Chapter 4

[T] Use technology to sketch the level curve of \(f(x, y)=x^{2}+4 y^{2}\) that passes through \(P(-2,0)\) and draw the gradient vector at \(P\).

For the following exercises, find the gradient vector at the indicated point. $$f(x, y)=x y^{2}-y x^{2}, P(-1,1)$$

For the following exercises, find the gradient vector at the indicated point. $$\quad f(x, y)=x e^{y}-\ln (x), P(-3,0)$$

For the following exercises, find the gradient vector at the indicated point. $$f(x, y, z)=x y-\ln (z), P(2,-2,2)$$

For the following exercises, find the gradient vector at the indicated point. $$f(x, y, z)=x \sqrt{y^{2}+z^{2}}, P(-2,-1,-1)$$

For the following exercises, find the derivative of the function. \(f(x, y)=x^{2}+x y+y^{2}\) at point (-5,-4) in the direction the function increases most rapidly.

For the following exercises, find the derivative of the function. \(f(x, y)=e^{x y}\) at point (6,7) in the direction the function increases most rapidly.

For the following exercises, find the derivative of the function. \(\quad f(x, y)=\arctan \left(\frac{y}{x}\right)\) at point (-9,9) in the direction the function increases most rapidly.

For the following exercises, find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. $$f(x, y)=x e^{-y}, \quad(1,0)$$

For the following exercises, find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. $$f(x, y)=\sqrt{x^{2}+2 y}(4,10)$$

For the following exercises, find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. $$\quad f(x, y)=\cos (3 x+2 y),\left(\frac{\pi}{6},-\frac{\pi}{8}\right)$$

For the following exercises, find equations of a. the tangent plane and b. the normal line to the given surface at the given point. The level curve \(f(x, y, z)=12\) for \(f(x, y, z)=4 x^{2}-2 y^{2}+z^{2}\) at point (2,2,2)

For the following exercises, find equations of a. the tangent plane and b. the normal line to the given surface at the given point. \(f(x, y, z)=x y z=6\) at point (1,2,3).

For the following exercises, find equations of a. the tangent plane and b. the normal line to the given surface at the given point. \(f(x, y, z)=x e^{y} \cos z-z=1\) at point (1,0,0).

Solve the problem. The temperature \(T\) in a metal sphere is inversely proportional to the distance from the center of the sphere (the origin: (0,0,0)\()\). The temperature at point (1,2,2) is \(120^{\circ} \mathrm{C}\). a. Find the rate of change of the temperature at point (1,2,2) in the direction toward point (2,1,3) . b. Show that, at any point in the sphere, the direction of greatest increase in temperature is given by a vector that points toward the origin.

Solve the problem. The electrical potential (voltage) in a certain region of space is given by the function \(V(x, y, z)=5 x^{2}-3 x y+x y z\) a. Find the rate of change of the voltage at point (3,4,5) in the direction of the vector \langle 1,1,-1\rangle b. In which direction does the voltage change most rapidly at point (3,4,5)\(?\) c. What is the maximum rate of change of the voltage at point (3,4,5)\(?\)

Solve the problem. If the electric potential at a point \((x, y)\) in the \(x y\) -plane is \(V(x, y)=e^{-2 x} \cos (2 y),\) then the electric intensity vector at \((x, y)\) is \(\mathbf{E}=-\nabla V(x, y)\) a. Find the electric intensity vector at \(\left(\frac{\pi}{4}, 0\right)\). b. Show that, at each point in the plane, the electric potential decreases most rapidly in the direction of the vector \(\mathbf{E}\)

In two dimensions, the motion of an ideal fluid is governed by a velocity potential \(\varphi .\) The velocity components of the fluid \(u\) in the \(x\) -direction and \(v\) in the \(y\) -direction, are given by \(\langle u, v\rangle=\nabla \varphi .\) Find the velocity components associated with the velocity potential \(\varphi(x, y)=\sin \pi x \sin 2 \pi y\).

For the following exercises, find all critical points. $$f(x, y)=1+x^{2}+y^{2}$$

For the following exercises, find all critical points. $$ f(x, y)=(3 x-2)^{2}+(y-4)^{2}$$

For the following exercises, find all critical points. $$f(x, y)=x^{4}+y^{4}-16 x y$$

For the following exercises, find the critical points of the function by using algebraic techniques (completing the square) or by examining the form of the equation. Verify your results using the partial derivatives test. $$f(x, y)=\sqrt{x^{2}+y^{2}+1}$$

For the following exercises, find the critical points of the function by using algebraic techniques (completing the square) or by examining the form of the equation. Verify your results using the partial derivatives test. $$f(x, y)=-x^{2}-5 y^{2}+8 x-10 y-13$$

For the following exercises, find the critical points of the function by using algebraic techniques (completing the square) or by examining the form of the equation. Verify your results using the partial derivatives test. $$f(x, y)=x^{2}+y^{2}+2 x-6 y+6$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=-x^{3}+4 x y-2 y^{2}+1$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=x^{2} y^{2}$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=x^{2}-6 x+y^{2}+4 y-8$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=2 x y+3 x+4 y$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=8 x y(x+y)+7$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=x^{2}+4 x y+y^{2}$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=x^{3}+y^{3}-300 x-75 y-3$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=9-x^{4} y^{4}$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=7 x^{2} y+9 x y^{2}$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=3 x^{2}-2 x y+y^{2}-8 y$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=3 x^{2}+2 x y+y^{2}$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=y^{2}+x y+3 y+2 x+3$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=x^{2}+x y+y^{2}-3 x$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=x^{2}+2 y^{2}-x^{2} y$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=x^{2}+y-e^{y}$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=e^{-\left(x^{2}+y^{2}+2 x\right)}$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=x^{2}+x y+y^{2}-x-y+1$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=x^{2}+10 x y+y^{2}$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=-x^{2}-5 y^{2}+10 x-30 y-62$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=120 x+120 y-x y-x^{2}-y^{2}$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=2 x^{2}+2 x y+y^{2}+2 x-3$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=x^{2}+x-3 x y+y^{3}-5$$

For the following exercises, use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $$f(x, y)=2 x y e^{-x^{2}-y^{2}}$$

For the following exercises, determine the extreme values and the saddle points. Use a CAS to graph the function. $$[T]f(x, y)=y e^{x}-e^{y}$$

For the following exercises, determine the extreme values and the saddle points. Use a CAS to graph the function. $$[T]f(x, y)=x \sin (y)$$

For the following exercises, determine the extreme values and the saddle points. Use a CAS to graph the function. $$f(x, y)=\sin (x) \sin (y), x \in(0,2 \pi), y \in(0,2 \pi)$$