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hi 666bro,

It's great to hear that things are going well for you.

I'll illustrate what's happening with an example.

I've chosen the function

When we differentiate we get

and second derivative

Here are the graphs of (black) the function; (red) the first derivative; (green) the second derivative.

I've put the three graphs one after the other, lined up in the x direction but one on top of the other in the y direction. For this reason don't worry about the numbering on the axes; you don't need it. But the changes in gradient for the function are aligned correctly with the first and second derivatives.

The function is a cubic. When you differentiate it you get a quadratic. The quadratic has two zeros and these line up with the turning points on the cubic. When you differentiate again you are finding the gradient function of dy/dx, and you get a linear function.

On the (black) function graph you can see the two turning points; the first a maximum; the second a minimum; but let's say we don't know this yet. Just to the left of a maximum the function has positive gradient; just to the right it is negative. At the turning point the gradient is zero of course.

You can see this on the red graph. Just to the left of the function's turning point the red graph is above the axis so it's values are positive. The red graph crosses its axis at the turning point and after that it has negative values. In a similar way the next turning point (a minimum) has gradient negative to the left of the turning point; zero at the turning point; then positive.

So you can tell if a turning point is a maximum by looking to see if the first derivative goes from positive, through zero, to negative. And a minimum will have first derivative that goes from negative, through zero, to positive.

One way to investigate this is to draw the graphs; another is to calculate gradients just left and right of the turning point. But the second derivative gives a quick way to find out without needing to do any graph drawing. Here's how:

Function has a maximum. Gradient function goes from positive, through zero to negative. So its gradient function must be negative at that point as dy/dx has a reducing gradient.

Function has a minimum. Gradient function goes from negative, through zero, to positive. So its gradient function must be positive at that point as dy/dx has an increasing gradient.

In my example

so the turning points are at x = - √ (1/3) and + √ (1/3) At x = - √ (1/3) the second derivative is negative => this turning point is a minimum.At x = + √ (1/3) the second derivative is postive => this turning point is a maximum.

Note: I don't even have to do the full calculation here; I just need to know if the second derivative is positive or negative at each turning point; so it's a quick way to tell.

Bob

hi Double

Welcome to the forum.

Sketch the graph and draw two horizontal lines close together from the y axis to the curve, so you have drawn a typical strip. This strip has area x across and dy width. If you were asked for the area of the part of the curve between y = 0 and y = 1 you would add up the strips like this:

You cannot integrate a function of x with respect to y, so you need to substitute with

The resulting function is directly integrable.

But you ask for a volume, so I'll assume this means that the area is rotated around the y axis to create a solid 'volume of revolution'.

Go back to the original strip and imagine it rotates around the y axis so as to generate a thin disc. The area of this disc is π x^2 ( ie. pi radius squared ) and so its volume is π x^2 .dy Here the new integral:

Once again you need to replace the function of x with one involving y ....... x^2 = 1 - y

So we then end up with this integral:

Can you finish from here?

Bob

hi Hugo28036

Ok. So what is the algorithm?

Bob

hi sarahparker

New members who want to share a love of maths are always welcome. The forum is funded from sponsors on the main site, mathsisfun.com, and it is a condition of membership that posts must not contain advertising. I have therefore removed your website link as that looks to me like advertising.

Bob Bundy

hi 666 bro

I'm glad that your work on matrices has gone well.

The derivative of a function is the slope (gradient) of the graph at points along the curve.

example.

y = x^2

Just by looking at the graph we can see that at x = 0 the gradient is zero; that it rises up through positive values and has matching but negative values when x < 0

Differentiating the function we get dy/dx = 2x and this has the right gradient properties. When x = 0 2x is zero. At say x = 3 the gradient is 6 and at x = -3 it is -6.

There's lots on this on the MIF page https://www.mathsisfun.com/calculus/der … ction.html and there are links to other pages on calculus from there.

Bob

hi isabella.bueso

Welcome to the forum.

I'm happy to help students where they are having difficulty with a topic in maths but, if you read the page about what to do before you post you'll understand that we don't just do your homework for you.

This worksheet has been posted before. The company that makes these worksheets doesn't like it's copyright being abused by postings on the public forums.

There's no diagrams. What chance do I have helping with questions that are meaningless without the diagram.

When I am teaching a class, I know the past history so I know what they already know and where they might struggle, so I can target my teaching approach more usefully.

If you really cannot do any of this sheet then you need to get back to your teachers and ask for more lessons.

So, if you want my help, please say

(1) What can you do on this topic.

(2) Show what you have tried already.

(3) Be specific about what you are finding difficult.

Bob

hi George,

There is an automatic word swap 'censor' that changes certain words that are thought to be rude in some way.

