Problem 28 - Single Variable Calculus

Question

In Exercises \(5-38\), sketch the graph of the function using the curve- sketching guidelines on page \(348 .\) $$ y=\cos ^{2} x, \quad-\pi \leq x \leq \pi $$

Step-by-Step Solution

Answer

The graph of the function \(y = \cos^2 x\) within the interval \(-\pi \leq x \leq \pi\) has a domain of \(-\pi \leq x \leq \pi\) and a range of \(0 \leq y \leq 1\). The function has critical points at \(x = -\pi, -\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi\) and is increasing for \(x \in \left(-\pi, -\frac{\pi}{2} \right) \cup \left(0, \frac{\pi}{2} \right)\) and decreasing for \(x \in \left(-\frac{\pi}{2}, 0 \right) \cup \left(\frac{\pi}{2}, \pi \right)\). The graph has no inflection points or changes in concavity and approaches the value of \(1\) at both endpoints of the domain. The graph is symmetric about the \(y\)-axis.

Step 1: 1. Determine the domain and range of the function

The given function is defined for all real values of \(x\). However, since we only need to sketch the graph within the interval \(-\pi \leq x \leq \pi\), the domain is: \[-\pi \leq x \leq \pi\] The function \(y = \cos^2 x\) takes the square of the cosine function, whose values range from \(-1\) to \(1\). Since squaring a value results in non-negative values, the range of the function is \[0 \leq y \leq 1\]

Step 2: 2. Find the critical points

To find the critical points, take the derivative of the function with respect to \(x\), and set it equal to zero. \(\frac{dy}{dx} = \frac{d(\cos^2 x)}{dx} = 2\cos x\cdot(-\sin x) = -2\cos x\sin x\) We will set \(\frac{dy}{dx} = 0\) and find the values of \(x\) within the domain \(-\pi \leq x \leq \pi\). \(-2\cos x\sin x = 0\) The above equation equals 0 when either \(\cos x = 0\) or \(\sin x = 0\). For \(\cos x = 0\), \(x = \frac{-\pi}{2}, \frac{\pi}{2}\) \\ For \(\sin x = 0\), \(x = -\pi, 0, \pi\)

Step 3: 3. Determine increasing and decreasing intervals

By analyzing the sign of the derivative, we can determine the intervals where the function is increasing or decreasing within the domain. - For \(x \in \left(-\pi, -\frac{\pi}{2} \right)\), \(\frac{dy}{dx} > 0\), so the function is increasing. - For \(x \in \left(-\frac{\pi}{2}, 0 \right)\), \(\frac{dy}{dx} < 0\), so the function is decreasing. - For \(x \in \left(0, \frac{\pi}{2} \right)\), \(\frac{dy}{dx} > 0\), so the function is increasing. - For \(x \in \left(\frac{\pi}{2}, \pi \right)\), \(\frac{dy}{dx} < 0\), so the function is decreasing.

Step 4: 4. Find inflection points and concavity

Since the function \(y = \cos^2 x\) is a quadratic function regarding the cosine function, it does not have any inflection points or changes in concavity.

Step 5: 5. Determine end behavior and limits

As \(x\) approaches the end of the domain \(-\pi\) or \(\pi\), the function approaches the value \(1\) since the cosine of these values is \(1\). This means that: \[\lim_{x \to -\pi} \cos^2 x = 1 \quad \text{and} \quad \lim_{x \to \pi} \cos^2 x = 1\]

Step 6: 6. Identify symmetry

The graph of the function \(y = \cos^2 x\) is symmetric with respect to the \(y\)-axis, as cosine is an even function. Therefore, it has symmetry about the \(y\)-axis. Now, with all the analyzed information, sketch the graph of the function \(y = \cos^2 x\) for the specified domain \(-\pi \leq x \leq \pi\), making sure to include critical points, increasing and decreasing intervals, and symmetry.

Key Concepts

Critical Points

Critical points are where a function's derivative is either zero or undefined. These points are essential because they indicate potential local maxima or minima, and can help in understanding the shape of a graph. For the function \(y = \cos^2 x\), we find its critical points by taking the derivative and setting it equal to zero:

• The derivative is \(-2 \cos x \sin x\).
• This equation equals zero when \(\cos x = 0\) or \(\sin x = 0\).

Within the domain \(-\pi \leq x \leq \pi\), we find critical points at:

• \(x = \frac{-\pi}{2}, \frac{\pi}{2}, -\pi, 0, \pi\).

These points are important to identify because they indicate changes in the direction of the curve, where the function may switch from increasing to decreasing.

Function Symmetry

Symmetry in functions helps simplify the graphing process. A symmetric function can be mirrored along an axis, saving you time and effort in analysis. For the function \(y = \cos^2 x\), it is instructive to know if it exhibits symmetry. Since cosine is an even function, and squaring is also an even operation, \(\cos^2 x\) is symmetric around the y-axis. This means that whatever happens on one side of the y-axis is mirrored on the other.Having this symmetry implies that you can study half of the interval (e.g., from 0 to \(\pi\)) and apply the results to the full graph. This feature simplifies the sketching of the graph, as you need less overall data to fully understand the curve's behavior.

Increasing and Decreasing Intervals

Understanding where a function increases or decreases is key to sketching its graph. This is found by analyzing the sign of the derivative:

• If \(\frac{dy}{dx} > 0\), the function is increasing.
• If \(\frac{dy}{dx} < 0\), the function is decreasing.

For \(y = \cos^2 x\):

• In \((-\pi, -\frac{\pi}{2})\), the function is increasing.
• In \((-\frac{\pi}{2}, 0)\), the function is decreasing.
• In \((0, \frac{\pi}{2})\), it increases again.
• Finally, in \((\frac{\pi}{2}, \pi)\), it decreases.

These intervals tell you where the function is going upwards or downwards, defining the "hills" and "valleys" of the curve on a plot.

Inflection Points

Inflection points are where a curve changes its concavity, switching from concave up to concave down or vice versa. For the quadratic nature of \(y = \cos^2 x\) concerning the cosine, this function has no inflection points in its domain. This means there are no changes in the curvature of this specific function on the examined interval.The absence of inflection points suggests that the graph is either concave up or concave down throughout, with no shifts. Knowing this makes it easier to anticipate the shape of the graph, limiting the complexity of its curvature.

Domain and Range

The domain and range are critical for understanding which values a function can accept and produce. For \(y = \cos^2 x\), the domain is defined as \(-\pi \leq x \leq \pi\), which means the function is examined within this interval only.The range of \(y = \cos^2 x\) results from the squaring of the cosine function, which itself ranges from -1 to 1. Hence, the range is all non-negative values produced by squaring, specifically:

• The range is \(0 \leq y \leq 1\).

This tells us that the output (y-values) will never fall below 0 nor exceed 1 within this interval, giving us a constricted and predictable interval for the vertical axis of the graph.