My thread of questions

[TrueTears]

Extended chain rule.

http://web.mit.edu/wwmath/calculus/differentiation/chain.html

[khalil]

Is this for spesh?

[TrueTears]

No.

[khalil]

Is it in the Essentials methods text?

[TrueTears]

For chain rule - yes.

[QuantumJG]

Why does 0! = 1?

Factorial (n!) is defined as the continued multiplication of 1...n (I.e. n! = 1*2*3*4*...*(n-2)*(n-1)*n, where n>0.

0! = 1, since, the product of no numbers is defined as equalling 1.

The only thing you need to know in maths methods is that 0! = 1. Even if that is required?

I hope in university I'll see how this is actually proved.

[kamil9876]

True, it's a convention so that things work nicely. As Quantum pointed out it's the product of no numbers and this can be a useful concept in certain problems (similairly like x^0=1 for all x besides 0). Say for example I have the number 10, and i want to multiply it by the first 3 consecutive numbers, I get 10*1*2*3. If i want to multiply it by the first n consecutive numbers I get 10*n!. But if i want to multiply it by no numbers(0 numbers) the number won't change, still be 10, hence it has the same affect as multiplying by 1. Also, it makes the binomial formula work well and things like that since there are is only one way of choosing nothing( similairly because there is only one way of choosing what not to choose in order to choose nothing).

[ryley]

If you want a more formal definition, this is way I was taught it (no, the following is not in methods).

[khalil]

Would this transformation be a dilation from the x or y axis.
Cos' when the graph is transformed it it 4/x

[Flaming_Arrow]

dilation of factor 4 from the y axis

[khalil]

Why not from the x-axis? It is dilated by 4 hence new equation 4/x

[TrueTears]

Would this transformation be a dilation from the x or y axis.
Cos' when the graph is transformed it it 4/x

Let be the coordinates of the transformed graph.





Thus the graph of has gone to



and

Thus is mapped onto

So this is a dilation of factor 4 from the y axis.

[kamil9876]

[khalil]

Then why does this work out also?

[kamil9876]

Note: The possible tranformations from one function to another is not unique.

e.g:

y=x^2 to y=4x^2

could be seen as dilating 4 from x axis. However by expressing the second function as y=(2x)^2 you get dilation by factor of 1/2 from y axis. Different transformations that yield same result.

A more simple and intuitive example:

y=x to y=(x+2)  could also be seen as y=x to (y-2)=x.  This means that Moving the line y=x 2 units left has the same affect as moving the line y=x 2 units up(draw this to see yourself). Hence different transformations can give same result.