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Ok let's have a go. The function is f (x) = ax^2 + bx + c . If you take the general point on the curve it has coordinates

(x, ax^2 + bx + c) I'll call that point A.

We then pick a new point B that is close to A. Newton was one of two mathematicians who invented calculus. The other was Leibniz. They both came up with the idea at about the same time and a controversy raged between them about who first had the idea and whether the other 'stole' it. I'll used Newton's notation. He invented a new symbol Δx to mean 'a little bit' in the x direction. It's a single algebraic symbol not two multiplied together.

So the coordinates of B are [x + Δx, a(x + Δx)^2 + b(x+ Δx) + c] I've only used square brackets for the outer to avoid confusion with the round brackets .

So the gradient of the chord AB is

( a(x + Δx)^2 + b(x+ Δx) + c - ax^2 - bx - c) / (x + Δx - x)

Have a go at simplifying that. It will probably help if you expand the bracket (x + Δx)^2 first.

You'll find there's lots that cancels out and finally you can divide Δx into the numerator.

Bob

I've never questioned why it is called that. Hhmm! It means an object travelling along a known straight path. I googled it and was surprised to discover that linear motion means along a path that need not be straight and rectilinear means the path is straight.

In a question, if they give you the equation of motion (eg. s = 5 - t^2) then it doesn't really matter what the path is.

Bob

ps. This morning I went through all your responses and answered the ones that need an answer at this stage (I hope I didn't miss one). When I click for active posts my answered ones jump to the top of list and the ones I hadn't looked at yet stayed mixed in with the ones I'm leaving until later. So it's getting harder and harder for me to keep track of whether I've done all I need to. (hope this is making sense)

So when you are ready to return to an unanswered one from your list, please bump it back to the top by posting that you want me to deal with it. Hope that makes sense.

The top table row starts with the label t and then the values 0.1, 0.01, 0.01 etc.

The second row will have the corresponding Distance (s) values calculated using the formula s = 5 - t^2

The final row will have the V values calculated using V = s / t

What you should observe is values of V getting closer and closer to a limit (the instantaneous velocity at the moment).

This process demonstrates the process known as differentiation from first principles.

Bob

Arh! Good question.

The units for V are determined by what units are used for D and T. So if you do 20/0.25 you have the velocity in ft / s.

To get an answer in mph you'll have to do the conversions.

20 ft converts to 20/(3 x 1760) miles and 0.25 s converts to 0.25/3600 hours.

Bob

In order to keep the algebra as simple as possible, replace y1 with mx1 + c, and y2 with mx2 + c

Bob

I know you were advised to split your posts but all the ones you have made are related in that they form an introduction to calculus. I suggest you get the other (easier) ones sorted first; then we can proceed to the harder ones.

Bob

This approach is exactly how differential calculus is developed.

The table will have a row for distance, one for time taken and one for velocity using V = D/T

Bob

speed = distance / time

B

I think this will have to be calculus. The other posts are an easier start so let's get them sorted first, then I'll show you this more complicated case.

Bob

This also can be done using calculus or you could try this:

Write down the coordinates of two points on the line (x1,y1) and (x2,y2) and compute

difference in y coords divided by difference in x coords.

Bob

hi

Have you met differential calculus yet?

It's here: https://www.mathsisfun.com/calculus/der … ction.html

Bob

Hello mathland,

Sorry he hear you're feeling sick. Usually it's a sign that your immune system is working so it'll most likely pass quickly. I had something similar with the astra zenica jab.

Best wishes,

Bob

hi clara95

Welcome to the forum.

Your triangle method and answer are correct. I'm not quite sure what you're doing for the trapezoid method. The usual formula for the area of a trapezoid is sum of the parallels divide by 2 and multiply by the distance between them. The parallels are 16 and 8; the distance between is the same as the apothem you have already worked out so the calculation becomes

(16+8)/2 x 4√3 x 2 as there are two such trapezoids.

Hope that helps,

Bob

hi mathland,

If you can use algebra to get a (x-c) term at the top of that expression, then these can be cancelled out and what's left will allow you to let x tend to c.

Now you can cancel the (x-c) and simplify the rest, put x=c and you're done.

Bob

hi individ

Thank you. It's been a steep learning curve for me but I'm getting the idea. Let's see what happens next.

