Thread: [SOLVED] Instantaneous rate of change

1. [SOLVED] Instantaneous rate of change

Which of the following will not give the instantaneous rate of change of h(x) with respect to x when x=3?

1. lim x->3 (f(x)-f(3))/(x-3)
2. lim h->0 (f(3+h)-f(3-h))/2h
3. lim x->3 (f(3)-f(x))/(3-x)
4. (f(3)-f(3-0.1))/0.1

I know for sure the 1 will give the instantaneous rate of change. Not sure about 2,3 and 4. I guessing that 4 is the answer, because it gives what seems like an average value instead of an instantaneous value.

2. Originally Posted by solars
Which of the following will not give the instantaneous rate of change of h(x) with respect to x when x=3?

1. lim x->3 (f(x)-f(3))/(x-3)
2. lim h->0 (f(3+h)-f(3-h))/2h
3. lim x->3 (f(3)-f(x))/(3-x)
4. (f(3)-f(3-0.1))/0.1

I know for sure the 1 will give the instantaneous rate of change. Not sure about 2,3 and 4. I guessing that 4 is the answer, because it gives what seems like an average value instead of an instantaneous value.
4.

An instantaneous rate of change is found using a limiting process.

4. gives the approximate rate of change at x = 3.

1., 2., and 3. are all equivalent and correct.

3. Originally Posted by solars
4. (f(3)-f(3-0.1))/0.1

I know for sure the 1 will give the instantaneous rate of change. Not sure about 2,3 and 4. I guessing that 4 is the answer, because it gives what seems like an average value instead of an instantaneous value.

Well done

Now as an exercise, can you reason out why the others are $f'(3)$?

P.S: There is an easy trick, do you know L'Hospital's rule?

4. Yup!