Question

Find the slopes of the surface in the \(x\) - and \(y\) -directions at the given point. $$ \begin{array}{l} g(x, y)=4-x^{2}-y^{2} \\ (1,1,2) \end{array} $$

Short answer

The slopes of the surface at the point \((1,1,2)\) in the \(x\) and \(y\) directions are both \(-2\).

Step-by-step solution

Find the Partial Derivatives

We first compute the partial derivative of \(g(x, y)\) with respect to \(x\) and \(y\). \n\nThe partial derivative with respect to \(x\) is: \n\( g_x = \frac{∂g}{∂x} = -2x \) \n\nThe partial derivative with respect to \(y\) is: \n\( g_y = \frac{∂g}{∂y} = -2y \)

Evaluate the Partial Derivatives at the Given Point

Next, we substitute \(x=1\) and \(y=1\) into the partial derivatives to find the slopes of the surface in the \(x\) and \(y\) directions at the point \((1,1,2)\). \n\nEvaluating \(g_x\) at \((1,1,2)\): \n\( g_x(1,1,2) = -2*1 = -2 \) \n\nEvaluating \(g_y\) at \((1,1,2)\): \n\( g_y(1,1,2) = -2*1 = -2 \)

Interpret the Result

The slopes in \(x\) and \(y\) directions at \((1,1,2)\) are both \(-2\). This means that if we move a small amount in either the \(x\) or \(y\) direction from this point on the given surface, the surface goes down with slope \(-2\).