Rate of change from 1 to 2 for f(x)=2x^3 + x

[Rusho]

Find the average rate of change from 1 to 2 for the function f(x)=2x^3 + x

so I did this:
[f(2) – f(1)] – [2x^3 + x] / 2-1

= 2-1-2x^3 + x / 1

= 1-2x^3 + x

= -2x^3 + x

Right?

 

[Galileo]

The average rate of change of f from a to b is
[f(b)-f(a)]/(b-a)

and it's (naturally) just a number. It doesn't depend on x. Check your definition.

 

[Rusho]

[2(2)^3 - 2(1)^3] - [2x^3 + x] / 2-1

16-1-2x^3 + x / 1

15-2x^3 + x

I don't understand what you are telling me

 

[Galileo]

You've got the definition of the average rate of change wrong.
You wrote something like (f(2)-f(1)-f(x))/(2-1).
By definition, the average rate of change of f on the interval [a,b] is:
$$\frac{f(b)-f(a)}{b-a}$$

So in your case, the average rate of change is:
$$\frac{f(2)-f(1)}{2-1}$$

 

[Rusho]

Ok, I am not sure what to do with 2x^3 + x . So I subtracted it from the f(b) - f(a).
If I had 2x^3 by it self, I can see just putting 2(2)^3 - 2(1)^3 / 2-1
but the "+x" is confusing me

 

[Rusho]

I think I got it

2(2)^3 + x - 2(1)^3 + x / 2-1

=16-2+x+x / 1

=14+2x

=-2x + 14

x = -7

 

[Galileo]

So you can solve it if the function is 2x^3, but not if it's 2x^3+x? What's the difference, conceptually?

f(x)=2x^3+x, so what is f(2)? And what is f(1)?

 

[BobG]

I think I got it

2(2)^3 + x - 2(1)^3 + x / 2-1

=16-2+x+x / 1

=14+2x

=-2x + 14

x = -7

Calculate f(2). Calculate f(1). Subtract the result of f(1) from f(2).

The solution for f(2) is not 16+x. You have to substitute '2' for x everywhere it appears, so the solution for f(2) is 16+2.

Also, your algebra is wrong (in addition to being not applicable in this case). If you have:

$$(3x^2 + 3x) - (2x^2 + 2x)$$
then the minus sign means both the 2x^2 and the 2x are negative:
$$3x^2 + 3x - 2x^2 - 2x$$
$$(3x^2 - 2x^2) + (3x - 2x)$$
etc.

 

[Rusho]

2(2)^3 + (2) -1 / 2-1

16+2-1 / 2-1

17/1

17

I'm sorry if I'm just not getting it

 

[Galileo]

Alright, let's take some steps back.
You are given a function f. It's a machine that eats a number and spits out a (usually different) number.
f(x)=2x^3+x tells you the value of the function at each point, it's an equality that holds for each number x.
For example:
f(1)=2(1)^3+1=2+1=3
f(5)=2(5)^3+5=2(125)+5=255

So if you want to calculate [f(2)-f(1)]/(2-1) you have to calculate f(2) and f(1). I already did f(1) for you above. Now you do f(2) and calculate [f(2)-f(1)]/(2-1)

 

[Rusho]

2(2)^3 + (2) - 2(1)^3 +1 / 2-1

=18-3 / 2-1

=15/1

=15

 

[Galileo]

Right that's correct.

BTW: Mind your brackets: -2(1)^3+1 is not the same as -(2(1)^3+1)

 

[Rusho]

Great! Thanks for your help!
I have another one, maybe I should start a new post