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So I have this multivariable function:

$$f(x,y)=\frac{e^{-x}}{y}$$

and I'm supposed to sketch the level curves f(x,y) = 1, f(x,y) = 3, f(x,y) = 5.

I can't post screenshots on here for some reason, so I'll just describe what the graphs look like.

The solution to this problem says that the level curves would look like an $e^x$ graph, but I assumed that the level curves would curve upwards to the left since that's how it looks when you graph $y=e^{-x}$ on a graphing calculator.

Am I just not understanding level curves or something?

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Yes the level curves look like an $e^x$ graph, indeed we have

$$f(x,y)=\frac{e^{-x}}{y}=k\iff y=\frac{e^{-x}}{k}$$

and $e^{-x}$ is $e^x$ reflected with respect to the $y$ axis.

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