Problem 90 - Thomas Calculus

Question

Graph \(f(x)=x \cos x\) and its second derivative together for \(0 \leq x \leq 2 \pi\) . Comment on the behavior of the graph of \(f\) in relation to the signs and values of \(f^{\prime \prime} .\)

Step-by-Step Solution

Answer

\(f(x)\) is concave up where \(f''(x) > 0\) and concave down where \(f''(x) < 0\), indicating regions of changing slopes affected by \(x\) and \(\cos x\).

Step 1: Understand the function

To graph the function, first understand that \(f(x) = x \cos x\) is a product of a linear function and a trigonometric function. The linear part \(x\) will stretch the cosine function as \(x\) increases.

Step 2: Find the first derivative

To find the behavior of the graph, we need the derivatives. The first derivative \(f'(x)\) can be found using the product rule: \(f'(x) = \cos x - x \sin x\).

Step 3: Find the second derivative

The second derivative \(f''(x)\) gives us information about concavity. Use the product rule again to find \(f''(x)\):\[ f''(x) = -2 \sin x - x \cos x \]

Step 4: Graph \(f(x)\) and \(f''(x)\)

Graph both \(f(x) = x \cos x\) and \(f''(x)\) on the same axes over the interval \(0 \leq x \leq 2\pi\). Notice where \(f(x)\) is concave up and down by observing the sign of \(f''(x)\).

Step 5: Analyze the behavior

By examining the signs of \(f''(x) = -2 \sin x - x \cos x\), determine the intervals of concavity:- \(f''(x) > 0\) indicates the graph of \(f(x)\) is concave up.- \(f''(x) < 0\) indicates the graph of \(f(x)\) is concave down. Note periodicity and transition of inflection points.

Key Concepts

Second Derivative

The second derivative of a function provides valuable insights into the behavior of its graph. It is essentially the derivative of the function's first derivative. In this context, understanding the function \( f(x) = x \cos x \), we need to apply the product rule multiple times to work our way to the second derivative. Once we derive it, the resulting expression \( f''(x) = -2 \sin x - x \cos x \) tells us about the function's concavity and points where the graph may have inflection points.

• When \( f''(x) > 0 \), the function is concave up, forming the shape of a smile.
• When \( f''(x) < 0 \), the function is concave down, resembling a frown.

The second derivative is crucial in determining changes in concavity, thus playing a central role in sketching and understanding curves. By examining these changes, we can better predict and comprehend the function's overall path.

Product Rule

The product rule is a fundamental differentiation technique used when an expression includes a product of two functions. For a function \( f(x) = u(x)v(x) \), where \( u \) and \( v \) are each functions of \( x \), the product rule states: \[ f'(x) = u'(x)v(x) + u(x)v'(x)\]
This is highly relevant to our function \( f(x) = x \cos x \), where \( u(x) = x \) and \( v(x) = \cos x \). When we apply the product rule, we find:

• The derivative of \( x \) is 1.
• The derivative of \( \cos x \) is \( - \sin x \).

By substituting these into the product rule formula, we obtain the first derivative \( f'(x) = \cos x - x \sin x \). This step is essential in proceeding to find the second derivative, impacting everything from identifying zeros and critical points to analyzing concavity and curvature.

Concavity Analysis

Concavity analysis involves using the second derivative to determine where a function's graph curves upward or downward. For our function \( f(x) = x \cos x \), having the second derivative \( f''(x) = -2 \sin x - x \cos x \) is key to this analysis. This information can help understand the behavior across the interval \( 0 \leq x \leq 2\pi \):

• Identify intervals where \( f''(x) > 0 \) to establish where the function is concave up.
• Conversely, find where \( f''(x) < 0 \) to see where the function is concave down.
• At points where \( f''(x) = 0 \), combined with a change in concavity, determine the locations of inflection points.

By using this analysis method, students can effectively visualize these segments on the graph of \( f(x) \). Understanding these can lead to identifying the overall shape and flow of the curve, which is crucial when sketching graphs or analyzing functional relationships in calculus.