Learning starts with a question
First time here? Checkout the FAQs!
x

*Math Image Search only works best with zoomed in and well cropped math screenshots. Check DEMO

0 like 0 dislike
25 views
1. Find the \(n^{\text {th }}\) Maclaurin polynomial for \(f(x)=e^x\).
2. Use \(p_5(x)\) to approximate the value of \(e\).
in Mathematics by Platinum (141,884 points) | 25 views

1 Answer

0 like 0 dislike
Best answer
1. We start with creating a table of the derivatives of \(e^x\) evaluated at \(x=0\).

\[
\begin{array}{lll}
f(x)=e^x & \Rightarrow & f(0)=1 \\
f^{\prime}(x)=e^x & \Rightarrow & f^{\prime}(0)=1 \\
f^{\prime \prime}(x)=e^x & \Rightarrow & f^{\prime \prime}(0)=1 \\
\vdots & & \vdots \\
f^{(n)}(x)=e^x & \Rightarrow & f^{(n)}(0)=1
\end{array}
\]
By the definition of the Maclaurin series, we have
\[
\begin{aligned}
p_n(x) &=f(0)+f^{\prime}(0) x+\frac{f^{\prime \prime}(0)}{2 !} x^2+\frac{f^{\prime \prime \prime}(0)}{3 !} x^3+\cdots+\frac{f^n(0)}{n !} x^n \\
&=1+x+\frac{1}{2} x^2+\frac{1}{6} x^3+\frac{1}{24} x^4+\cdots+\frac{1}{n !} x^n .
\end{aligned}
\]
2. Using our answer from part 1, we have
\[
p_5(x)=1+x+\frac{1}{2} x^2+\frac{1}{6} x^3+\frac{1}{24} x^4+\frac{1}{120} x^5 .
\]
To approximate the value of \(e\), note that \(e=e^1=f(1) \approx p_5(1)\). It is very straightforward to evaluate \(p_5(1)\) :
\[
p_5(1)=1+1+\frac{1}{2}+\frac{1}{6}+\frac{1}{24}+\frac{1}{120}=\frac{163}{60} \approx 2.71667 .
\]
by Platinum (141,884 points)

Related questions

1 like 0 dislike
1 answer
asked May 23, 2021 in Mathematics by Student SIlver Status (11,274 points) | 504 views
1 like 0 dislike
0 answers
2 like 0 dislike
1 answer
0 like 0 dislike
1 answer
asked Nov 22 in Mathematics by MathsGee Platinum (141,884 points) | 15 views
3 like 0 dislike
1 answer
1 like 0 dislike
1 answer
0 like 0 dislike
1 answer

Join the MathsGee Q&A forum where you get STEM education support to succeed from our community. Connect and Learn.


On the MathsGee Q&A Forum, you can:


1. Ask questions


2. Answer questions


3. Vote on questions and answers


4. Start a fundraiser


5. Tip your favorite community members


6. Create Live Video Tutorials (Paid/Free)


7. Join Live Video Tutorials (Paid/Free)


8. Earn points by participating



MathsGee Q&A forum post


1. Remember the human


2. Act like you would in real life


3. Find original source of content


4. Check for duplicates before publishing


5. Read the community guidelines




FORUM RULES


1. Answers to questions will be posted immediately after moderation


2. Questions will be queued for posting immediately after moderation


3. Depending on the number of messages we receive, you could wait up to 24 hours for your message to appear. But be patient as posts will appear after passing our moderation.




USEFUL LINKS


Acalytica | Web Analytics | SEO Reports | Social Proof Tool | Email Marketing


MathsGee Android Q&A