Problem 44 - Calculus A New Horizon

Question

Use a graphing utility to generate the graphs of \(f^{\prime}\) and \(f^{\prime \prime}\) over the stated interval, and then use those graphs to estimate the \(x\) -coordinates of the relative extrema of \(f .\) Check that your estimates are consistent with the graph of \(f\). $$f(x)=\sin \frac{1}{2} x \cos x, \quad-\pi / 2 \leq x \leq \pi / 2$$

Step-by-Step Solution

Answer

Graph \(f'(x)\) and \(f''(x)\) to estimate relative extrema of \(f(x)\) and verify using the graph of \(f\).

Step 1: Identify the functions

We start by identifying the functions we need to graph. We are given the function \(f(x) = \sin \frac{1}{2} x \cos x\). We need to find the first derivative, \(f'(x)\), and the second derivative, \(f''(x)\), to find the relative extrema.

Step 2: Differentiate the function

Calculate \(f'(x)\) using the product rule. The product rule states: \( (uv)' = u'v + uv' \), where \(u = \sin \frac{1}{2}x\) and \(v = \cos x\).Differentiate the components:* \(u' = \frac{1}{2} \cos \frac{1}{2}x\)* \(v' = -\sin x\)Thus, \(f'(x) = \frac{1}{2} \cos \frac{1}{2}x \cdot \cos x + \sin \frac{1}{2}x \cdot (-\sin x)\).

Step 3: Differentiate again to find f''(x)

Differentiate \(f'(x)\) again to find \(f''(x)\). This will involve using both the product and chain rules since \(f'(x)\) itself is a product of functions. Calculate both parts separately for \(f''(x)\).

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

Use a graphing utility to plot \(f'(x)\) and \(f''(x)\) over the given interval \([-\pi/2, \pi/2]\). These graphs will show where \(f'(x) = 0\) (indicating potential relative extrema) and the concavity from \(f''(x)\).

Step 5: Analyze graphs for extrema

Examine the graph of \(f'(x)\) to locate where it crosses the x-axis (where \(f'(x) = 0\)), as these points are potential relative extrema. Verify these points by examining the sign of \(f''(x)\) at the determined x-values to confirm whether they are maxima or minima.

Step 6: Confirm with the graph of f

Graph the original function \(f(x)\) over the same interval to confirm the relative extrema by checking that increases and decreases correspond to where \(f'(x)\) is positive or negative.

Key Concepts

Relative Extrema

In calculus, relative extrema refer to points on a graph where a function reaches a relative maximum or minimum. These occur at critical points where the derivative of the function, denoted as \( f'(x) \), is zero or undefined. When we graph \( f'(x) \), the locations where \( f'(x) \) cross the x-axis indicate potential relative extrema. To confirm these points, we analyze the sign changes of \( f'(x) \):

• A change from positive to negative indicates a relative maximum.
• A change from negative to positive indicates a relative minimum.

Once we identify the critical points using \( f'(x) \), we must verify them using the second derivative, \( f''(x) \), to determine the nature of these extrema.

Product Rule

The product rule is an essential technique in calculus used to differentiate products of two functions. When given a function that is a product of two differentiable functions \( u(x) \) and \( v(x) \), the derivative is calculated as:\[(uv)' = u'v + uv'\]In the context of the problem, we apply the product rule to \( f(x) = \sin \frac{1}{2}x \cdot \cos x \), where \( u = \sin \frac{1}{2}x \) and \( v = \cos x \). Differentiating each component yields:

• \( u' = \frac{1}{2} \cos \frac{1}{2}x \)
• \( v' = -\sin x \)

Thus, the derivative \( f'(x) \) can be expressed as the sum of these product combinations, showing how changes in each part affect the overall function.

Derivative Graph Analysis

Graphing the derivatives of a function provides visual insights into its behavior. For this exercise, plotting \( f'(x) \) and \( f''(x) \) helps to:

• Identify points where \( f'(x) = 0 \), suggesting potential relative extrema.
• Observe the overall slope of the function, with positive values indicating increasing intervals and negative values indicating decreasing intervals.
• Check the concavity of the function using \( f''(x) \).

By comparing these graphs in the interval \([-\pi/2, \pi/2]\), students can see how the derivative shapes the function's progression and allows prediction of its high and low points.

Concavity and Inflection Points

Concavity describes how a function curves, affected by the second derivative \( f''(x) \). If \( f''(x) > 0 \), the function is concave up (resembling a cup), and if \( f''(x) < 0 \), it's concave down (like a cap).
To determine concavity and locate inflection points, analyze the graph of \( f''(x) \):

• Inflection points occur where the sign of \( f''(x) \) changes. These points indicate a switch in concavity.
• Verify inflection points by observing them in the graph of the original function \( f(x) \).

Understanding concavity helps in analyzing the global shape of a graph, providing insights into potential behaviors and guiding accurate predictions of extrema.