In many cases, when using Simpson's rule
to approximate a deaf integral, there will be an error.
However, as n increases, the error will decrease.
And remember, for Simpson's rule, n must be even.
So, in this question, we want to determine how large
should n be to guarantee that Simpson's rule approximation
of the given deaf integral, is accurate
within zero point one or one tenth.
So, to answer this question, we'll be using
the error bounds formula given here for Simpson's rule,
where the error is less than or equal to
the quantity b minus a raised to the fifth power,
or a and b on the limits of integration,
divided by 180 n to the fourth times the absolute value
of the maximum function value of the fourth derivative
on the closed interval from a to b.
So, there's really two steps to this problem.
Once we set up this inequality, we'll have to work
to find this maximum function value.
Once we find this value, we'll then solve
this inequality for n.
So, to begin to set this up, looking at the right side
of this inequality, we would have the quantity
b minus a to the fifth, or one minus
negative two to the fifth,
divided by 180 n to the fourth,
times the absolute value of the maximum function value
of the fourth derivative,
on the closed interval from negative two to positive one.
We want this to be accurate within one tenth,
and, therefore, we want this to be less than
or equal to one tenth, which I'll add as a fraction.
Now, let's go to the next slide and work on
finding this maximum function value.
Remember, it's going to be
the absolute value of this maximum.
So, if it happens to be negative,
we are going to take the absolute value.
So, we'll first start with our function, f of x,
which is the integran function,
and work to find our fourth derivative.
So, the first derivative
would be negative four x to the third,
minus six x squared,
plus 24 x plus two minus zero.
And the second derivative function would be
negative 12 x squared,
minus 12 x plus 24
plus zero.
And the third derivative function would be
negative 24 x minus 12.
And notice how our fourth derivative function
is going to be a constant function.
It's equal to negative 24.
Remember, our goal here is to find the absolute value
of the maximum function value
of the fourth derivative function
on the closed interval from negative two to positive one.
But because the fourth derivative is a constant function,
we would just have a horizontal line
through y equals negative 24.
And therefore, in this case, the absolute value
of the maximum function value
would just be the absolute value of negative 24,
which would be positive 24.
So, we're going to use positive 24 in our air bounds formula.
So, going back to the previous slide,
we'll now substitute 24 for this value,
and then solve this for n.
So, here we'd have three to the fifth,
which is 243
divided by 180 n to the fourth
times 24,
is less than or equal to one tenth.
243 times 24 is equal to 5,832.
And now, from here, let's clear
the fractions from the inequality.
One way of doing this would be to multiply both sides
of the equation by 180 n to the fourth.
Notice on the left side, this simplifies nicely to one,
and on the right side, these two do share
a common factor of 10.
This simplifies to one, this simplifies
to 18 n to the fourth.
So, now we have 5,832 is less than or equal to
18 n to the fourth.
Now, if we divide both sides by 18,
we would have 342 is less than or equal to n to the fourth.
And now, we'll take the fourth root
of both sides of the equation.
We'll have to get a decimal approximation for the left side.
So, we would have N is greater than or equal to,
approximately, four point two four.
Or if we want, we can write this as n on the left side,
n I greater than or equal to four point two four.
Remember, n is the number of sub intervals
using Simpson's rule, and n must be even.
So, notice hour four would not satisfy this inequality,
and we can't use five, because five is not even.
And, therefore, to answer the question,
we would have to use n equals six
to guarantee that Simpson's rule approximation
of this integral would be accurate within one tenth.
So, that is one thing to be careful about:
when determining n for Simpson's rule,
n must be even.
I hope you found this explanation helpful.