During the first couple of weeks
of a new flu outbreak, the disease spreads
according to the equation
I of t equals 2000 times e
raised to the power of 0.082t,
where I of t is the
number of infected people
t days after the outbreak
was first identified.
Find the rate at which the
infected population is growing
after 10 days.
Include the appropriate units.
Because we are looking for the
rate of growth after 10 days,
we need to find the derivative of I of t,
or I prime of t.
Notice I of t is an exponential
function with base e,
and because the exponent is not just t,
we will need to apply the chain rule,
which is built into the
derivative formula here,
where the derivative of e
to the u with respect to x
equals e to the u times u prime.
So beginning with the given function,
we have I of t equals 2000
times e raised to the power of 0.082t.
Looking at the derivative formula,
the exponent is u, so
we have u equals 0.082t.
And now we need to find u prime or du/dt.
U prime is equal to the
derivative of 0.082t
with respect to t, which is 0.082.
And now we can find I prime of t.
I prime of t
is equal to 2000 times the
derivative of e to the 0.082t,
which gives us 2000
times e to the u times u prime,
which is e to the 0.082t times 0.082.
And now simplifying, 2000 times 0.082
equals 164, giving us I prime of t equals
164 times e raised to the power of 0.082t.
And now we can use this function
to determine the rate of growth
of the infected population
after t days.
Because we are looking for the
rate of growth after 10 days,
we need to find I prime of 10.
I prime of 10
is equal to 164 times e
raised to the power of 0.082 times 10.
Simplifying, we have 164 times e
raised to the power of 0.82,
and now we will go to the calculator
to get a decimal approximation.
Notice how this gives
us approximately 372.36.
This is the rate of growth
of the infected population
after 10 days.
Let's focus on the units of this rate.
Another way to write I prime of t
is to use Leibniz's notation,
which would be dI/dt.
This reminds us that the
derivative function value
is the ratio of the change
of the function value I of t
to the change in t,
where I of t represents the
number of infected people
and t represents the number of days.
And therefore, I prime of t, or dI/dt,
represents the ratio of the
instantaneous rate of change
of the number of people infected
to the change in days.
In our case, if we round to
the nearest whole number,
we have 372, which we can
write as a fraction as
372 over one, where the 372
represents the change in people
and the one represents the change in days.
And therefore, the units
would be 372 people
per one day.
Let's write this as a complete sentence.
After 10 days, the infected
population is growing
at a rate of approximately 372 people
per one day or just per day.
I hope you found this helpful.
