# User:W035tjl/sandbox

Suppose, a doctor says for every cigar you smoke you will lose three days of your life and for every glass of whisky you drink you will lose two days of your life. And, your expected life expectancy is 70 years old.

${\displaystyle L(0,0)=70}$

Using your mathematical knowledge, you come up with this equation.

${\displaystyle L=70-\left({\frac {3days}{cigar}}x\right)-\left({\frac {2days}{whisky}}y\right)}$

Where L is a function that takes two arguments. x = number of cigars and y = number of whiskies

${\displaystyle L=L(x,y)}$

Your calculus teacher then gives you this equation. He also says assume dL is small and delta L is big.

${\displaystyle dL={\frac {\partial L}{\partial x}}dx+{\frac {\partial L}{\partial y}}dy}$

If y is a constant

${\displaystyle {\frac {\partial L}{\partial x}}dx{\Big |}_{y}^{.}=-3}$

If x is a constant

${\displaystyle {\frac {\partial L}{\partial y}}dx{\Big |}_{x}^{.}=-2}$

Perhaps

${\displaystyle L=70-3x-2y-xy}$

If so. For small values of x and y, xy becomes unimportant. However, if x and y are bigger values. xy really matters. In which case this new equation should be used.

Old math problem, for earlier link.

${\displaystyle I=\int e^{-x}\cos x\,dx}$

${\displaystyle {\tilde {I}}=\int e^{-x}e^{ix}\,dx}$

${\displaystyle {\tilde {I}}=\int e^{-x}(\cos x+i\sin x)\,dx}$

${\displaystyle I=\Re \{{\tilde {I}}\}}$

${\displaystyle L=\int e^{x}\,dx}$

${\displaystyle L=e^{x}}$