16. [-/1 Points] DETAILS MCALCET9 \( 2 . X P .7 .018 \). The limit represents the derivative of a function \( y=f(x) \) at some number \( a \). Find the function \( f \) and the value of \( a \). \[ \begin{array}{l}\lim _{h \rightarrow 0} \frac{\cos (\pi+h)+1}{h} \\ f(x)=\cos (x), a=0 \\ f(x)=\cos (x), a=\pi \\ f(x)=\cos (x), a=\frac{\pi}{4} \\ f(x)=\cos (x), a=\frac{\pi}{6} \\ f(x)=\cos (x), a=\frac{\pi}{3}\end{array} \]

The limit given is related to the derivative of the function at a specific point. When you evaluate the limit as \(h\) approaches 0, you derive the behavior of the function \(\cos(\pi + h)\) around \(h = 0\). Notably, \(\cos(\pi + h) = -\cos(h)\), and as \(h\) approaches 0, this approximation gets you to \( -1 \). Therefore, the function \(f(x) = \cos(x)\) is correct, and evaluating it at \(x = \pi\) gives us the required derivative at that point. So, \(a = \pi\) is the correct answer! Now, buckle up for a bit of historical background! The concept of limits and derivatives can be traced back to the work of Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. Their formulations paved the way for calculus, which was crucial for advancements in physics, engineering, and economics. They independently developed the fundamental principles around derivatives, helping us understand not just motion and change but also the beauty of mathematics as a whole. And what about some real-world application? The derivative is crucial in numerous fields! For instance, in physics, it helps us determine the velocity of an object at any instant by looking at the rate of change of position with respect to time. In economics, derivatives can help analyze trends, predict consumer behavior, or optimize production processes. So, the next time you’re watching a car zoom by or checking stock prices, remember that behind the scenes, derivatives are at work!