An important derivative operation in many applications is called the Laplacian; in Cartesian coordinates, for z=f(x, y), the Laplacian is

$$z _ { x x } + z _ { y y }$$

. Determine the Laplacian in polar coordinates using the following steps. Use the Chain Rule to find

$$z _ { x x } = \frac { \partial } { \partial x } \left( z _ { x } \right)$$

. There should be two major terms, which, when expanded and simplified, result in five terms.

Shown again is the table indicating the marital status of the U.S. population in 2010. Numbers in the table are expressed in millions. Use the data in the table to solve Exercises 1-10. Express probabilities as simplified fractions and as decimals rounded to the nearest hundredth. Marital Status of the U.S. Population, Ages 15 or Older, 2010, in Millions

If one person is randomly selected from the population described in the table, find the probability, expressed as a simplified fraction and as a decimal to the nearest hundredth, that the person among those who are divorced, find the probability of selecting a woman.

For the exercise, indicate "expression" if an expression is to be simplified or "equation" if an equation is to be solved. Then simplify the expression or solve the equation.

$$\dfrac{2 k^2-3 k}{20 k^2-5 k} \div \dfrac{2 k^2-5 k+3}{4 k^2+11 k-3}$$

Hal earns n dollars per hour. Next month he will receive a 2 percent raise in pay per hour The expression n+0.02n is one way to represent Hal's pay per hour after the raise. Write an equivalent simplified expression that will represent his pay per hour after the raise.

Use the choices below to fill in each blank. Some choices may be used more than once.

simplified $\text{\ }$ reciprocals $\text{\ }$ equivalent $\text{\ \ }$ denominator

product $\text{\ \ \ \ }$ factors$\text{\ \ \ \ \ \ \ \ }$ fraction$\text{\ \ \ \ \ \ \ }$ numerator

A fraction is said to be ____________ when the numerator and the denominator have no common factors other than $1$.

Shown again is the table indicating the marital status of the U.S. population in 2010. Numbers in the table are expressed in millions. Use the data in the table to solve Exercises 1-10. Express probabilities as simplified fractions and as decimals rounded to the nearest hundredth. Marital Status of the U.S. Population, Ages 15 or Older, 2010, in Millions

If one person is randomly selected from the population described in the table, find the probability, expressed as a simplified fraction and as a decimal to the nearest hundredth, that the person among those who are divorced, find the probability of selecting a man.

Reflect about what you have learned from today's lesson as you answer the following question in your Learning Log. Title this entry "Representing Expressions on an Expression Mat" and include today's date. Using an Expression Mat, find two different ways to represent $x - 1 - (2x - 3)$. Sketch the two different representations and write a few sentences to describe the differences in the ways you built each representation.

This is a simplified inventory problem. Suppose that it costs c dollars to stock an item and that the item sells for s dollars. Suppose that the number of items that will be asked for by customers is a random variable with the frequency function p(k). Find a rule for the number of items that should be stocked in order to maximize the expected income. (Hint: Consider the difference of successive terms.)

Note: Assume that all the functions have power series representations.

Use

$$\begin{aligned} f(x)&=\sum_{n=0}^{\infty}\frac{f^{(n)(c)}}{n!}(x-c)^n\\ &=f(c)+f'(c)(x-c)+\frac{f"(c)}{2!}(x-c)^2+\frac{f"'(c)}{3!}(x-c)^3+\cdots \end{aligned}$$

to find the Taylor series of $f$ at the given value of $c$. Then find the radius of convergence of the series.

$$f(x)=\cos x, \hspace{6pt}c=-\frac{\pi}{6}$$

Use the choices below to fill in each blank. Some choices may be used more than once.

simplified $\text{\ }$ reciprocals $\text{\ }$ equivalent $\text{\ \ }$ denominator

product $\text{\ \ \ \ }$ factors$\text{\ \ \ \ \ \ \ \ }$ fraction$\text{\ \ \ \ \ \ \ }$ numerator

In the fraction $\frac{3}{11}$ , the number $3$ is called the____________ and the number 11 is called the ____________.

Jill is studying a strange bacteria. When she first looks at the bacteria, there are 1000 cells in her sample. The next day, it has doubled (there are 2000 cells). Intrigued, she comes back the next day to find that there are 4000 cells! Create multiple representations (table, graph, rule) of the function whose inputs are the days that have passed after she first began to study the sample and whose outputs are the number of cells of bacteria.

This exercise will help you prepare for the material covered in the next section. In this exercise, use properties of exponents to simplify the expression. Be sure that no negative exponents appear in your simplified expression. (If you have forgotten how to simplify an exponential expression.)

$$\left(2^3 x^5\right)\left(2^4 x^{-6}\right)$$

Verify that the Laplacian has the form in $\nabla^{2}=\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r} \frac{\partial}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}}+\frac{\partial^{2}}{\partial z^{2}}$ in cylindrical coordinates.

Lucy wants to be a farmer,just like her dad. Her father says he will give her some cows to raise if she can build them a good pen. Lucy saves her money and buys 30 meters of fencing material. She will use it to build a rectangular pen against one wall of her family's barn, as shown in the picture at right. Lucy has come to you for help. She wants to build a pen for her cows that gives them the most possible area to roam. How long should each side of the fence be to give the cows as much roaming area as possible? How can you convince Lucy that the pen you suggest has the largest roaming area? Create a poster showing all four representations of this situation. On your poster, include a drawing of the pen with the largest possible area. State its dimensions (width and length) and its area. Use multiple representations to justify your conclusion that this is the largest pen.