We can use differentials (which is the same as linear approximation) to estimate some complicated functions.
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Example 1:
Estimate $\sqrt{4.036}$ without a calculator. Solution: We are looking at the function $f(x)=\sqrt{x}$. Notice that $f'(x)=\displaystyle{\frac{1}{2\sqrt{x}}}$. Take $a=4$ and $x=4.036$. Since $f(a)=2$ and $f'(a) = \frac{1}{4}$, we can estimate \begin{eqnarray*}f(x) &\approx& f(a) + f'(a) (x-a) \cr &=& 2 + \frac{.0036}{4} \cr & = & 2.009. \end{eqnarray*} This is an extremely accurate approximation. |
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Example 2: Estimate $e^{.03}$ without a calculator.
Solution: This time we take $f(x)=e^x$, $f'(x)=e^x$, and $a=0$. Then \begin{eqnarray*}f(x) &\approx& f(a) + f'(a) (x-a) \cr &=& 1 + 1(.03) \cr & = & 1.03. \end{eqnarray*} This is also an accurate approximation. |