x)=\sin 3 x-\sin x\) (e) \(f(x)=\sin 2 x-\cos 3 x\)

Step-by-Step Solution

Answer

Determine periods and plot each function over one period. Find extrema and inflection points numerically or analytically for precise coordinates.

Step 1: Determine the Periods of the Functions

Identify the period of each function by examining the trigonometric components. For functions in the form of \(f(x) = a \sin(bx) + c\), the period is given by \(\frac{2\pi}{b}\). Similarly, for \(f(x) = a \cos(bx) + c\).- For \(f(x) = 2 \sin x + \cos^2 x\), the period is \(2\pi\) because both \(\sin x\) and \(\cos x\) have periods of \(2\pi\).- For \(f(x) = 2 \sin x + \sin^2 x\), the period is \(2\pi\) as well.- For \(f(x) = \cos 2x - 2\cos x\), the period is \(\pi\), since \(\cos 2x\) has a period of \(\pi\).- For \(f(x) = \sin 3x - \sin x\), the period is computed as the least common multiple of the periods of individual terms, resulting in \(2\pi\).- For \(f(x) = \sin 2x - \cos 3x\), the period is \(2\pi\), being the least common multiple of \(\pi\) (\(\sin 2x\)) and \(\frac{2\pi}{3}\) (\(\cos 3x\)).

Step 2: Plot the Functions with the Center at the Origin

Using a graphing calculator or software tool, plot each function over one complete period centered at the origin:- For \(f(x) = 2 \sin x + \cos^2 x\): Plot from \(-\pi\) to \(\pi\).- For \(f(x) = 2 \sin x + \sin^2 x\): Plot from \(-\pi\) to \(\pi\).- For \(f(x) = \cos 2x - 2\cos x\): Plot from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).- For \(f(x) = \sin 3x - \sin x\): Plot from \(-\pi\) to \(\pi\).- For \(f(x) = \sin 2x - \cos 3x\): Plot from \(-\pi\) to \(\pi\).

Step 3: Determine the Global Extrema and Inflection Points

Use analytic or numerical methods to find extrema and inflection points:- **Extrema**: Analyze derivative \(f'(x)\) for critical points where \(f'(x) = 0\) or is undefined. These are potential locations for maxima and minima.- **Inflection Points**: Solve the second derivative \(f''(x) = 0\) to find points of concavity change.Example for \(f(x) = 2 \sin x + \cos^2 x\):- Calculate first and second derivatives.- Solve \(2\cos x - 2\sin 2x = 0\) for critical/extrema points.- Solve \(f''(x) = -2\sin x - 4\cos 2x = 0\) for inflection points.Repeat similarly for all functions to find precise coordinates using accurate methods or graph-based approximation.

Step 4: Summarize the Results

Compile the calculated global extrema and inflection points for each function, including their coordinates to one decimal place accuracy. Document this in a concise format for each part:Example for part (a), after precise calculations or using graph:- Global Maximum at \((x, f(x)) \approx (\pm \frac{\pi}{2}, 2.5)\)- Global Minimum at \((x, f(x)) \approx (\pm \pi, 1)\)- Inflection Points at \(x \approx 0, \pm \frac{\pi}{4}\)Repeat for parts (b) through (e), ensuring all coordinates reflect at least one decimal of accuracy from calculation or well-set graph determination.

Key Concepts

Trigonometric Functions

Trigonometric functions, like sine and cosine, are foundational in mathematics and are periodic by nature, meaning they repeat their values over specific intervals. This periodicity helps in predicting their behavior over time. In this exercise, we're dealing with functions built from these basic trigonometric forms:

• In the form of sine functions, \(f(x) = a \sin(bx) + c\), the period is calculated using \(\frac{2\pi}{b}\).
• Similarly, cosine functions take the form \(f(x) = a \cos(bx) + c\), also with a period determined by \(\frac{2\pi}{b}\).

For example, the function \(f(x)=2 \sin x+\cos^2 x\) repeats its values every \(2\pi\) interval, typical of both \(\sin x\) and \(\cos x\). Such periodicity is crucial for analyzing cyclical trends in real-world phenomena, like sound waves or tidal patterns.

Extrema

To find the extrema of a function, which are the maximums and minimums, we look at where its derivative equals zero (or is undefined). This is where the function changes direction. For each of the periodic functions given:

• The first derivative \(f'(x)\) is used to pinpoint the critical points.
• At these critical points, the tangent to the curve is horizontal, indicating potential local maxima or minima.
• Analyzing the sign of \(f'(x)\) around these points helps confirm the nature of these extrema.

For instance, in the function \(f(x)=2 \sin x+\cos^2 x\), by solving the first derivative equation \(2\cos x - 2\sin 2x = 0\), we discover precise x-values where these extrema occur. Maxima or minima align with patterns in the graph, reflecting the repetitive nature of these functions.

Inflection Points

Inflection points are places on a graph where the curvature changes direction, from concave up to concave down, or vice versa. These are determined by the second derivative of the function:

• Calculating \(f''(x) = 0\) helps find where these transitions happen.
• A change in the sign of \(f''(x)\) confirms the presence of an inflection point.

For our function \(f(x)=2 \sin x+\cos^2 x\), solving \(-2\sin x - 4\cos 2x = 0\) provides the specific x-values where the curvature changes. Inflection points are crucial in understanding the graph's shape since they tell us how the slope changes across different segments of the function.

Graphing Techniques

When graphing trigonometric functions, it's important to capture a full period of the function to understand its behavior. Here's how:

• Start by setting the graphing interval according to the period of the function. For instance, for \(f(x)=2 \sin x+\cos^2 x\), plot from \-\pi\ to \pi\.
• Include critical information such as global extrema and inflection points in the graphed period. This highlights changes in direction and curvature.
• Use precise graphing methods like graphing calculators or CAS (Computer Algebra Systems) for accuracy. These tools assist in visualizing the nuances of the function's behavior.

By following these graphing techniques, you not only sketch an accurate representation but also deepen your understanding of how trigonometric elements interact and present in periodic functions.