Math Is Fun Forum

the figure is representing a 3d cartesian space, and the squares represent a section of each axial plane, the yellow square representing part of the xy plane, red the yz plane, and green the xz plane.

As far as i am aware there is no name for that exact shape.

bobbym, surely that only holds for cubic equations of real coeffecients, surely the cubic (x-i)^3 has no real roots for example?

finding solutions to problems like these usually tend to come from natural intuition and experimentation, thinking about the problem; thinking of a possible solution, and seeing if you can improve it, or come up with an even totally different solution etc.

let the centre stone be 'x'

either side of the centre stone you will 7 stones, with values of:

x - 7×50, x - 6×50 ... x - 50, x, x - 25 ... x - 6×25, x - 7×25    etc for the rest

the ones on the left can be written as x - n×50, and ones on the right as x - n×25, with n from 1 to 7

you can then write the sum of all stones as being:

which simplified is:

you can evaluate that by hand if you do not know the formula for the sum, otherwise it is:

since the total sum is given as 4650, 15x - 2100 = 4650, 15x = 6750, x = 450

so the centre stone is worth $450

i 'could' but it wouldn't run very fast at all

thankyou very much.

since we have 2 2D vectors a quick and simple method to show they are linearly dependant is to check that

therefore they are linearly dependant

this method of using determinants of course only works if you have 'n' number of n'th dimensional vectors

Or to explain how it works:

The normal to a curve always has a gradient of -1/m from the tangent, it is calculating the tangent normal that is the complex part.
Method 1 is the standard approach that you would be taught in school, method 2 is a much simpler way of doing it that relies on a different kind of differentiation that usually isn't taught at high-school level, atleast not at this stage.

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Method 1:

Remembering the chain rule:

in this case, we need to be able to differentiate a function of y, in terms of x

so wherever there is a function of y, differentiate with respect to y, and multiply by dy/dx
when you have as above, a function of x multiplied with a function of y, you must then use the product rule:

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Method 2:

without an explanation of why this is, it is immediately apparent that it is true with the worked example above, if you are not familiar with partial derivatives, they are found by differentiating with respect to one variable, whilst assuming that all other variables are true, (as oposed to implicit where we allow that all variables can vary)

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either way, once you have the derivitive, plug in your values of x and y, and find the negative reciprocal -1/m for the gradient of the normal

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Another interesting result of partial differentiation, is that the vector formed with the partial derivitives as the components gives you a direct equation for the normal to a curve (as a vector, not a line), in which case the normal to the curve in my worked example would be (non normalised)

in which case it's gradient would be dy/dx as

which you can immediately see is -1/m from dy/dx for the tangent

you could also turn this around, and use it as an argument for why the gradient of the curve is given by:

which could also be seen as an argument for why the 3 medians MUST all intersect at a common point, because the triangle must have a single centre of mass

pretty much yeh, only that is it also multipled by the distance from the point being rotated by

evaluating that 'determinant' as you call it (whilst true, shouldn't really be calling it that) the 'perp-dot product' follows the same rules as the cross product in 3d: namely that:

thus evaluating the perp-dot gives you the distance from the axis of rotation, multiplied by the magnitude of the force, and the sine of the angle between the two vectors, which gives you the magnitude of the force perpendicular to the radial vector, multiplied by the distance from the axis

you should be able to get an expression for the nth derivitive:

the bottom of the fraction is simple, the top follows the pattern:

1 -> 2 x 1
2 -> 2x2 x 1
3 -> 2x2x2 x 1x2
4 -> 2x2x2x2 x 1x2x3
5 -> 2x2x2x2x2 x 1x2x3x4
6 -> 2x2x2x2x2x2 x 1x2x3x4x5

n -> 2^n(n-1)!

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if you are instead allowed to use the mclaurin series for ln(1-x) then you have:

then:

which gives you:

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in either case (using windows calculator)

oops ^^, (fixed it)