91Ó°ÊÓ

Let $$f(x, y)=4-x^{2}-y^{2}$$ Compute \(f_{x}(1,1)\) and \(f_{y}(1,1)\), and interpret these partial derivatives geometrically.

Find the linear approximation of $$ f(x, y)=e^{x+y} $$ at \((0,0)\), and use it to approximate \(f(0.1,0.05) .\) Using a calculator, compare the approximation with the exact value of \(f(0.1,0.05)\)

Find a unit vector that is normal to the level curve of the function $$ f(x, y)=x^{2}-y^{3} $$ at the point \((1,3)\).

Can a continuous function of two variables have two maxima and no minima? Describe in words what the properties of such a function would be, and contrast this behavior with a function of one variable.

Show that the equilibrium \(\begin{array}{ll}0 & \text { of } \\ 0\end{array}\) $$ \left[\begin{array}{l} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{ll} 4.2 & -3.4 \\ 2.4 & -1.1 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ is unstable.

Show that $$f(x, y)=\left\\{\begin{array}{cc} \frac{3 r^{2} y}{\left(2 x^{4}+y^{2}\right)} & \text { for }(x, y) \neq(0,0) \\\ 0 & \text { for }(x, y)=(0,0)\end{array}\right.$$ is discontinuous at \((0,0) .\)

Find a unit vector that is normal to the level curve of the function $$ f(x, y)=x y $$ at the point \((2,3)\).

Suppose \(f(x, y)\) has a horizontal tangent plane at \((0,0)\). Can you conclude that \(f\) has a local extremum at \((0,0)\) ?

Let $$f(x, y)=\sqrt{4-x^{2}-y^{2}}$$ Compute \(f_{x}(1,1)\) and \(f_{y}(1,1)\), and interpret these partial derivatives geometrically.

Find the linear approximation of $$ f(x, y)=\sin (x+2 y) $$ at \((0,0)\), and use it to approximate \(f(-0.1,0.2) .\) Using a calculator, compare the approximation with the exact value of \(f(-0.1,0.2)\)