Question

what is difference ?

f(x1) - f(x0)
----------
x1 - x0


f(x1) - f(x0)
lim ----------
x1->x0 x1 - x0

Answer 1

In the first one
\[ \text{x1 may or may not be equal to x0}\]
but in the second one x1 is approaching x0 and it's actually a limit test

Answer 2

are both "rate of change" ?

Answer 3

No, first is not rate of change
If you look carefully, the second one is actually f'(x0)
which is slope of f at x0

Answer 4

ash.... why first one is not rate of change..... whereas second one is rate of change...

Answer 5

Okay just wait for 5 minutes. I'll reconfirm

Answer 6

sure...........................

Answer 7

Yeah Jatin both are rate of change

Answer 8

But there is a difference between those two!!!

Answer 9

then what is the difference.. i'm so confused... i know that the second one is differentiation.

Answer 10

what is the difference ?

Answer 11

If f is a linear function like f(x)= mx+c
then both will give the same results,
say we have y=f(x)=2x
let x1= 3 and x0=1
so first one will give
\[\frac{f((x1)-f(x0)}{x1-x0}=\frac{6-2}{3-1}=2\]
now you find the second one it's f'(3)

Answer 12

yah... if f is linear function then this is true .. but what is the difference i really need to know

Answer 13

@jatinbansalhot the first one is the average rate of change
the second one is instantaneous rate of change

Answer 14

I'll explain with a diagram

Answer 15

ohhhhhhhhhhhhhhhhhhh......................... omg.. that is what i wanted thanks thanks really thanks a lot :) :) :)

Answer 16

Say we have f which is not a linear function

Answer 17

sure if u explain with dia then really thanks :)

Answer 18

Yeah. Now I'll mark two points (x1, f(x1)) and (x0, f(x0)) (x1, x0 are very close to each other) on this curve
and the first one which is the average rate of change considers this function as a line and just give the slope of this line
You can see the line 12
1 is (x0, f(x0))
2 is (x1, f(x1))
The first relation gives the slope of this line or simply average rate of change