Use polar coordinates to find the limit. [Hint: Let x = r cos and y = r sin , and note that (x, y) (0, 0) implies r 0.] lim_((x,y)->(0,0)) (x^2-y^2)/sqrt(x^2+y^2)

Lynbrook West , an apartment complex , has 100 two-bedroom units. The montly profit (in dollars) realized from renting out x apartments is given by the following function.
${\displaystyle P\left(x\right)=-12x^2+2136x-41000}$
To maximize the monthly rental profit , how many units should be rented out?
What is the maximum monthly profit realizable?

What is the advantage of saying that to multiply a number by 100, we shift the digits two places to the left, instead of saying move the decimal places to the right?

Solve the following equations. Give exact simplified values and decimal approximations to three decimal places , as appropriate. Solutions may be complex , and complex solutions must be written in standard form.
$x(2-2x)=3$

Which of the following is true?
${\displaystyle \frac{dy}{dt}+\frac{y}{1-t}=3t+4}$ is best solved with undetermined coefficients.
${\displaystyle \frac{dy}{dt}=\frac{t}{y-y{t}^2}}$ is best solved with separation of variables.
${\displaystyle \frac{dy}{dt}=\frac{t}{y}}$ is best solved with integrating factors.
${\displaystyle \frac{dy}{dt}=2y-\sin \left(5t\right)}$ is best solved with separation of variables.
${\displaystyle \frac{dy}{dt}=\frac{t}{y-y{t}^2}}$ is best solved with undetermined coefficients.

The function ${\displaystyle f\left(x\right)=2x^3-33x^2+168x+9}$ has one local minimum and one local maximum.
Use a graph of the function to estimate these local extrema.
This function has a local minimum at ${\displaystyle x=?}$
with output value:
and a local maximum at ${\displaystyle x=?}$
with output value:

Solve for x in the equation 3-3x=w+1.