Let P[t] represent a population of a city
in millions of people, and t represent
the number of years since 2010.
Interpret the meaning of P'[2]=1/2.
Before we do this, let's
also write this derivative
function value using Leibniz notation.
P prime of two equals dp/dt,
when t equals two, which
we're given is equal to 1/2.
Leibniz notation reminds us
that this derivative function
value indicates derivative
change of the population,
with respected time, when t equals two.
The first thing to notice is
that the derivative function
value is positive, which indicates P[t]
is increasing, at t equals two.
Now to interpret this
meaning more precisely,
let's write the derivative function value
of 1/2 as a fraction.
I know 1/2 is already a fraction,
but let's write it as a
fraction as 1/2 over one.
We'll say P prime of
two equals 1/2 over one.
Now, we'll include units
for the one half of the one,
to better interpret the meaning
of this derivative function value.
The 1/2 represents the
change in the population,
which is measured in millions of people.
The 1/2 represents 1/2 a million people.
The denominator of the one
represents the change in time,
and time is measured in years.
This would be one year.
Now we know when t equals two,
the population is increasing
at a rate of 1/2 a
million people per year.
The last step is to interpret
the meaning of t equals two.
Remember t is the number
of years since 2010,
and therefore t equals two,
corresponds to the year 2010,
the base year, plus the t value of two.
2010 plus two equals 2012.
Now we know in 2012, the
population is increasing
at a rate of 1/2 a
million people per year.
Let's write this as a complete sentence.
Now, let's look at our choices.
The first two choices cannot be correct,
because the year is given as 2010.
Let's look at the last two choices.
Here, we have the population in 2012
is growing by 1/2 a million people.
This cannot be correct,
because notice how this is showing
an amount of growth, not a rate of growth.
The fourth choice is
the population in 2012
is growing by 1/2 a
million people per year.
Which is the same as saying
in 2012, the population is
increasing at a rate of 1/2
a million people per year.
The correct choice is this sentence.
I hope you found this helpful.
