[To be turned in on Wednesday]
- Write each of the following systems of equations as a single function (using the full function notation we developed in class). For each, indicate the spaces (i.e., the $\R^n$) where you would find the level sets, image, and graph of your function.
a. $\displaystyle w = {x + y \over z}$ b. $y = x\tan x$ c. $\begin{aligned} x &= \cos u \cos v\\ y &= \cos u \sin v\\ z &= \sin u \end{aligned}$ d. $\begin{aligned} x &= \sin\theta-\cos\theta\\ y &= \cos2\theta\\ \end{aligned}$ e. $\begin{aligned} x &= u + 2v\\ y &= -u^2\\ \end{aligned}$ - Let $f\colon\R^2\to\R$ be defined by $\displaystyle f(x,y)={1\over\sqrt{x^3-xy}}$. Determine the natural domain for $f$ and sketch it. Be sure to explain your reasoning.
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