Preview Activity 11.5.1.
We want to find the equation of the plane, using the form given in Key Idea 11.5.2, that best describes the surface given by \(z=f(x,y)=6-\frac{x^2}2 - y^2\) for input values around \((x_0,y_0) = (1,1)\text{.}\) In particular, we will need to find how the values of \(z_0\text{,}\) \(a\text{,}\) and \(b\) are related to \(f(x,y)\text{.}\)
(a)
(b)
We want the plane to match the height of the surface at \((1,1)\text{.}\) For this to happen, the value \(z_0\) from Key Idea 11.5.2 must be the \(z\)-coordinate value where the plane intersects the surface. What is the \(z\)-coordinate of the point where the tangent plane and the surface intersect?
(c)
For the plane to have the same behavior as the surface \(z=f(x,y)\) near \((1,1)\text{,}\) the plane must match the behavior of the traces \(x=1\) and \(y=1\) near this point.
Sketch the traces of \(f(x,y) = 6 - \frac{x^2}2 - y^2\) for \(y=y_0=1\) and \(x=x_0=1\) below. Draw the tangent lines to the each of the traces when the appropriate input is 1.
(d)
What are the slopes of the tangent lines to the traces that you drew in the previous part? Write a couple of sentences to explain why the tilt of the tangent plane in the \(x\)-direction is given by the partial derivative \(f_x\) and the tilt of the tangent plane in the \(y\)-direction is given by the partial derivative \(f_y\text{.}\) Your answer should discuss how each of these slopes/partial derivatives relates to the traces of the surface and the plane illustrated below. You should identify whether the red/orange traces along the plane and the black/blue traces along the surface correspond to constant values of \(x\) or \(y\text{.}\)
(e)
Fill in the blanks below with the proper values to give the tangent plane to the graph of \(f(x,y)=6-x^2/2 - y^2\) at the point \((x_0,y_0)=(1,1)\text{.}\)
\begin{equation*}
z=z_0 + a(x-x_0) + b(y-y_0)= \fillinmath{X} + \fillinmath{X}(x-\fillinmath{X}) +\fillinmath{X}(y-\fillinmath{X})\text{.}
\end{equation*}
