Calculating Average Rate Of Change...

[DerkPY]

I'm sick of these maths papers... none of them have enough detail in them so that I understand what i'm doing!

I found this example online:

Average rate of change =

f(4) - f(2)/4 - 2 = 17 - 1/2 = 8

http://people.hofstra.edu/faculty/Stefan_W.../Calcsumm3.html

The examples on the right hand side of the page since the example above is a bit wierd.

I just need to know how they got 17-1 as the second part of the sum... then I'm away!

None of the sites or my books seem to tell me, they just do it!

[dreamz Veteran]

you're calculating the slope using two points. they substituted in x = 2 and x = 4 and calculated f(4) - f(2).

instantaneous rate of change is the slope at a point.

[kylejn]

[f(b)-f(a)] / (b-a) = avg. rate of change from x=a to x=b

f(x) = 2x^2 - 4x + 1

Let b = 4 and a = 2

[f(4)-f(2) / (4-2)

=(17-1) / 2

=8

EDIT: Clarification

[DerkPY]

Great! Thanks.... I just got help off a friend just as you posted that! I figured out the right answers though... so I've got a grip of my work now!

[kylejn]

Glad to hear. :)

[PseudoRandomDragon]

The formula for average rate of change is

f(x1) - f(x2)

-------------

x1 - x2

That formula comes from the change in y over the change in x. Ya know, that formula you used to find the slope of a line.

mSec (slope of a line) formula:

y1 - y2

-------

x1 - x2

Do you see how they are exactly the same, only y has been replaced with f(x)? That is because y is merly a function of x and the average rate of change is merly the slope of the line.

I hope that explains it.

[DerkPY]

I've done all bar one - well I've done it, but I don't know if it's right:

the function f(x) = 2/x+1 over intervals:

[x , x + Delta x]

I worked this:

2 / [x + Delta x + 1] - 2 / (x)+1

= Delta X / (Delta X + x + 1)

Is that anywhere near right? I really got lost with all of the delta's!

[PseudoRandomDragon]

It was hard to read your answer, but it's probably wrong.

Here is what the answer should read:

[DerkPY]

Is that the final answer or can I cancel that down?

[PseudoRandomDragon]

Yes the answer can be reduced. See if you can do it.

[DerkPY]

I ended up with Delta X / X - X + Delta X

Once I had subtracted all of the top. Is that right?

[PseudoRandomDragon]

X - X + deltaX = deltax, you can subtract the two X's.

The top is wrong. You probably made a mistake in following the rules of subtracting the two fractions. The two fractions do not have a common denominator. You need to find it, then subtract the two fractions on the top. Once that is done you will be able to reduce even further.

[ariad]

:blink: :blink: :blink: :blink: :blink: :blink:

[DerkPY]

X - X + deltaX = deltax, you can subtract the two X's.

The top is wrong. You probably made a mistake in following the rules of subtracting the two fractions. The two fractions do not have a common denominator. You need to find it, then subtract the two fractions on the top. Once that is done you will be able to reduce even further.

585598536[/snapback]

Oh yer... i didn't see that bottom one!

As for the top, can I not simply take them away from each other? They are minus each other. I'm stumped!

- Oh! I need to make the bottom the same don't I on either side? - so do I times the first one by Delta X... which would give me the same fraction on either side?

[PseudoRandomDragon]

Oh great you are learning average rate of change and you dont have a grasp on subtracting fractions. The denominators are different, so no you cant simply subtract them. The first thing you do is find the common denominator, then you subtract. How do you find the common denominator? Well look below.

[DerkPY]

Looking through this I got 2 answers, and to be honest... I think their both wrong! I'm really crap at maths!

I got:

-X + Delta X + 1

Delta X

Or

X + -Delta X + 1

Delta X

[PseudoRandomDragon]

Yep they are both wrong. You're just going to have to review subtracting fractions, otherwise any function that has a fraction in it will give you difficulty.

Maybe giving you a simpler example will give you the ability to solve the top part correctly. In the attached image I made up two fractions that had to be added. Look at the steps closely.

[DerkPY]

I got it as the whole thing = Delta X...

On the top I got 2(X + Delta X + 1) / X + Delta X + 1 (X + 1)

Then on the other fraction I got 2 (X + 1) / X + Delta X + 1 (X + 1)

That makes both of the bottom the same (like the example above).

I then canceled both of the 2's from the top, followed by 2 X's and 2 1's. Leaving me with Delta X.

Because both the bottom were the same, I could just remove it completely giving me Delta X.

Is this right? I'm sure someone else got Delta X!

[PseudoRandomDragon]

I then canceled both of the 2's from the top, followed by 2 X's and 2 1's.  Leaving me with Delta X.

585600798[/snapback]

I think you got it. The top part does come out to be deltax/(x+1)(x+deltax+1). The whole fraction should be in the image I attached, which looks somewhat funny because it looks like a double fraction. That fraction can be reduced one more time. See if you can figure it out (hint: it is a fraction being divided by deltax)

[DerkPY]

I handed it in on Friday. In the end I got Delta X over Delta X which gave me the answer of 1. A lot of other people seemed to of got the same answer, and even if its wrong, you get most of the marks for the working out.

Thanks for all your help.

[PseudoRandomDragon]

Thats cool. Don't feel bad about having trouble with that problem, you'll get it. When you study just start with easy functions that don't have fractions.

Looking back I think I made a mistake somewhere, so I just redid the problem and post the answer so you have something to look at.

[DerkPY]

Kind of! My method's the same... I think I slipped up somewhere. Thanks for your help :)

[PseudoRandomDragon]

You are welcome. :)