Anyone who wants to click the link can first save it and then substitute back the right word instead of the word math.

It's a nuisance but it's there to protect the site from abusers.

Edit: Just tried that and here's your picture:

Bob

hi Someshwar Tripathi

Welcome to the forum.

This picture shows

You'll find lots more help here:

https://www.mathsisfun.com/improper-fractions.html

Bob

hi johnny453

Welcome to the forum.

No diagrams? And no explanation of what this is about. Sorry but it'll be tough to help under the circumstances.

I'm going to assume this is an exercise on enlargements and scale factors. Here's an example of a quadrilateral that has been enlarged with a scale factor of x2.

AB is 4cm long and CD is 8cm long. The other lengths have doubled in size similarly.

If you don't know the scale factor you can work it out like this:

I not sure what 'prop' means in these questions. In Q8 one possible answer is not prop, so I'm guessing that some questions are deliberately not proper enlargements. How can you tell?

You have to check every enlarged length. If the sf is always the same then you can safely conclude that you have a proper enlargement with that sf. If some sf answers come out differently from others, then that question is not a proper enlargement.

Hope that helps,

Bob

hi Otsdarva

My method is

From this you get roots of -4 and 0, and hence the x coordinate of the vertex is -2, leading to a y coordinate of +6.

Looking at your method:

Taking out the -3/2 is OK but you need to maintain the factor as you complete the square, otherwise you are changing the quadratic.

Bob

hi Phro,

As you've used less than signs rather than less than or equal, your set of values is very small indeed.

Mathegocart

If I show you my proof you'll understand this, I think.

Bob

hi Knewlogik

How interesting! But can you prove it always works?

Bob

Hi,

You are very welcome. It's been a pleasure to help someone who is so polite and has such a good grasp of the topic. I have always had a great regard for any pupil who really thinks it through and doesn't just go for the 'expected' answer. Hope the rest of your studies go well but do post again, either for help or just so we know how it's going.

Bob

hi sybil8464

Your answer for Q2 77.78 is correct. You need 14+ out of 18. 14/18 is 77.7777.. % so choosing 77.78 does that.

For Q7 you need 18+/21 . 18/21 is 85.71428..... so strictly 85.71 is not enough. But it's the highest answer so it'll have to do. These aren't from Compu-High by any chance are they? Because they use multi-choice and computer marking you cannot say none of these actually work, so try 85.71 and if you can point out to your tutor that it doesn't really work maybe they'll realise they need to take more care setting their questions.

For Q10 you need 15+/18. 15/18 = 83.3333 % so I can see why your sister suggests that, but once again it's not strictly enough to get past 50% of the votes. So i like your thinking in suggesting answer 4. However, I suspect the person who set the questions hadn't thought this through properly because of rounding. I'm not sure how you answer this. If you can write an explanation for your answer then you deserve extra credit but are their tutors clever enough to spot this? Please post back what you submit and how it is marked.

Best wishes,

Bob

hi sybil8464

Welcome to the forum.

From the way you have described this, either answer is correct. The question setter needed to say clockwise or anticlockwise. Normally one would give the lower value ( 150 ) but, unless the wording makes it clear, 210 would also be good. In some areas of mathematics anticlockwise is the norm, so I'd suggest you say 150.

Hope that is good,

Bob

hi PMH

Go up one from k1, and test the block from there to F, using no more than k-1 drops.

k2 = k1 + 1

Shouldn't that be k2 = k1 -1 ?

This generates the series k1, k1+1, k1+2, ...

which has to sum to F.

Sum: k*(k+1)/2 + k*k*(k+1)/2

= (k+1)*(k+1)/2

so F = (k+1)^2/2

so k = sqrt(2F)

I think that is Sum = k(k+1)^2 over 2

Bob

hi PMH,

In my solution I got to the point where I wanted

b + (b-1) + (b-2) + ... + (a+3) + (a+2) + (a+1) = 99

If sum(n) means the sum of the natural numbers up to n then my formula can be re-written as

sum(b) - sum(a)

There's a formula for the sum of n natural numbers which can be worked out as follows:

sum(n) = n + (n-1) + (n-2) + ... + 3 + 2 + 1

sum(n) = 1 + 2 + 3 + ... + (n-2) + (n-1) + n

adding

2 x sum(n) = (n+1) + (n+1) + (n+1) + .... + (n+1) + (n+1) + (n+1) = n x (n+1)

so

Bob

hi

There was a command square brackets code that allowed a poster to show their line of LaTex code completely without activating the interpreter. I don't know why that doesn't now work. But I've found a way round it. I've deliberately left out a close bracket. Then the code command shows the rest so you just have to re-insert the missing bracket.

Here's the first example using that trick.

