91Ó°ÊÓ

The derivative of the sine function is a fundamental concept in calculus. Mathematically, the derivative of \( \sin x \) with respect to \( x \) is \( \cos x \). This means that at any point \( x \), the rate at which \( \sin x \) is changing is equal to the value of \( \cos x \). For instance, at \( x = \pi \), the derivative \( \cos(\pi) = -1 \). This derivative tells us how the sine function is behaving at any given point. It's essential to understand that this derivative provides the "instantaneous" rate of change of \( \sin x \) at \( x \). In practical terms, it helps us understand how a wave oscillates as represented by \( \sin x \). The sinusoidal nature of the function is reflected in its derivative, showcasing peaks and troughs at predictable intervals.

The difference quotient is a crucial tool used to approximate the derivative of a function. It represents the average rate of change of the function over a small interval. The general form of a difference quotient for a function \( f(x) \) is given by:

• \[ \frac{f(x + \Delta x) - f(x)}{\Delta x} \]

This expression calculates the slope of the secant line passing through the points \( (x, f(x)) \) and \( (x + \Delta x, f(x + \Delta x)) \). As \( \Delta x \) approaches zero, the difference quotient converges to the derivative of the function at \( x \). In the context of the exercise, using the difference quotient with \( f(x) = \sin x \) allows us to find an approximation for the derivative \( \frac{dy}{dx} \) at \( x = \pi \). By setting \( \Delta x = 0.01 \), we apply the difference quotient to the sine function and find that it closely approximates the derivative at that point.

The instantaneous rate of change is a concept that describes how a function changes at a specific point, and it is synonymous with the derivative. For example, in physics, it can represent the velocity of an object at a particular moment. In mathematics, when we talk about the instantaneous rate of change of a function \( y = f(x) \), we're referring to the derivative \( \frac{dy}{dx} \). At any specific value \( x \), the derivative tells us the rate at which \( y \) is changing with respect to \( x \). In our exercise, the instantaneous rate of change at \( x = \pi \) for \( y = \sin x \) gives us the slope of the tangent line to the curve at that point. This is equivalent to evaluating the derivative at \( x = \pi \), yielding \( -1 \). Thus, the rate at which the sine function changes instantaneously at \( x = \pi \) is \(-1\), indicating it is decreasing at that point.