Calculus I - Differentials

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Section 4.12 : Differentials

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

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

Δ y = e^{3.01^{2}} - e^{3^{2}} = 501.927

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

Next, we’ll need the differential.

d y = 2 x e^{x^{2}} d x

$dy = 2x\,{{\bf{e}}^{{x^{\,2}}}}dx$ 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,

d y = 2 (3) e^{3^{2}} (0.01) = 486.185

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

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