Answer:
Let \[x=2\]and \[\Delta \,x=-\,0.001\] \[[\because 2-0.001=1.999]\] Again, let \[y={{x}^{5}}\] On differentiating both sides w.r.t x, we get \[\frac{dy}{dx}=5{{x}^{4}}\] \[\therefore \] \[\Delta \,y=\frac{dy}{dx}\cdot \Delta \,x=5{{x}^{4}}\times \Delta \,x\] \[=5\times {{2}^{4}}\times [0.001]\] \[=-\,80\times 0.001=-\,0.080\] Now, \[{{(1.999)}^{5}}=y+\Delta \,y={{2}^{5}}+(-\,0.080)\] \[=32-0.080=31.920\]
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