In estimating the definite integral
of cosine-4X from negative one to four
using Simpson's Rule with N equals 10,
we can estimate the error involved in the approximation
using the Error Bounds formula
given here below.
When using Simpson's Rule,
the error will be less than or equal to
the quantity B minus A to the fifth power,
where A and B are the limits of integration,
divided by 180 N to the fourth
times the absolute value of the maximum
of the fourth derivative function
on the close interval from A to B.
To find this maximum function value on the close interval,
we'll have the option of either analyzing
the fourth derivative function graphically,
or find the critical numbers
on this close interval,
and then evaluating the function at the critical numbers
as well as the end points
because we have a closed interval.
So, let's begin to set this up.
We'll have the error
is less than or equal to,
again, we have the quantity B minus A to the fifth,
where A is negative one and B is four,
so we would have the quantity
four minus negative one raised to the fifth
divided by 180 N to the fourth
where N is given as 10.
Then, we're going to multiply this by
the absolute value of the maximum function value
of the fourth derivative on this close interval.
Where our close interval will be
from negative one to positive four.
Now, let's work on finding this maximum function value,
and then we'll come back and complete this calculation.
Starting with the integrated function,
we would have F of X
equals cosine for X,
and now, we'll work on finding
our fourth derivative function.
The first derivative
requires the Chain Rule,
so the derivative of cosine-4X would be negative sine
4X times four, or negative four sine-4X.
Now, we'll find the second derivative,
which would be negative times the derivative of sine-4X,
which would be a negative four cosine-4X times four,
or negative 16 cosine-4X.
Now we can find the third derivative,
which should be negative 16 times
the derivative of cosine-4X,
which would be negative 16 times
negative sine-4X times four,
or 64 sine-4X.
And now finally, we can find the fourth derivative function,
which should be 64 times the derivative of sine-4X,
which should be 64 cosine-4X times four,
or 256 cosine-4X.
And again, our goal is to find the maximum function value
of the fourth derivative function
on the close interval from
negative one to four.
To do this using calculus techniques,
we would have to find the derivative of this function here,
which would be the fifth derivative.
Set it equal to zero and solve for X
to find the critical numbers on this interval,
and then evaluate the fourth derivative function
at the critical numbers and the end points
to determine the maximum function value.
But we should be able to recognize
that the graph of our fourth derivative function
would have an amplitude of 256,
which means the maximum function value would be 256.
So, let's go ahead and verify this graphically.
Here's the graph of the function
on the close interval from negative one to four,
and notice how
the maximum function value,
which would be right here,
which is 256,
actually occurs three times over this interval,
here, here, and here.
Also, notice if we evaluate this function at the end points,
F of negative one
would be here,
and the absolute value of this would still be
less than 256,
and F of four
would be here,
and notice how the absolute value of this
would still be less than 256
because we know that the minimum function value here
would be negative 256.
In this case, one reason why it's easier
to analyze the graph is because
on this close interval, there would actually be
one, two, three,
four, five, six, seven different critical numbers
because that's where the maximum end values occur.
Now that we know the maximum function value
on this close interval,
we can finish our error calculation.
We'll simply substitute 256 for this value here.
We'll have the error is less than or equal to,
this is going to be five to the fifth
divided by
180 times,
this is going to be 10,000,
and then we have times 256.
If we want, we can put this over one.
This is going to be equal to 800,000
divided by one million 800,
which does simplify nicely to four-ninths.
In estimating this definite integral
using Simpson's Rule with N equals 10,
the error will be less than or equal to four-ninths.
I hope you found this explanation helpful.