Yeah, I just wanted to say hi. I'm a bit fatigued.

]]>I can't remember.

If so, the area A=1.5*2.06 is equal to 3.09 or 3.

Is this correct???]]>

I think I have found the flaw in this post. This is the corrected version.

The test of any differentiation process is "Does it give the correct gradient function?" So I thought I would investigate this thoroughly.

The function is:

It can also be expressed:

and also re-arranged to make y the subject:

I have tried all three using the equation grapher and all give the same result:

So I then described the gradient function by experimenting with this graph. The gradient function has these properties:

It tends to negative infinity as x approaches -1 from either side.

There is a local maximum at about -2.4 and a local minimum at about +0.4.

The gradient tends to 1 as x tends to infinity and negative infinity.

Here are the three versions of a possible gradient function:

By quotient rule:

By product rule:

By direct differentiation:

This has the right properties and so I claim is the correct answer.

My difficulty lies with the way I was using the equation grapher.

So I have used algebra instead and managed to show that all three are in fact equivalent.

By quotient rule:

Substituting

the quotient version becomes

By product rule:

So all three versions are, in fact, the same.

Bob

]]>It's a regular octagon so you can work out the central angles such as GAC. That triangle is isosceles so you can work out ACH or use the right angle.

Bob

]]>I'll introduce some 'notation' to make it clearer.

With the crate in front of our position so we're facing one side, I'll call the left-right direction x, the back-front direction y, and the up-down direction z.

And to indicate which way round a box should be placed, I'll give its dimensions x first, then y then z.

Start by placing boxes 40x20x10. You can make two entire walls of boxes like this taking up 80cm of the x direction. This packs 50 x 2 boxes.

But no more can be placed with 40 in the x direction because 100 / 40 goes twice with a remainder of 20cm.

So now place boxes 20x40x10. This fills across the front and we can make two columns of these packing another 10 x 2 boxes. Then we hit the remainder problem again … there's a 20cm gap at the back in the y direction. So no more boxes can be placed so that the 40 goes in the y direction.

Only choice left is to pack them with the 40 in the z direction: 10x20x40. You can get another 2x2 like this and then there's only a 20cm gap left so no more can be placed with the 40 in the z direction. That's all three directions filled to the maximum extent so maximum number of boxes is 50 + 50 + 10 + 10 + 2 + 2 = 124.

QED.

Bob

]]>If z = 1/2 + √3/2 i then 'n' isn't any value. Specially it is 6 because only this complex number raised to the power 6 (or any higher multiple of 6) will actually give 1. Use the angle made with the X axis ... It's π/3 so needs six of these to make 2π.

So, best answer I can come up with: n must be a multiple of 6.

I think I can sleep on that.

Bob

]]>thanks

https://imgur.com/a/mmBujDP

]]>It's a result of vector theory. If a coordinate (in any system of axes) is written as a vector then it is a 2x1 matrix. After a transformation the result is too. So only a 2x2 matrix may be a candidate for the transformation … 2x2 multiplied by 2x1 gives 2x1. If you try multiplying the origin (0,0) by such a matrix you will get (0,0) again, so this will only work if the origin is an invariant point. So what follows can be applied to rotations around the origin, reflections in a line that goes through the origin and shears, where the invariant line goes through the origin. (You could also make up fancy transformations that would also work.)

(1,0) and (0,1) are used as a basis for the coordinate system. When you assemble a 2x2 matrix from these vectors you get the identity matrix. That's why this method works for those two points. Those transformations listed above are all 'linear' meaning that straight lines are transformed to straight lines. That's important too, because it means, once you have a matrix that works for the base vectors it will work for all other points too.

So decide where you want (1,0) and (0,1) to transform to. Let's say (a,b) and (c,d)

Then the transformation is:

That is (transformation matrix M) x (vectors) = (transformed vectors)

As 1, 0, 0, 1 is the identity matrix that means that:

So, to get the matrix for a rotation of 60 degrees I wrote down the vectors for four points in a parallelogram shape, before and after the rotation, and wrote out the matrix multiplication. To get M, I realised that I only needed two of the four vectors so the correct matrix was easy to write down. The other two points in my parallelogram were the origin (obviously stays the same) and the point (1,1) which maps to (-1,2) which helps to confirm that I've got the right matrix.

Bob

]]>Your reply (both parts) is correct.

]]>Hi,

The Answers A, C, and D are partially correct.

That is,

A) First diff. = 24, 24, 24; Second Diff. = 0 : Linear. Common difference is the same in first difference. Arithmetic Progression.

C) First diff. = 18, 34, 50, 66; second diff. 16, 16, 16 : Common difference in first difference is increasing in Arithmetic Progression : Quadratic.

D) First diff. = -7, -17, -27, -37. -47. Second diff. = 0. Common difference is the same, that is -10, in first difference. Linear.

Regarding B,

The first differences ought to be 3, 27, 243, 2187, 19683.

First difference = 24, 216, 1944, 17496. (Not 17676).

Second diff. = 9, 9, 9 :

The series is Exponential, in first difference.

Thanks much

I am glad i posted the questions i thought i was right!! ...

Thanks for correcting my answers

Regards

Dan

]]>If the end points of the line (if it's straight) are (x1, y1) and (x2, y2) then the gradient is

(y2 - y1) / (x2 - x1)

That's (how much the temperature has changed) / (how much time has changed)

https://www.mathsisfun.com/data/straigh … graph.html

Bob

]]>You first have to deduce the standard deviation which is the squre root of the variance, so from the given infronation its value is 10 ml. Second you have to measure the filling amount of coffee in terms of the standard deviation after putting the normal distribution in the standard form (i.e. putting the average of the filling amount of coffee equals to zero). Therefore, applying the previous procedures, the required answers are:

a- (235-230)/10=0.5 ⇒ 1-0.6915=0.3085

b- (245-230)/10=1.5 ⇒ 0.9332-0.6915=0.2417

c- 350×0.6915=242.025 cups

Good luck

]]>The Distance formula :

Distance =

In #11,

.

In a similar manner, the others can be found.

]]>