Activity 46.
For each section perform the calculations then use them to explain what happens at that critical value.
(a)
Here we use a simple relative maximum to identify what the gradient does near a maximum.
The surface is
\begin{equation*}
f(x,y)=1-x^2-y^2\text{.}
\end{equation*}
(i)
Graph the surface.
(ii)
Find the single critical value using the partial derivatives.
(iii)
Use the extrema theorem to identify this as a maximum.
(iv)
Calculate the gradient at points one unit to the left, right, forward, and backward of the critical value. For example if the critical value occurs at (1,2), calculate the gradient at (0,2), (2,2), (1,1), and (1,3).
(v)
Around a relative maximum what should the gradient do?
Hint.
From each of these points graph the gradient vector and compare to the critical point.
(b)
Around a relative minimum what should the gradient do?
(c)
Here we use a saddle point to identify what the gradient does near a saddle point.
The surface is
\begin{equation*}
g(x,y)=x^2-y^2\text{.}
\end{equation*}
(i)
Graph the surface.
(ii)
Find the single critical value using the partial derivatives.
(iii)
Use the extrema theorem to identify this as a saddle point.
(iv)
Calculate the gradient at points one unit to the left, right, forward, backward and left/forward and right/forward of the critical value. For example if the critical value occurs at (1,2), calculate the gradient at (0,2), (2,2), (1,1), (1,3), (0,1), and (2,1).
(v)
Around a saddle point what should the gradient do?
Hint.
From each of these points graph the gradient vector and compare to the critical point.
(d)
Here we analyze a point that is not an extreme point nor a saddle point.
The surface is
\begin{equation*}
h(x,y)=x^3+y^3\text{.}
\end{equation*}
(i)
Graph the surface.
(ii)
Find the single critical value using the partial derivatives.
(iii)
Use the extrema theorem to determine this is not a maximum, minimum, nor saddle point.
(iv)
Calculate the gradient at points one unit to the left, right, forward, backward and left/forward and right/forward of the critical value. For example if the critical value occurs at (1,2), calculate the gradient at (0,2), (2,2), (1,1), (1,3), (0,1), and (2,1).
(v)
Calculate both \(\frac{\partial^2 h}{\partial x^2}\) and \(\frac{\partial^2 h}{\partial y^2}\) at the same points as above.
(vi)
What is special about this critical point?
(e)
Here we analyze another point that is not an extreme point nor a saddle point.
The surface is
\begin{equation*}
j(x,y)=x^3+y^2\text{.}
\end{equation*}
(i)
Graph the surface.
(ii)
Find the single critical value using the partial derivatives.
(iii)
Use the extrema theorem to determine this is not a maximum, minimum, nor saddle point.
(iv)
Calculate the gradient at points one unit to the left, right, forward, backward and left/forward and right/forward of the critical value. For example if the critical value occurs at (1,2), calculate the gradient at (0,2), (2,2), (1,1), (1,3), (0,1), and (2,1).
(v)
Calculate both \(\frac{\partial^2 h}{\partial x^2}\) and \(\frac{\partial^2 h}{\partial y^2}\) at the same points as above.
(vi)
What is special about this critical point?
