Question

Use a linear approximation (or differentials) to estimate the given number.35. \({e^{0.1}}\)

Short answer

The required value is: \({e^{0.1}} = 1.1\)

Step-by-step solution

 Step 1: Differentials

The differentialscan be taken as the substitute method of the Linear Approximation in calculus to approximate the value of any function.

Calculating values using differential operators:

The givennumber is:

\(f\left( x \right) = {e^{0.1}} = {e^x}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,..........\left( {{\rm{say}}} \right)\)

Now, solving the linear approximation at\(a = 0\):

\(\begin{aligned}{c}dy&= \left\{ {f'\left( a \right)} \right\}dx\\dy&= \left\{ {\left( {{e^a}} \right)\left| {_{a = 0}} \right.} \right\}\left( {0.1} \right)\\&= \left( 1 \right)\left( {0.1} \right)\\&= 0.1\end{aligned}\)

So, for \(x= 0.1\):

\(\begin{aligned}{c}{e^{0.1}}&= f\left( 0 \right) + 0.1\\&= 1 + 0.1\\&= 1.1\end{aligned}\)

Hence, the required value is:

\({e^{0.1}} = 1.1\)