Chapter 13

Write the approximate change formula for a function \(z=f(x, y)\) at the point \((a, b)\) in terms of differentials.

Consider the function \(f(x, y)=8-x^{2} / 2-y^{2},\) whose graph is a paraboloid (see figure). a. Fill in the table with the values of the directional derivative at the points \((a, b)\) in the directions \(\langle\cos \theta, \sin \theta\rangle\) $$\begin{array}{|l|l|l|l|} \hline & (a, b)=(2,0) & (a, b)=(0,2) & (a, b)=(1,1) \\ \hline \theta=\pi / 4 & & & \\ \hline \theta=3 \pi / 4 & & & \\ \hline \theta=5 \pi / 4 & & & \\ \hline \end{array}$$ b. Sketch the \(x y\) -plane and indicate the points and the direction of the directional derivative for each of the table entries in part (a).

Describe in words the level curves of the paraboloid \(z=x^{2}+y^{2}\).

Use Theorem 7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. $$d z / d t, \text { where } z=x^{2}+y^{3}, x=t^{2}, \text { and } y=t$$

Find the first partial derivatives of the following functions. $$f(x, y)=3 x^{2}+4 y^{3}$$

Lagrange multipliers in two variables Use Lagrange multipliers to find the maximum and minimum values of \(f\) (when they exist) subject to the given constraint. $$f(x, y)=x^{2}+y^{2} \text { subject to } 2 x^{2}+3 x y+2 y^{2}=7$$

What is the procedure for locating absolute maximum and minimum values on a closed bounded domain?

Let \(R\) be the unit disk $$\left\\{(x, y): x^{2}+y^{2} \leq 1\right\\}$$ with (0,0) removed. Is (0,0) a boundary point of \(R ?\) Is \(R\) open or closed?

Write the differential \(d w\) for the function \(w=f(x, y, z)\)

Consider the function \(f(x, y)=2 x^{2}+y^{2}\) whose graph is a paraboloid (see figure). a. Fill in the table with the values of the directional derivative at the points \((a, b)\) in the directions \(\langle\cos \theta, \sin \theta\rangle\) $$\begin{array}{|l|l|l|l|} \hline & (a, b)=(1,0) & (a, b)=(1,1) & (a, b)=(1,2) \\ \hline \theta=0 & & & \\ \hline \theta=\pi / 4 & & & \\ \hline \theta=\pi / 2 & & & \\ \hline \end{array}$$ b. Sketch the \(x y\) -plane and indicate the points and the direction of the directional derivative for each of the table entries in part (a).

What is the name of the surface defined by the equation \(y=\frac{x^{2}}{4}+\frac{z^{2}}{8} ?\)

How many axes (or how many dimensions) are needed to graph the level surfaces of \(w=f(x, y, z) ?\) Explain.

Use Theorem 7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. $$d z / d t, \text { where } z=x y^{2}, x=t^{2}, \text { and } y=t$$

Find the first partial derivatives of the following functions. $$f(x, y)=x^{2} y$$

Lagrange multipliers in two variables Use Lagrange multipliers to find the maximum and minimum values of \(f\) (when they exist) subject to the given constraint. $$f(x, y)=x y \text { subject to } x^{2}+y^{2}-x y=9$$

Find all critical points of the following functions. $$f(x, y)=1+x^{2}+y^{2}$$

At what points of \(\mathbb{R}^{2}\) is a rational function of two variables continuous?

Find an equation of the plane tangent to the following surfaces at the given points. $$x^{2}+y+z=3 ;(1,1,1) \text { and }(2,0,-1)$$

Computing gradients Compute the gradient of the following functions and evaluate it at the given point \(P\). $$f(x, y)=2+3 x^{2}-5 y^{2} ; P(2,-1)$$

What is the name of the surface defined by the equation \(x^{2}+\frac{y^{2}}{3}+2 z^{2}=1 ?\)

The domain of \(Q=f(u, v, w, x, y, z)\) lies in \(\mathbb{R}^{n}\) for what value of \(n ?\) Explain.

Use Theorem 7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. $$d z / d t, \text { where } z=x \sin y, x=t^{2}, \text { and } y=4 t^{3}$$

Find the first partial derivatives of the following functions. $$f(x, y)=3 x^{2} y+2$$

Lagrange multipliers in two variables Use Lagrange multipliers to find the maximum and minimum values of \(f\) (when they exist) subject to the given constraint. $$f(x, y)=x-y \text { subject to } x^{2}+y^{2}-3 x y=20$$

Find all critical points of the following functions. $$f(x, y)=x^{2}-6 x+y^{2}+8 y$$

Evaluate $$\lim _{(x, y, z) \rightarrow(1,1,-1)} x y^{2} z^{3}$$

Find an equation of the plane tangent to the following surfaces at the given points. $$x^{2}+y^{3}+z^{4}=2 ;(1,0,1) \text { and }(-1,0,1)$$

What is the name of the surface defined by the equation \(-y^{2}-\frac{z^{2}}{2}+x^{2}=1 ?\)

Give two methods for graphically representing a function with three independent variables.

Use Theorem 7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. $$d z / d t, \text { where } z=x^{2} y-x y^{3}, x=t^{2}, \text { and } y=t^{-2}$$

Find the first partial derivatives of the following functions. $$f(x, y)=y^{8}+2 x^{6}+2 x y$$

Lagrange multipliers in two variables Use Lagrange multipliers to find the maximum and minimum values of \(f\) (when they exist) subject to the given constraint. $$f(x, y)=e^{2 x y} \text { subject to } x^{2}+y^{2}=16$$

Find all critical points of the following functions. $$f(x, y)=(3 x-2)^{2}+(y-4)^{2}$$

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(2,9)} 101$$

Find an equation of the plane tangent to the following surfaces at the given points. $$x y+x z+y z-12=0 ;(2,2,2) \text { and }(2,0,6)$$

Computing gradients Compute the gradient of the following functions and evaluate it at the given point \(P\). $$g(x, y)=x^{2}-4 x^{2} y-8 x y^{2} ; P(-1,2)$$

Find an equation of the plane that passes through the point \(P_{0}\) with a normal vector \(\mathbf{n}\). $$P_{0}(0,2,-2) ; \mathbf{n}=\langle 1,1,-1\rangle$$

Find the domain of the following functions. $$f(x, y)=2 x y-3 x+4 y.$$

Use Theorem 7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. $$d w / d t, \text { where } w=\cos 2 x \sin 3 y, x=t / 2, \text { and } y=t^{4}$$

Find the first partial derivatives of the following functions. $$f(x, y)=x e^{y}$$

Lagrange multipliers in two variables Use Lagrange multipliers to find the maximum and minimum values of \(f\) (when they exist) subject to the given constraint. $$f(x, y)=x^{2}+y^{2} \text { subject to } x^{6}+y^{6}=1$$

Find all critical points of the following functions. $$f(x, y)=3 x^{2}-4 y^{2}$$

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(1,-3)}(3 x+4 y-2)$$

Find an equation of the plane tangent to the following surfaces at the given points. $$x^{2}+y^{2}-z^{2}=0 ;(3,4,5) \text { and }(-4,-3,5)$$

Computing gradients Compute the gradient of the following functions and evaluate it at the given point \(P\). $$p(x, y)=\sqrt{12-4 x^{2}-y^{2}} ; P(-1,-1)$$

Find an equation of the plane that passes through the point \(P_{0}\) with a normal vector \(\mathbf{n}\). $$P_{0}(1,0,-3) ; \mathbf{n}=\langle 1,-1,2\rangle$$

Find the domain of the following functions. $$f(x, y)=\cos \left(x^{2}-y^{2}\right).$$

Use Theorem 7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. $$d z / d t, \text { where } z=\sqrt{r^{2}+s^{2}}, r=\cos 2 t, \text { and } s=\sin 2 t$$

Find the first partial derivatives of the following functions. $$f(x, y)=\ln (x / y)$$

Lagrange multipliers in two variables Use Lagrange multipliers to find the maximum and minimum values of \(f\) (when they exist) subject to the given constraint. $$f(x, y)=y^{2}-4 x^{2} \text { subject to } x^{2}+2 y^{2}=4$$