Paul's Online Notes
Home / Calculus I / Applications of Derivatives / Differentials
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### Section 4.12 : Differentials

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

Show All Steps Hide All Steps

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.