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Section 4.12 : Differentials
5. Compute \(dy\) and \(\Delta y\) for \(y = {x^5} - 2{x^3} + 7x\) as \(x\) changes from 6 to 5.9.
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Start SolutionFirst let’s get the actual change, \(\Delta y\).
\[\Delta y = \left( {{{5.9}^5} - 2\left( {{{5.9}^3}} \right) + 7\left( {5.9} \right)} \right) - \left( {{6^5} - 2\left( {{6^3}} \right) + 7\left( 6 \right)} \right) = - 606.215\] Show Step 2Next, we’ll need the differential.
\[dy = \left( {5{x^4} - 6{x^2} + 7} \right)dx\] Show Step 3As \(x\) changes from 6 to 5.9 we have \(\Delta x = 5.9 - 6 = - 0.1\) and we’ll assume that \(dx \approx \Delta x = - 0.1\). The approximate change, \(dy\), is then,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{dy = \left( {5\left( {{6^4}} \right) - 6\left( {{6^2}} \right) + 7} \right)\left( { - 0.1} \right) = - 627.1}}\]Don’t forget to use the “starting” value of \(x\) (i.e. \(x = 6\)) for all the \(x\)’s in the differential.