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Example Questions
Example Question #1 : Rate Of Change Problems
Find the average rate of change of the function over the interval from
to
.
The average rate of change will be found by .
Here, , and
.
Now, we have .
Example Question #2 : Rate Of Change Problems
Let a function be defined by .
Find the average rate of change of the function over .
We use the average rate of change formula, which gives us .
Now , and
.
Therefore, the answer becomes .
Example Question #1 : Rate Of Change Problems
Suppose we can model the profit, , in dollars from selling
items with the equation
.
Find the average rate of change of the profit from to
.
We need to apply the formula for the average rate of change to our profit equation. Thus we find the average rate of change is .
Since , and
, we find that the average rate of change is
.
Example Question #1 : Rate Of Change Problems
Let the profit, , (in thousands of dollars) earned from producing
items be found by
.
Find the average rate of change in profit when production increases from 4 items to 5 items.
Since , we see that this equals
. Now let's examine
.
which simplifies to
.
Therefore the average rate of change formula gives us .
Example Question #1 : Rate Of Change Problems
Suppose that a customer purchases dog treats based on the sale price
, where
, where
.
Find the average rate of change in demand when the price increases from $2 per treat to $3 per treat.
Thus the average rate of change formula yields .
This implies that the demand drops as the price increases.
Example Question #1 : Rate Of Change Problems
A college freshman invests $100 in a savings account that pays 5% interest compounded continuously. Thus, the amount saved after
years can be calculated by
.
Find the average rate of change of the amount in the account between and
, the year the student expects to graduate.
.
.
Hence, the average rate of change formula gives us .
Example Question #1 : Rate Of Change Problems
Find the average rate of change of between
and
.
The solution will be found by the formula .
Here gives us
, and
.
Thus, we find that the average rate of change is .
Example Question #2 : Rate Of Change Problems
Find the average rate of change of over the interval from
to
.
The average rate of change will be .
.
.
This gives us .
Example Question #1 : Rate Of Change Problems
Find the average rate of change of over the interval from
to
.
The average rate of change will be .
Now.
We also know .
So we have .
Example Question #1 : Rate Of Change Problems
Why can we make an educated guess that the average rate of change of , between
and
would be
?
We know is vertical on that interval.
We know is symmetrical on that interval.
We know is a polynomial.
We know is odd on that interval.
We know is horizontal on that interval.
We know is symmetrical on that interval.
Because is symmetrical over the y axis, it increases exactly as much as it decreases on the interval from
to
. Thus the average rate of change on that interval will be
.
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