Answer:
To find the slope of the curve at the specified point, we can use the concept of differentiation. We'll need to find the derivative of the equation \(x = \cos(y)\) with respect to \(y\), and then evaluate it at the point \((x, y) = (\cos(-3\pi), -3\pi)\).
First, let's find the derivative of \(x\) with respect to \(y\):
\[\frac{dx}{dy} = \frac{d}{dy}(\cos(y))\]
Using the chain rule, we get:
\[\frac{dx}{dy} = -\sin(y)\frac{dy}{dy} = -\sin(y)\]
Now, we need to evaluate this derivative at the point \((x, y) = (\cos(-3\pi), -3\pi)\).
Since \(\cos(-3\pi) = \cos(3\pi) = 1\) and \(\sin(-3\pi) = \sin(3\pi) = 0\), we have:
\[\frac{dx}{dy} \bigg|_{(x, y) = (\cos(-3\pi), -3\pi)} = -\sin(-3\pi) = 0\]
So, the slope of the curve at the point \((x, y) = (\cos(-3\pi), -3\pi)\) is \(m = 0\).
None of the given options (A, B, C, D) correspond to \(m = 0\), so it seems there might be a mistake in the answer choices provided.
Explanation: