Chapter 4

For the following exercises, find an equation of the level curve of \(f\) that contains the point \(P\). $$g(x, y)=e^{x y}\left(x^{2}+y^{2}\right), P(1,0)$$

The strength \(E\) of an electric field at point \((x, y, z)\) resulting from an infinitely long charged wire lying along the \(y\) -axis is given by \(E(x, y, z)=k / \sqrt{x^{2}+y^{2}},\) where \(k\) is a positive constant. For simplicity, let \(k=1\) and find the equations of the level surfaces for \(E=10\) and \(E=100\).

A thin plate made of iron is located in the \(x y\) -plane. The temperature \(T\) in degrees Celsius at a point \(P(x, y)\) is inversely proportional to the square of its distance from the origin. Express \(T\) as a function of \(x\) and \(y\).

For the following exercises, find the limit of the function. $$\lim _{(x, y) \rightarrow(1,2)} x$$

For the following exercises, find the limit of the function. $$\lim _{(x, y) \rightarrow(1,2)} \frac{5 x^{2} y}{x^{2}+y^{2}}$$

Show that the limit \(\lim _{(x, y) \rightarrow(0,0)} \frac{5 x^{2} y}{x^{2}+y^{2}}\) exists and is the same along the paths: \(y\) -axis and \(x\) -axis, and along \(y=x .\)

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\). If the limit does not exist, state this and explain why the limit does not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{4 x^{2}+10 y^{2}+4}{4 x^{2}-10 y^{2}+6}$$

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\). If the limit does not exist, state this and explain why the limit does not exist. $$\lim _{(x, y) \rightarrow(11,13)} \sqrt{\frac{1}{x y}}$$

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\). If the limit does not exist, state this and explain why the limit does not exist. $$\lim _{(x, y) \rightarrow(0,1)} \frac{y^{2} \sin x}{x}$$

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\). If the limit does not exist, state this and explain why the limit does not exist. $$\lim _{(x, y) \rightarrow(0,0)} \sin \left(\frac{x^{8}+y^{7}}{x-y+10}\right)$$

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\). If the limit does not exist, state this and explain why the limit does not exist. $$\lim _{(x, y) \rightarrow(\pi / 4,1)} \frac{y \tan x}{y+1}$$

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\). If the limit does not exist, state this and explain why the limit does not exist. $$\lim _{(x, y) \rightarrow(2,5)}\left(\frac{1}{x}-\frac{5}{y}\right)$$

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\). If the limit does not exist, state this and explain why the limit does not exist. $$\lim _{(x, y) \rightarrow(4,4)} x \ln y$$

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\). If the limit does not exist, state this and explain why the limit does not exist. $$\lim _{(x, y) \rightarrow(4,4)} e^{-x^{2}-y^{2}}$$

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\). If the limit does not exist, state this and explain why the limit does not exist. $$\lim _{(x, y) \rightarrow(0,0)} \sqrt{9-x^{2}-y^{2}}$$

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\). If the limit does not exist, state this and explain why the limit does not exist. $$\lim _{(x, y) \rightarrow(1,2)}\left(x^{2} y^{3}-x^{3} y^{2}+3 x+2 y\right)$$

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\). If the limit does not exist, state this and explain why the limit does not exist. $$\lim _{(x, y) \rightarrow(\pi, \pi)} x \sin \left(\frac{x+y}{4}\right)$$

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\). If the limit does not exist, state this and explain why the limit does not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x y+1}{x^{2}+y^{2}+1}$$

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\). If the limit does not exist, state this and explain why the limit does not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+y^{2}}{\sqrt{x^{2}+y^{2}+1}-1}$$

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\). If the limit does not exist, state this and explain why the limit does not exist. $$\lim _{(x, y) \rightarrow(0,0)} \ln \left(x^{2}+y^{2}\right)$$

For the following exercises, use algebraic techniques to evaluate the limit. $$\lim _{(x, y) \rightarrow(2,1)} \frac{x-y-1}{\sqrt{x-y}-1}$$

For the following exercises, use algebraic techniques to evaluate the limit. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{4}-4 y^{4}}{x^{2}+2 y^{2}}$$

For the following exercises, use algebraic techniques to evaluate the limit. $$\quad \lim _{(x, y) \rightarrow(0,0)} \frac{x^{3}-y^{3}}{x-y}$$

For the following exercises, use algebraic techniques to evaluate the limit. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}-x y}{\sqrt{x}-\sqrt{y}}$$

For the following exercises, evaluate the limits of the functions of three variables. $$ \lim _{(x, y, z) \rightarrow(1,2,3)} \frac{x z^{2}-y^{2} z}{x y z-1}$$

For the following exercises, evaluate the limits of the functions of three variables. $$\lim _{(x, y, z) \rightarrow(0,0,0)} \frac{x^{2}-y^{2}-z^{2}}{x^{2}+y^{2}-z^{2}}$$

For the following exercises, evaluate the limit of the function by determining the value the function approaches along the indicated paths. If the limit does not exist, explain why not. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x y+y^{3}}{x^{2}+y^{2}}\) a. Along the \(x\) -axis \((y=0)\) b. Along the \(y\) -axis \((x=0)\) c. Along the path \(y=2 x\)

For the following exercises, evaluate the limit of the function by determining the value the function approaches along the indicated paths. If the limit does not exist, explain why not. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} y}{x^{4}+y^{2}}\) a. Along the \(x\) -axis \((y=0)\) b. Along the \(y\) -axis \((x=0)\) c. Along the path \(y=x^{2}\)

Discuss the continuity of the following functions. Find the largest region in the \(x y\) -plane in which the following functions are continuous. $$f(x, y)=\sin (x y)$$

Discuss the continuity of the following functions. Find the largest region in the \(x y\) -plane in which the following functions are continuous. $$f(x, y)=\ln (x+y)$$

Discuss the continuity of the following functions. Find the largest region in the \(x y\) -plane in which the following functions are continuous. $$f(x, y)=e^{3 x y}$$

Discuss the continuity of the following functions. Find the largest region in the \(x y\) -plane in which the following functions are continuous. $$f(x, y)=\frac{1}{x y}$$

For the following exercises, determine the region in which the function is continuous. Explain your answer. $$f(x, y)=\frac{x^{2} y}{x^{2}+y^{2}}$$

For the following exercises, determine the region in which the function is continuous. Explain your answer. $$f(x, y)=\left\\{\begin{array}{ll}\frac{x^{2} y}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{array}\right\\}$$

For the following exercises, determine the region in which the function is continuous. Explain your answer. $$f(x, y)=\frac{\sin \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}}$$

Determine whether \(g(x, y)=\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\) is continuous at (0,0) .

Create a plot using graphing software to determine where the limit does not exist. Determine the region of the coordinate plane in which \(f(x, y)=\frac{1}{x^{2}-y}\) is continuous.

Determine the region of the \(x y\) -plane in which \(f(x, y)=\ln \left(x^{2}+y^{2}-1\right)\) is continuous. Use technology to support your conclusion. (Hint: Choose the range of values for \(x\) and \(y\) carefully!)

At what points in space is ist what \(g(x, y, z)=x^{2}+y^{2}-2 z^{2}\) continuous?

At what points in space is \(g(x, y, z)=\frac{1}{x^{2}+z^{2}-1}\) continuous?

[T] Evaluate \(\lim _{(x, y) \rightarrow(0,0)} \frac{-x y^{2}}{x^{2}+y^{4}}\) by plotting the function using a CAS. Determine analytically the limit along the path \(x=y^{2}\).

[T] a. Use a CAS to draw a contour map of \(z=\sqrt{9-x^{2}-y^{2}}\) b. What is the name of the geometric shape of the level curves? c. Give the general equation of the level curves. d. What is the maximum value of \(z\) ? e. What is the domain of the function? f. What is the range of the function?

True or False: If we evaluate \(\lim _{(x, y) \rightarrow(0,0)} f(x)\) along several paths and each time the limit is 1 , we can conclude that \(\lim _{(x, y) \rightarrow(0,0)} f(x)=1\).

Use polar coordinates to find \(\lim _{(x, y) \rightarrow(0,0)} \frac{\sin \sqrt{x^{2}+y^{2}}}{\sqrt{x^{2}+y^{2}}}\). You can also find the limit using L'Hôpital's rule.

Use polar coordinates to find \(\lim _{(x, y) \rightarrow(0,0)} \cos \left(x^{2}+y^{2}\right)\).

Given \(\quad f(x, y)=x^{2}-4 y, \quad\) find \(\lim _{h \rightarrow 0} \frac{f(x+h, y)-f(x, y)}{h}\).

Given \( f(x, y)=x^{2}-4 y, \quad\) find \(\lim _{h \rightarrow 0} \frac{f(1+h, y)-f(1, y)}{h}\).

For the following exercises, calculate the partial derivative using the limit definitions only. $$ \frac{\partial z}{\partial x}\( for \)z=x^{2}-3 x y+y^{2}$$

For the following exercises, calculate the partial derivative using the limit definitions only. $$ \frac{\partial z}{\partial y}\( for \)z=x^{2}-3 x y+y^{2}$$

For the following exercises, calculate the partial derivatives. \(\quad \frac{\partial z}{\partial x}\) for \(z=\sin (3 x) \cos (3 y)\).