Bob

hi individ

Your posts here about diophantine equations are still on the MIF site and won't be deleted. That's not the issue.

I don't understand how NFT works and you haven't explained it.

After creating a token, it will announce its solution to the problem.

What does this mean?

Let me try asking specific questions.

(1) Let's say you have a 'paper' on some area of equations that you want to publish. How does that work on opensea ? Do you have to pay to post there ? Would I have to pay if I wanted to view your work ?

(2) How do you expect mathematicians and scientists to know about your publication ?

(3) What is a 'token' ?

(4) Are you just interested in spreading the knowledge or are you hoping to make money from this ?

the rules say wrote:

No Spam. Spam includes messages that have no relevance to the topic, that are annoying, repetitious or

promotional in nature. Overuse of short comments is also spam.

Make Yourself Understood.It is ok to sometimes use shortened words such as gr8, but too much of this will classify your post as spam.Possible Actions: At first you will be gently warned or have your message edited or deleted. More serious cases may result in banning or other measures.

I have underlined some key phrases. If someone makes a link to a site that sells things then I consider that is promotional and I remove the link. If I think the member has only joined to make this link then I ban them.

I think your use of the abbreviation NFT is an example of an unhelpful shortened word. I didn't know what it meant and I still don't; even though I have read a definition. I need to be confident about what you are posting so you'll have to accept that your original post is likely to be edited unless you provide a better explanation.

I'm trying to help you; please now help me.

Thanks,

Bob

hi individ

One of my tasks in moderating the forum is removing posts (and sometimes the poster) that break our rules. The disadvantage of post moderation is we get all sorts of adverts: trainers, cars, taxi services, adult services, and so on. So I check them all out and do my best to keep the forum free of such things. Recently I removed a post that was advertising bit coin for example.

I read your post twice but still have no idea what you are talking about. I googled NFT and then fungible and still only have a vague idea what this is about. I looked at Opensea and it seems to be a page where you can buy and sell stuff. That's clearly against our rules. All I could see was 'art' (I use the term loosely as it's hardly Constable, so I searched for diophantine and found some of your work. But these are already posted on MIF aren't they; so what are you expecting to happened on Opensea?

Some cases are borderline. If a person asks for a recommendation of a good maths book and gets a reply, is that OK?

But I know what Diophantine equations are; and that's certainly ok. So now I'm in a dilemma. Is your post OK or not? It would help if you made it clear to 'old fogeys' like me what this is about and particularly if financial transactions are involved. If you want to claim copyright to a piece I think the post on MIF is sufficient.

Also, why Science HQ ? D. equations are maths first and foremost aren't they?

Best wishes,

Bob

hi mathland,

This is not a topic I have worked on before, so I've waited a while in the hope someone else might give an answer.

As that hasn't happened yet I've started by researching the method. I found this pdf which I found very clear.

https://math.berkeley.edu/~willij/1a/epsilonics.pdf

Then I did the following. You'll have to decide if it is a proper solution:

As x is close to 1 we may assume that 3-x > 0 therefore

So for any epsilon I have found an epsilon dependant delta . I think that is what the method requires.

Let me know what you think.

Bob

hi mathland,

One of the advantages of the forum is that you may get different methods for solving a problem and you are free to choose what suits you best.

For the limit problems you have posted so far I find it easy enough to consider just left (or right) of the x value and so determine whether the y goes to + or - infinity. I can do that 'in my head' so it's quick and easy. Later the problems will be harder and you'll need other techniques.

A long time ago on an old server it was possible to upload small images directly to the server. But we lost that with an upgrade. The forum server has been upgraded again very recently. Haven't found out if image upload is possible; I'll ask.

What I've done for a while is uploaded to imgur.com You can get the bccode for imbedding into your post and then the image appears at the point where you put the code.

That's what I did here http://www.mathisfunforum.com/viewtopic … 79#p417979 to show the scatter plot.

The link is this:

https://i.imgur.com/dPeu971.gif

I'm not sure if you can view that directly, as you won't be logged in to my account, but it shows the method. edit. Tried it and it does load.

Please be careful copying parts of a textbook to the forum. A while back we had a complaint from an on-line teaching company because a student had posted a worksheet. It is a breach of copyright so we asked the student to (1) only put a single question into a request for help and (2) to type the question in their own words to avoid direct copying. In the UK the copyright law allows copies when used for fair criticism so perhaps you could preface your post with a suitable comment such as 'This comes from <name of book> which I really like as it has good explanations and questions'

Is it Easter? The days just blur into one what with lockdown restrictions. Arrh! Just noticed I've got a choccy egg in front of me. That must mean something.

