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I have a question below but I missed this day of class maybe someone can show me how to approach?

Find $dy$ and evaluate $dy$ for the given values of $x$ and $dx$

  1. $\displaystyle y=e^\frac{x}{10}$
  2. $\displaystyle x = 0$ and $dx = 0.1$
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Well, it seems like you should recall $$ \mathrm{d}y=\frac{\mathrm{d}y}{\mathrm{d}x}\mathrm{d}x$$ From there, it should be straightforward to solve... –  Alex Nelson Nov 23 '12 at 21:59

1 Answer 1

With differentials, it's an abuse of notation, but the answer is that $$ \mathrm{d}y=\frac{\mathrm{d}y}{\mathrm{d}x}\mathrm{d}x$$ Since $\mathrm{d}x$ is impossible to use, we make the approximation $\Delta x\approx\mathrm{d}x$.

For us, $\Delta x=0.1$. Now, we find the derivative $$\frac{\mathrm{d}}{\mathrm{d}x}e^{x/10}=\frac{1}{10}e^{x/10}.$$ Then we plug in the data: $$\mathrm{d}y\approx \left.\left(\frac{1}{10}e^{x/10}\right)\Delta x\right|_{x=0}$$ which means we are evaluating the parenthetic term when $x=0$, and multiply by $\Delta x=0.1$.

Now we have to just plug these things in to find: $$\begin{align}\left.\left(\frac{1}{10}e^{x/10}\right)\Delta x\right|_{x=0}&=\left(\frac{1}{10}e^{0}\right)0.1\\ &=0.01\end{align}$$ Thus $\mathrm{d}y\approx 0.01$.

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Precisely how is it an abuse of notation? I recall reading about this, but it has now escaped me. Oh, and wouldn't it be best to write: $$ \begin{align} dy&\approx\left. \left(\frac{1}{10}e^{\frac{x}{10}}\right)\Delta x\right|_{x=0}\\ \left. \left(\frac{1}{10}e^{\frac{x}{10}}\right)\Delta x\right|_{x=0}&=\left(\frac{1}{10}e^{0}\right)0.1\\ \left(\frac{1}{10}e^{0}\right)0.1&=1. \end{align} $$ ? –  000 Nov 23 '12 at 22:11
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Because the notation treats differentials like $\mathrm{d}x$ "as if" they were numbers, and could have $$\frac{\mathrm{d}y}{\color{red}{\mathrm{d}x}}\color{red}{\mathrm{d}x}$$ which is illegal mathematics. The result is right, the reasoning fallacious :( –  Alex Nelson Nov 23 '12 at 22:14
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Oh, right. They're not even algebraic quantities since they're simply a notational convention just like $\int$. Am I following you? –  000 Nov 23 '12 at 22:17
    
Exactly right! :) –  Alex Nelson Nov 23 '12 at 22:17
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Isn't the answer $\frac1{10}0.1=0.01$? –  robjohn Nov 23 '12 at 22:25

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