Simple quick graph question, not a problem

[neutron star]

What does this mean on a graph?

http://img16.imageshack.us/img16/7046/picture19pc.png

I was given a y=f(x) graph and it has a multiple choice of like f(3) is greater than f(4) etc. But then it says f^r(4) is greater than f^r(3) what does that mean? I'm going to skip this one for now until I know what those mean.

 

[Galadirith]

Hi neutron star.

Now its slightly ambiguous a question as f^r(x), could mean raising the function to a power of r at x, ie [f(x)]^r and I wouldn't be completely sure without know the context the question is asked in. However generally f^r(x), means the r^th derivative of f(x) (this is know as Lagrange's notation for differentiation).

So these two mean the same thing:

<br /> f^{5}(x) = \frac{d^{5}y}{dx^5}<br />

<br /> f^{r}(x) = \frac{d^{r}y}{dx^r}<br />

obviously where y=f(x).

So qualitatively f^r(a) is the r^th derivative of f^r(x) evaluated at x=a (ie plonk a into the derivative in place of x :D)

Now hopefully that makes sense to you. It might be that the former explanation is actually the correct one, but it should be evident to you now know what it could possibly mean.

 

[neutron star]

Ok, I have it in another problem and this problem doesn't make sense. It looks like this.

http://img25.imageshack.us/img25/1750/picture20ag.png

Now, this made sense to me until I looked at a practice problem just like it in the book.

Instead it said If f(x)=x^3+4x
estimate f^l(3) using a table with values of x near 3, spread by 0.001.

The answer shown in the back of the book says f^l(3)=approx. 31

So I plugged in 3 and 2.999 to x^3+4x and got around 39. Why is this what is that x^l thing?

 

[neutron star]

I found someone else's question with the same symbol here https://www.physicsforums.com/showthread.php?t=342522
It's ' whatever that means?

 

[Galadirith]

Ah rite neutron star, now I think I can give you a bit more help.

Rite I think you are reading the notation wrong. Now you wrote f^l(3) however that is not what the question will be it will be f'(3), where that little "dash" in-between the f and the (3) bit is just that, a dash, read http://en.wikipedia.org/wiki/Notation_for_differentiation#Lagrange.27s_notation" about lagrange's notation (in fact it has a nice blown up image of what you are confused with, so you will clearly be able to see what it is :D). I think that if your not understanding the question simply because you confusing the notation just go talk with you teacher about it, and they will be able to explain much better any problems you have with notation.

In light of this as well when you said f^r(x) in your original post, you again I think actually couldn't see the notation properly, so mistook ' for ^r, which if it was quite small writing is understandable, but now you should be able to discern what it should be in the context.

Now in light of this new information, first work out the first derivative of f(x) = x³ + 4x. Now put your values into that. Notice anything :D. Hope that helps Neutron

 

[neutron star]

I'm still confused because I don't know what to do with the +4x. Can you explain that. I've been working with f(x)=x^n and f'(x)=nx^n-1. But I don't see how that would work out. And this problem is different than the other ones. The other problems were either just something like x^3 at x=2, and x^3+5 at x=1. But even in the second one I'm confused still about the +5 in it. Can you help?

 

[Galadirith]

Ok sure thing neutron. I was assuming that you had already covered basic calculus skills, but that's kl. So let's first look at the case where its in the form ax. So a represents and integer, just like the example you asked about 4x.

Now let's we say the f(x) = 10x. Now let's look at the ones you have been dealing with f(x)=xⁿ and f'(x)=nx^n-1. Now what if I then say 10x = 10x¹, we just don't write that 1 in normally, in fact no one ever does because it can't be anything else. Now have another look at that and see if you can solve it :D.

Now looking at constants, values that aren't multiplied my x like above, like the example you gave of 5. Now I could also write 5 as 5 = 5x⁰. again using the form that you know already how can you take this further :D. Have a real think about this now neutron, as the more you can reason yourself the better you'll be at it and the more you'll remember. It may take time, don't think you'll crack this in five minutes, it can take days or weeks or even longer some times to get a good grasp of the basics.

 

[neutron star]

Alright, so for this problem: http://img25.imageshack.us/img25/1750/picture20ag.png

It would be f'(5)=5^2=25*3=75+7=82 right?

 

[Slimsta]

Alright, so for this problem: http://img25.imageshack.us/img25/1750/picture20ag.png

It would be f'(5)=5^2=25*3=75+7=82 right?

look at the function, of f(x).
x^3 + 7x
derivative can be expressed in the form of:
lim [f(x+h) - f(x)] / h
h>0
if you don't know limits yet,
do this
f(x) = x^n + nx
f'(x) = nx^n-1 + n
so you take the power of x down and reduce one from the original power of x, for the constant, nx, you just remove the x.

i don't know if that helps, i really have to explain it in person for you to get it..