Answer
$$ \approx 5.20405$$
Work Step by Step
$$\eqalign{
& 5{e^{0.04}} \cr
& {\text{Using the result of Taylor polynomial of degree }}4{\text{ at }}x = 0{\text{ found }} \cr
& {\text{in the exercise 18 above }}\left( {page{\text{ 662}}} \right){\text{. }}f\left( x \right) = 5{e^{2x}}{\text{ we found that}} \cr
& {P_4}\left( x \right) = 5 + 10x + 10{x^2} + \frac{{20}}{3}{x^3} + \frac{{10}}{3}{x^4} \cr
& {\text{To approximate }}5{e^{0.04}},{\text{ we must evaluate }}5{e^{2\left( {0.02} \right)}},{\text{ }} \cr
& {\text{then taking }}x = 0.02 \cr
& {P_4}\left( {0.02} \right) = 5 + 10\left( {0.02} \right) + 10{\left( {0.02} \right)^2} + \frac{{20}}{3}{\left( {0.02} \right)^3} + \frac{{10}}{3}{\left( {0.02} \right)^4} \cr
& {P_4}\left( {0.02} \right) = 5 + 0.20 + 0.004 + 0.000053333 + 0.000000533 \cr
& {P_4}\left( {0.02} \right) = 5.20405 \cr
& {\text{Thus}}{\text{, }}5{e^{0.04}} \approx 5.20405 \cr} $$
