Functions of Several Variables

Identify the domain and range of each of the following functions. Sketch each domain in the $xy$-plane.

1. $f(x,y) = x^2+y^2$

2. $f(x,y) = \sqrt{x^2+y^2}$

3. $\displaystyle Q(x,y) = \frac{x+y}{x^2-y^2}$

4. $\displaystyle s(x,y) = \frac{1}{\sqrt{1-xy^2}}$

The Ideal Gas Law, $PV = RT$, relates the pressure ($P$, in pascals), temperature ($T$, in Kelvin), and volume ($V$, in cubic meters) of 1 mole of a gas ($R = 8.314 \ \frac{\text{J} }{\text{ mol } \ \text{K} }$ is the universal gas constant), and describes the behavior of gases that do not liquefy easily, such as oxygen and hydrogen. We can solve the ideal gas law for the volume and hence treat the volume as a function of the pressure and temperature:

1. Explain in de…

When people buy a large ticket item like a car or a house, they often take out a loan to make the purchase. The loan is paid back in monthly installments until the entire amount of the loan, plus interest, is paid. The monthly payment that the borrower has to make depends on the amount $P$ of money borrowed (called the principal), the duration $t$ of the loan in years, and the interest rate $r$. For exa…

Consider the function $h$ defined by $h(x,y) = 8 - \sqrt{4 - x^2 - y^2}$.

1. What is the domain of $h$? (Hint: describe a set of ordered pairs in the plane by explaining their relationship relative to a key circle.)

2. The range of a function is the set of all outputs the function generates. Given that the range of the square root function $g(t) = \sqrt{t}$ is the set of all nonnegative real numbers, what do you think is the range of $h$? Why?

3. Choose 4 differen…