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Section 4.11 : Linear Approximations

5. Without using any kind of computational aid use a linear approximation to estimate the value of \({{\bf{e}}^{0.1}}\).

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Hint : This is really nothing more than Problem 3 and 4 from this section. The only difference is that you need to determine the function and the point for the linear approximation.

The function should be pretty obvious given the value we are asked to estimate. There should also be a pretty obvious point to use given that we aren’t supposed to use calculators/computers.

Start Solution

This is really the same problem as Problems 3 & 4 from this section. The difference is that we need to determine the function and point for the linear approximation.

Given the value we are being asked to estimate it should be fairly clear that the function should be,

\[\underline {f\left( x \right) = {{\bf{e}}^x}} \]

The point for the linear approximation should also be somewhat clear. With the function in hand it’s now clear that we are being asked to use a linear approximation to estimate \(f\left( {0.1} \right)\). So, we’ll need a point that is close to \(x = 0.1\) and one that we can evaluate in the function without a calculator. It therefore seems fairly clear that \(x = 0\) would be a really nice point use for the linear approximation.

Show Step 2

At this point finding the linear approximation shouldn’t be too bad so here is the work for that.

\[f'\left( x \right) = {{\bf{e}}^x}\hspace{0.5in}f\left( 0 \right) = 1\hspace{0.5in}f'\left( 0 \right) = 1\]

The linear approximation is then,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{L\left( t \right) = 1 + \left( 1 \right)\left( {x - 0} \right) = x + 1}}\] Show Step 3

The estimation of \({{\bf{e}}^{0.1}}\) is then,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{{{\bf{e}}^{0.1}} \approx L\left( {0.1} \right) = 1.1}}\]

For comparison purposes the exact value is \(f\left( {0.1} \right) = 1.10517\) and so we have an error of 0.467884 %.