00:01
We will use the error bound formula for the trapezoidal rule to determine n so that if the interval from 0 to 10 of the exponential of negative 2x is approximated using the trapezoidal rule with n sub intervals, the error is guaranteed to be less than 10 to the negative 4.
00:22
So to apply this formula, we define the function f of x equal e to the negative 2x on the close interval.
00:33
010.
00:34
This function is continuous there.
00:37
It has derivative of all orders.
00:40
They are all continuous.
00:42
And we can apply then the error bound formula for the terms of the rule, which needs the second derivative to be continuous on the interval of integration, but in this case, all derivative of all orders are continuous there, particularly the second order derivative.
01:01
So that formula says that if we approximate the interval using trapezoid rule formula with ends of intervals, the error will be exactly equal to formulae's negative b minus a h square over 12 times the second derivative of f at some point z for some see on the open interval from a to b where a, b are the limits where a, b, it's a better, it's in both integration and so see in number c that exists is in the open interval, ab, and h, the step size h is b minus a over the number of intervals n.
02:37
So that's the formula of the error.
02:39
So that is the error is exactly equal to that.
02:43
Or what is the same, the difference between the function, which is the interval, sorry, and the formula of the terpsozor rule, is exactly equal to that expression.
02:54
This expression, as we can see, depends on h, but h is defined through n, the numbers of intervals, this way.
03:01
So it depends on n.
03:02
And the only thing we have here is that we don't know the value c.
03:08
We only know that it's in the open interval of integration.
03:12
Okay, so now we translate this to our problem.
03:15
Here we have a equals 0, e equal 10, because the interval of integration is 010.
03:22
We want to find n, and the second derivative of f, the first derivative is the same exponential function times derivative of the exponent, which is negative 2.
03:35
And the second derivative is the same exponential function, the same constant negative 2 times the derivative of despondent again.
03:46
That's 4, because it's negative 2 times negative 2.
03:51
That's it.
03:52
So the error, the trapezoidal rule with ends of intervals, is negative 10 minus 0 over 12...