Question

Evaluate f(x+h)-f(x)/h and simplify if f(x)=x^2-2x

Answer 1

Ok so first write f(x+h)

Answer 2

K got it

Answer 3

What did you get?

Answer 4

I wrote that down?

Answer 5

I just wrote down f(x+h) like you said ha

Answer 6

Oh, no I mean evaluate it. Sorry

f(x+h) = ?

Answer 7

So where there is an x in f(x) put an x+h.

Answer 8

Okay so it's f(x+h)=(x+h)^2-2(x+h)?

Answer 9

Right! Now multiply all that out

Answer 10

So it's -2(x+h)^3?

Answer 11

Nope..\[f(x+h) = (x+h)^2 - 2(x+h)\] \(=(x+h)(x+h) - 2(x+h)\)
\(=(x+h)(x+h) - 2(x+h)\)
\(=x^2 + 2xh + h^2 - 2(x+h)\)
\(=x^2 + 2xh + h^2 - 2x-2h\)

Answer 12

You have to foil out the (x+h)^2

Answer 13

Oh goodness

Answer 14

It's not as scary as it looks

Answer 15

So now we want to know what \(f(x+h) - f(x)\) is because we want to divide it by h

Answer 16

So what is -f(x) ?

Answer 17

Okay hold on uno momento, I dont' understand where you got the last part of the foiled equation?? How did it turn into x^2+2xh+h^2?

Answer 18

Ok.
\[(x+h)^2\]\[=(x+h)(x+h)\]If you know FOIL you take ther First, Outer, Inner, and Last terms and multiply them, then sum the results.

\(=(x\cdot x) + (x\cdot h) + (h \cdot x) + (h\cdot h)\)
first outer inner last
\[=x^2 + hx + hx + h^2\]\[= x^2 + 2hx + h^2\]

Answer 19

Okay so it was two different steps? It was (x+h)(x+h)-2(x+h)+(x+h)(x+h)-2(x+h) and then x\[^{2}\]+2xh+h\[^{2}\]=x\[^{2}\]+2xh+h\[^{2}\]-2x-2h?

Answer 20

Whoa that was weird...

Answer 21

I'm not sure I follow all that.

Answer 22

Hold on

Answer 23

So it was two different steps? (x+h)(x+h)-2(x+h)=(x+h)(x+h)-2(x+h) and then x^2+2xh+h^2-2(x+h)=x^2+2xh+h^2-2x-2h?

Answer 24

Yes, those are separate steps

Answer 25

So now we have an expression for f(x+h) now from that we must subtract f(x)

Answer 26

So subtract all of the f(x+h) from the equation?

Answer 27

\[f(x+h) = x^2 + 2xh + h^2 - 2x - 2h\]\[f(x) = x^2 - 2x\]
\[\implies f(x+h) - f(x) = [x^2 + 2xh + h^2 - 2x - 2h] - [x^2 - 2x]\] ^This is f(x+h) ^This is f(x)

Answer 28

So what do we have left after we subtract?

Answer 29

(2xh+h^2-2h)?

Answer 30

f(x+h)-F(x)=(2xh+h^2-2h)

Answer 31

That's exactly right. Sorry someone called.

Answer 32

So now we divide that by h (cancelling one factor of h from each term)

Answer 33

Oh it's alright!

Answer 34

So it's f(x+h)-f(x)=(2x+h-2)?

Answer 35

(f(x+h)-f(x))/h=2x+h-2

Answer 36

So I'm right then myininaya?

Answer 37

with that one little correct yes

Answer 38

correction*

Answer 39

Okay but why is it divided by h?

Answer 40

you want to find
\[\frac{f(x+h)-f(x)}{h}\]

Answer 41

Because you had it that way in the original problem

Answer 42

Right, duh. Sorry my mind is kinda trashed

Answer 43

We found that \[f(x+h) - f(x) = 2xh + h^2 - 2h\]

Therefore \[\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 - 2h}{h}=2x + h - 2\]

Answer 44

Excellent! Thank you so much