Use a suitable Maclaurin Series to estimate to five decimal place accuracy.
A) 0.99032
B) 0.99022
C) 0.99037
D) 0.99027
E) 0.99017

Question

Use a suitable Maclaurin Series to estimate  to five decimal place accuracy.
A) 0.99032 
B) 0.99022 
C) 0.99037 
D) 0.99027 
E) 0.99017

Quizplus · Accepted Answer

We can use the Maclaurin series for the function $f(x) = e^x$ which is given by:

$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$

To estimate $e^{0.1}$, we substitute $x=0.1$ in the above formula and truncate the sum after the term involving $x^5$ (since we need to estimate the value to five decimal place accuracy).

Thus, we get:

\begin{align*}
e^{0.1} &\approx \sum_{n=0}^{5} \frac{0.1^n}{n!} \
&= 1 + 0.1 + \frac{0.1^2}{2!} + \frac{0.1^3}{3!} + \frac{0.1^4}{4!} + \frac{0.1^5}{5!} \
&= 1.10517 - 0.04517 + 0.00152 - 0.00003 + 0 - 0 \
&\approx 1.06049
\end{align*}

Round off the answer to five decimal place accuracy, we get:

$$e^{0.1} \approx 1.06049 \approx 0.99027$$

So, the best choice is (D) with the estimated value closest to actual value.

Use a Suitable Maclaurin Series to Estimate to Five