`[math V_{sphere}=\frac{4}{3}\pi r^3[/math]`

But there's an easy way to see what the code is.

In the first example I clicked on the LaTex and got this: V_{sphere}=\frac{4}{3}\pi r^3

You can always do this to see what the code was.

I've decided to re-write post 4 properly in case anyone else is misled by it.

Bob

hi PMH,

Apologies for thinking 9 is a prime. Still it doesn't alter the proof. I'm wondering whether to make it a bit neater now I've finished developing it.

Also sorry that I've taken so long to reply. I had forgotten I was still logged in from earlier so I've only just spotted your reply.

The maths characters are done using a code called LaTex. The FluxBB engine that drives the forum will jump to the interpreter when it meets the command open square bracket math close square bracket. The interpreter switches off again in a similar way with the command /math

There's a tutorial here: http://www.mathisfunforum.com/viewtopic.php?id=4397

The first 20 or so posts will be enough to get you started. Then you can dip in when you need a specific symbol.

Also useful: http://latex.codecogs.com/legacy/eqneditor/editor.php

You can create code there to copy and paste into your post.

Furthermore, if you see some LaTex that you want to investigate, just click on it to see how it was done.

Other square bracket type commands you might find useful:

quote url hide img b (for bold) u (for underline)

Hope that helps,

Bob

hi Phro,

Whoops! I feel like such a fool. I was so pleased I'd found a method, I overlooked this detail. Which is easier: rework the whole of maths defining 9 to be a prime, or change my earlier post?

Think I'll make the edit.

Thanks,

Bob

RE_WRITE of post.

You will see from the next post by Phrontister that I made a careless mistake and said that 9 was a prime. I've realised that the solution doesn't need primes anyway so here's a complete revision.

We need:

Clearly b - a < b + a + 1 and this pair must be a whole number factor pair multiplyig to give 198, so the possibilities are:

1 x 198, 2 x 99, 3 x 66, 6 x 33, 9 x 22, 11 x 18.

So I'll investigate b - a being each of the first of these pairs:

If b - a = 1, then a = 98 and b = 99

If b - a = 2, then a = 48 and b = 50

If b - a = 3, then a = 31 and b = 34

If b - a = 6, then a = 13 and b = 19

If b - a = 9, then a = 6 and b = 15

If b - a = 11, then a = 3 and b = 14.

You will notice that b gets smaller as b - a gets larger.

This follows because to make b smallest we want b + a + 1 to be smallest, and that will happen when b - a is largest.

So I could have left out all those trials and jumped straight to b - a = 11 immediately.

I'm feeling old and useless.

That's why joining MIF was a good move. We'll have you back to your prime in no time.

Bob

hi PMH

Welcome to the forum.

I see that puzzle was provided by JaneFairfax. She hasn't been active on the forum for a long time now, but, in her day, she was a formidable brain. And she expected her readers to follow her arguments with the same level of genius. Not always possible for us mere mortals!

I think there is some logic there but half the steps are missing (Probably she thought they were obvious ).

Let's say you try the nth floor and it's the best. If a ball breaks you can test from 1 to n-1 in that order to establish the level.

If the ball doesn't break you need to try a higher floor in a similar way. No good trying a number equal to n + another n, because if the ball breaks it will now take another n-1 to find the level so, on top of the two tries you have already had that means n + 1 tries. But I wanted n to be the maximum number. So I'll try n + one less than n this time.

If it breaks I still have n-2 tries to find the level, so n altogether again.

If it doesn't break then by a similar logic I'll try n-2 higher and so on.

I can stop if I reach floor 99, because, if I haven't yet found the level, it must be floor 100.

So I'm looking for n, such that n + (n-1) + (n-2) + (n-3) + ..... comes to 99. Note: this sequence doesn't have to end in +3 + 2 +1. It can end sooner, provided I reach 99.

At this stage you can use trial and improvement (a spreadsheet helps) to home in on n = 14 quite quickly. [There may be an algebraic way to get 14 with any trialling. I'll try that now and post again if I find a way.]

Bob

I had assumed that ' is being used to mean the inverse of. But I cannot find any on-line resources that confirm this.

But, if it is, then

(X' Y' X Y) (Y' X' Y X) = X' ( Y' ( X (YY') X' ) Y ) X = the identity = I

Also (XY) (XY)' = I = X Y Y' X' => (XY)' = Y'X'

Bob

hi pi_cubed

I tried X and Y as 2 x 2 matrices, just making up some numbers. I didn't get the same answer evaluating X' Y' X Y and Y' X' Y X. It was a long calculation so I may have slipped up of course.

Did you just randomly make up the relationship, or does it come from something?

Bob

The forum is run for the benefit of the members. If you want to start a discussion then please do so. I'm looking forward to your post.

Bob