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Derivatives play a crucial role in calculus, serving as tools to determine the rate at which a function changes. Essentially, the derivative measures how much the value of the function increases or decreases as its input changes. In our example with the function \( f(x) = \sin\left(\frac{1}{2}x\right) \cos(x) \), we applied the product rule to find the first derivative.

• Identify the parts to differentiate: treat the function as a product \( u \cdot v \).
• For \( u = \sin\left(\frac{1}{2}x\right) \), the derivative \( u' \) is \( \frac{1}{2}\cos\left(\frac{1}{2}x\right) \).
• For \( v = \cos(x) \), the derivative \( v' \) is \( -\sin(x) \).

To find \( f'(x) \), apply the product rule: \( u'v + uv' \), yielding \( f'(x) = \frac{1}{2}\cos\left(\frac{1}{2}x\right)\cos(x) - \sin\left(\frac{1}{2}x\right)\sin(x) \).We then simplify this using trigonometric identities, resulting in \( f'(x) = \frac{1}{2}\cos\left(\frac{3}{2}x\right) \). This derivative tells us about the slope of the original function \( f(x) \). A positive derivative suggests \( f(x) \) is increasing, while a negative derivative suggests it's decreasing.

Relative extrema refer to the turning points on the graph of a function, where the function reaches local minimums or maximums. These points are crucial because they represent values where the function's rate of change switches direction—from increasing to decreasing or vice versa. To find these points, we look where the first derivative \( f'(x) \) equals zero and change signs.To determine the relative extrema for \( f(x) \), we:

• Find where \( f'(x) = 0 \) within the interval \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\).
• Solving \( \frac{1}{2}\cos\left(\frac{3}{2}x\right) = 0 \), we get the values \( x = -\frac{\pi}{3}, 0, \frac{\pi}{3} \).

At these \( x \)-coordinates, the function changes its direction:- At \( x = -\frac{\pi}{3} \) and \( x = \frac{\pi}{3} \), these points likely indicate relative minimums or maximums based on the sign change in \( f'(x) \).- At \( x = 0 \), depending on the context (increasing or decreasing), it could either be a maximum or minimum point.Determining whether these points are indeed relative extrema can be cross-referenced with the graph of the function \( f(x) \).

Graphing utilities are incredibly valuable tools in calculus used to visualize functions and their derivatives. They allow us to see the behavior of specific mathematical expressions over a given interval, aiding in understanding complex calculus concepts.To apply a graphing utility:

• Enter the expression into the utility. For instance, \( f'(x) = \frac{1}{2}\cos\left(\frac{3}{2}x\right) \) for the derivative graph.
• Set appropriate viewing windows, such as \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\), to encompass critical points of interest.
• Observe where the graph crosses the x-axis, as these correspond to zero points of the derivative.

This visualization helps identify the \( x \)-coordinates of potential relative extrema by locating where \( f'(x) \) transitions from positive to negative.By checking the graphs of both the function and its derivatives, we ensure that estimated extrema points align appropriately. Graphing utilities reinforce analytical findings, providing a visual confirmation of the changes and behaviors in the function's graph.