# Graphing advanced functions

MartynaJ
Homework Statement:
Sketch a possible graph for the function that satisfies all of the following criteria and justify
your sketch by clearly labelling each of the given information.
□ Average rate of change = 5 for t belongs to [1, 5]
□ Average rate of change = -20 for t belongs to [1,10]
□ Average rate of change = 0 for t belongs to [2 , 9]
□ Instantaneous rate of change = -10 at t = 2
□ Instantaneous rate of change = 0 at t =3
Relevant Equations:
So I attempted this problem and to satisfy the first condition (for t in the range of [1, 5]), I drew the straight line that has a slope of 5 (i.e. f(x)=5x). I just don't understand how I can have the same function with a different slope (average rate of change) for the interval [1,10] or for [2 , 9]... Any help please!

## Answers and Replies

Gold Member
As for the first statement it means, say f(t)
$$f(5)-f(1)=5(5-1)=20$$ and so on.

WWGD, Delta2 and DaveE
Gold Member
By "average rate of change" they mean the average slope over the specified interval. So, for example, you can have an average slope of 5 over [1, 5] with a line that is flat on [1, 3.5] and has a slope of 10 on [3.5, 5].

Frankly I'm not sure what "instantaneous rate of change" means. I would guess it means "constant rate of change" or "rate of change at each point".

Homework Helper
Gold Member
2022 Award
you can have an average slope of 5 over [1, 5] with a line that is flat on [1, 3.5] and has a slope of 10 on [3.5, 5].
Wouldn't that give an overall change of 15 instead of 20?
what "instantaneous rate of change" means
It just means the slope at that point.

Gold Member
Wouldn't that give an overall change of 15 instead of 20?
Oops! Yes. I guess I'm better at math than arithmetic, LOL. I didn't split the interval in half as I intended.

It just means the slope at that point.
Yes, I missed that it was defined at a point not the whole interval.