Rate of Change Help

[cole92]

The question asks me to find the average rate of change of the function over the indicated interval. Then I am supposed to compare that rate of change with the instantaneous rates of change at the endpoints of the interval. My question is how I find the instantaneous rates of change? I have a feeling the answer is right under my nose but I don't see it.

For example:

Function: f(t)= t^2 - 3
Interval: [2,2.1]

If I am correct I do [f(2.1) - f(2)]/2.1 - 2 in which case I get 4.1 as my avg. rate of change. How do I know the inst. rates of change?

 

[mmm4444bot]

… If I am correct I do [f(2.1) - f(2)]/2.1 - 2 …

… How do I know the inst. rates of change?

Yes, that's correct for the average rate of change. (You forgot to type square brackets around the denominator.)

Do you "see" that the average rate of change is simply the slope of the line through the two given points on the graph of function f ? This is always the case, with average rates of change.

The instantaneous rate of change is given by the first derivative of f. What have you learned about derivatives?

 

[cole92]

Re:

… If I am correct I do [f(2.1) - f(2)]/2.1 - 2 …

… How do I know the inst. rates of change?

Yes, that's correct for the average rate of change. (You forgot to type square brackets around the denominator.)

Do you "see" that the average rate of change is simply the slope of the line through the two given points on the graph of function f ? This is always the case, with average rates of change.

The instantaneous rate of change is given by the first derivative of f. What have you learned about derivatives?

The derivative would be 2t, so do I plug in 2 and 2.1 for t?
If so that would make the rates 4 and 4.2, correct?

 

[mmm4444bot]

The derivative would be 2t, so do I plug in 2 and 2.1 for t?
If so that would make the rates 4 and 4.2, correct?

Yes and yes. (Instantaneous rates of change always occur at distinct values in the domain.)