How Can Variable Slopes Affect Average Rate of Change in 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

above please

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!

 

[anuttarasammyak]

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

 

[DaveE]

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".

 

[haruspex]

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.

 

[DaveE]

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.