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Example Questions
Example Question #2 : 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 #3 : 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 #5 : Derivatives
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 #1 : 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 #2 : 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 horizontal on that interval.
We know is odd on that interval.
We know is a polynomial.
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
.
Example Question #51 : Introductory Calculus
If the average rate of change of between
and
, where
, is positive, then what can be said about
on that interval?
is constant
is decreasing
is increasing
is odd
is increasing
If the average rate of change is positive, then the formula gives us , so
. We know
because it is a given in the proble, so
. Hence
and
. This shows that
must increase over the interval from
to
.
Example Question #12 : Rate Of Change Problems
If the average rate of change of between
and
, where
, is negative, then what can be said about
on that interval?
is constant
is an odd function
is negative
is decreasing
is increasing
is decreasing
If the average rate of change is negative, then the function is changing in a negative direction overall. Hence, the graph of the function will be decreasing on that interval.
and since
,
is decreasing
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