ask questions - get instant answers - receive/send p2p payments - get tips - get points - get fundraising support - meet community tutors
First time here? Checkout the FAQs!
1 like 0 dislike
Find all solutions of \(5 x+\ln x=10000\), correct to 4 decimal places; use the Newton Method.
in Mathematics by Platinum (108k points) | 1.2k views

1 Answer

1 like 0 dislike
Best answer
Let \(f(x)=5 x+\ln x-10000\). We need to approximate the root(s) of the equation \(f(x)=0\). The function \(f\) is only defined for positive \(x\). Note that the function is steadily increasing, since \(f^{\prime}(x)=5+1 / x>0\) for all positive \(x\). It follows that the function can be 0 for at most one value of \(x\). It is easy to verify that \(f(1)<0\) and \(f(2000)>0\), and therefore the equation has a root in the interval \((1,2000)\).

The Newton Method iteration is easy to set up. We get
x_{n+1}=x_{n}-\frac{5 x_{n}+\ln x_{n}-10000}{5+1 / x_{n}} .
We could simplify the right hand side somewhat. This is probably not worthwhile.

Now we need to choose \(x_{0}\). The idea is that even when \(x\) is large, \(\ln x\) is by comparison quite small. So as a first approximation we can forget about the \(\ln x\) term, and decide that \(f(x)\) is approximately \(5 x-10000\). Thus the root of our original equation must be near \(x=2000\).

Shall we choose \(x_{0}=2000\) ? It is sensible to do so. But we can do better. Note that \(\ln (2000)\) is about 7.6. So we can take \(5 x_{0} \approx\) \(10000-7.6\). Let \(x_{0}=1998.48\).

A quick computation gives \(x_{1}=1998.479972\). This agrees with \(x_{0}\) to 4 decimal places, so the answer, correct to 4 decimal places, should be 1998.4800. If we feel like it, we can show by the usual "sign change" procedure that this answer is indeed correct to 4 places.

Note. If we start with \(x_{0}=2000\), it turns out that \(x_{1}=1998.479972\), so perhaps the extra thinking that went into starting with \(1998.48\) was unnecessary. But it illustrates the fact that in some cases we can get an extremely accurate estimate of a root without bringing out heavy machinery.
by Platinum (108k points)

Related questions

1 like 0 dislike
1 answer
1 like 0 dislike
0 answers
2 like 0 dislike
1 answer
1 like 0 dislike
1 answer
0 like 0 dislike
2 answers
0 like 0 dislike
2 answers
2 like 0 dislike
1 answer
1 like 0 dislike
1 answer
asked May 9, 2021 in Mathematics by Student Bronze Status (5.8k points) | 6.2k views

Join MathsGee for AI-powered Q&A, tutor insights, P2P payments, interactive education, live lessons, and a rewarding community experience.

On MathsGee, you can:

1. Ask Questions on Various Topics

2. Request a Tutor

3. Start a Fundraiser

4. Become a Tutor

5. Create Tutor Session - For Verified Tutors

6. Host Tutor Session - For Verified Tutors

7. Join Tutor Session

8. Enjoy our interactive learning resources

9. Get tutor-verified answers

10. Vote on questions and answers

11. Tip/Donate your favorite community members

12. Earn points by participating

Posting on the MathsGee

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

MathsGee Rules

1. Answers to questions will be posted immediately

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.

MyLinks On Acalytica | Social Proof Widgets | Web Analytics | SEO Reports | Learn | Uptime Monitoring