In estimating the definite integral
of cosine four x from negative one to four
using the Trapezoid Rule with n equals 10,
we can estimate the error involved in the approximation
using the Error Bounds formula
which is given here below.
The error is going to be less than or equal
to the quantity b minus a-cubed
where a and b are the limits of integration
divided by 12 n-squared times the absolute value
of the maximum of the second derivative function
on the closed interval from a to b.
So to determine this maximum function value
on the closed interval, we can either analyze
the graph of the second derivative function
on this interval or we can find the critical numbers
and then evaluate the second derivative function
of the critical numbers as well as the endpoints
because we do have a closed interval.
So let's begin to set this up.
The error is going to be less than or equal to.
We'll have the quantity b minus a cubed
where a is negative one and b is four.
So we'll have the quantity four minus negative one cubed
divided by 12 n-squared where n is given as 10
and then times the absolute value of the maximum
of the second derivative function
on the closed interval from negative one to four.
So let's go find this maximum function value
on the next slide and then we'll come back
and finish this calculation.
So starting with the integrand function,
f of x equals cosine four x,
we want to find the second derivative.
So we'll first find the first derivative.
To find the derivative of cosine four x,
we have to apply the chain rule.
So our derivative will be negative sine four x times four
or negative four sine four x.
And now the second derivative
would be equal to negative four times
the derivative of sine four x
which would be negative four cosine four x times four
or negative 16 cosine four x.
Now we want to find the maximum function value
of our second derivative function
on the closed interval from negative one to four.
So what we could do is define the derivative
of this function which would be the third derivative,
set it equal to zero, and then solve
to find the critical numbers
and then evaluate the function at the critical numbers
and the endpoints to determine the maximum function value.
But in this case, we should recognize
that the graph of our second derivative function
would have an amplitude of 16.
So in this case, let's just analyze the graph
of our second derivative function
to determine the maximum function value
on this closed interval.
So here's the graph of our second derivative function.
And again, because we have an amplitude of 16,
we can easily verify the maximum function value
on the closed interval from negative one to four
would be positive 16 which actually occurs four times
over this interval: here, here, here, and here.
Notice f of negative one
would be some value here less than 12
and f of four would be some value here
less than 16.
So again, this graph verifies that the absolute value
of the maximum function value
of our second derivative function
on this closed interval is 16.
So going back to our error calculation,
we can now substitute 16 for this maximum function value.
So we'd have the error is less than
or equal to, this would be five cubed
or 125 divided by, this would be 12 times 100 or 1,200.
Again, times 16, which would give us 2,000
divided by 1,200.
These two have a common factor of 400,
so this simplifies nicely to five-thirds.
So what we've discovered is
when using the Trapezoid Rule
to evaluate this definite integral
when n equals 10, the error will be less than
or equal to five-thirds.
Now even though the directions do say
keep at least two decimal places accuracy,
since we have the exact value here
there's no reason to convert this
to a rounded decimal.
Let's go ahead and leave this as the error
is less than or equal to five-thirds.
I hope you found this explanation helpful.