Best wishes to you too.

Bob

There are some Latex commands that are not implemented on this forum but most are. That thread shows what is possible. You can also construct an expression using this:

https://latex.codecogs.com/legacy/eqneditor/editor.php

Each line must start with:

`[math]`

and end with a similar /math command.

I'm having trouble getting this bit to display properly so I hope you can read this ok.

If someone has used Latex in a post and you want to see their commands, just click on the Latex. You can copy and paste the commands and then edit as you need for your own post.

Bob

hi mathland

That's exactly how I'd do this and yes, you are correct.

You can also use the graph to see what is happening.

Here's a link to a function grapher https://www.mathsisfun.com/data/function-grapher.php

You'll need to use the zoom control to zoom out enough to see what happens either side of x =2.

Bob

As you know that

all you need is

But one is the reciprocal of the other so that follows straight away.

Bob

hi camicat

The formulas

and

are really just re-arrangements of the same result. Let's go back to the right angled triangle.

If you divide through by H squared you get:

If you divide this formula by cos squared you get

So it's 'in the spirit' of what your teacher has told you to use any of the many versions of the formula to help with these problems.

eg. 12. Express cosecant in terms of tangent.

cosecant is 1/sin so I need a version of the formula that has 1/sin in it.

So I'll start with formula (1) and divide through by sin squared.

A full set of identities is on this page: https://www.mathsisfun.com/algebra/trig … ities.html about 2/3 of the way down the page.

Hope that clears it up for you,

Bob

Following post 8. Sorry for the slow reply but I've been away and out of touch with my laptop.

1. What can I do to help make this site a great place for students and for people that love math?

You've joined and become a regular. That's a good start. See also other answers below.

2. Can I post math practice problems in the Exercise section?

Certainly!

3. Would you like me to help answer questions that are not beyond my level in the forums?

Again, certainly. There are no rules about this (except courtesy etc of course)

4. How many tutors here actually help with math questions?

I've never counted. Nobody is paid to help on this site. We just do it because we enjoy it. Some folk just chip in occasionally. Others are more regular contributors.

5. Are you the main math helper here?

Again , who is counting? If I see an unanswered question that I can do then I'll try to help. If someone else has already answered then I'll only post if I feel I can add something useful. If I'm not sure about a topic then I'll wait and hope to learn something myself from a poster who can help. In a few rare cases, when no one else has answered at all I might do some research and see if I can figure out what is needed. I have had a few long 'conversations' with posters where we have worked together towards an answer.

6. When will this site include Calculus 3 lessons? How about Linear Algebra lessons?

Don't know ... I wouldn't recommend 'holding your breath while you wait'. MathsIsFun himself started with the teaching site and Mathopolis. Then he created the forum for folk like us to talk maths. Largely he leaves us to it as he is busy making a science site.

I'm in the UK and I'm not even sure what's in Calculus 3. I did linear algebra in my first year at University .. didn't enjoy it much and was glad when we were given options for later years.

If you post something specific I'll have a go but see 5 above.

In looking back to find which post number your questions came from I see you had earlier questions too. So here goes with them:

1. Are you a math professor?

Ha ha! No. After University I taught maths in UK 11-18 schools for many years.

4. What do you think about my passion for math at 55 about to be 56?

Well it will help to keep your brain active. I'm a tad older than that. I cannot restore a Rubik cube in a few seconds but I do know the underlying maths for why the various methods work and how you can create new tricks with the colours. At the moment I'm trying to get my head around contour integration. There's a long standing and never solved question in amongst the posts that I still have hopes of solving one day.

Bob

hi camicat

In a right angled triangle and using A for adjacent, O for opposite and H for hypotenuse:

Pythagoras: O^2 + A^2 = H^2

Divide by H^2: sin^2 (x) + cos^2 (x) = 1 **

This useful identity can be used for many things including all of your tasks.

eg.

Divide by sin^2 (x) : 1 + 1/tan^2(x) = cosec^2 (x)

** For this to work x has to be an angle in a right angled triangle. But trig values beyond the range 0-90, are defined in such a way that the result still holds.

Bob