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Section 4.12 : Differentials

4. Compute \(dy\) and \(\Delta y\) for \(y = {{\bf{e}}^{{x^{\,2}}}}\) as \(x\) changes from 3 to 3.01.

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Start Solution

First let’s get the actual change, \(\Delta y\).

\[\Delta y = {{\bf{e}}^{3.01{\,^2}}} - \,{{\bf{e}}^{3{\,^2}}} = 501.927\] Show Step 2

Next, we’ll need the differential.

\[dy = 2x\,{{\bf{e}}^{{x^{\,2}}}}dx\] Show Step 3

As \(x\) changes from 3 to 3.01 we have \(\Delta x = 3.01 - 3 = 0.01\) and we’ll assume that \(dx \approx \Delta x = 0.01\). The approximate change, \(dy\), is then,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{dy = 2\left( 3 \right)\,{{\bf{e}}^{{3^{\,2}}}}\left( {0.01} \right) = 486.185}}\]

Don’t forget to use the “starting” value of \(x\) (i.e. \(x = 3\)) for all the \(x\)’s in the differential.