I'm not sure what content is in the current General course - I've never taught it. But Simpson's rule was in the old Maths in Society course (pre 2001).(Unless your emphasis was about 'areas' (the integral is exact, but this may not be the 'area').)
Why isn't it possible for the area to be exact with some arcs concave up, and some concave down? I mean, it's unlikely to be true in practice, but theoretically we could have exact area. We could imagine the following: the x-axis is the base, we start at x = 0 and finish at x = 1. The measurements at x = 0, x = ½, and x = 1 could all be the same (say 1 metre), and the measurement at x = ¼ could be say 1.5 m, with a measurement of 0.5 m at x = ¾. Then we have one parabola being concave down, and the other concave up, and using Simpson's rule for each parabolic arc would give the exact area. (Probably an unrealistic fence structure, but still possible mathematically to get exact area.)Many teachers taught that it gave the exact area if the boundary of the field was made up of parabolic arcs. This is nonsense, unless all the arcs are "concave outwards".
I'm not talking about concave up/down.Why isn't it possible for the area to be exact with some arcs concave up, and some concave down? I mean, it's unlikely to be true in practice, but theoretically we could have exact area. We could imagine the following: the x-axis is the base, we start at x = 0 and finish at x = 1. The measurements at x = 0, x = ½, and x = 1 could all be the same (say 1 metre), and the measurement at x = ¼ could be say 1.5 m, with a measurement of 0.5 m at x = ¾. Then we have one parabola being concave down, and the other concave up, and using Simpson's rule for each parabolic arc would give the exact area. (Probably an unrealistic fence structure, but still possible mathematically to get exact area.)
Oh, I see.I'm not talking about concave up/down.
I'm talking about (for example) the area of the region between y=x^2 and the Y-axis. (ie. y = sqrt(x) and the x-axis)
That is, it gives the exact area on the convex side of the the parabola, but not on the concave side.
Edit: Oops ... I now see the confusion .... I'm saying it all wrong.
It's not about concave convex ... it's about whether the axis that bounds the region is parallel or perpendicular to the axis of the parabola.
Thanks!In that formula, n is the number of sub-intervals, you're right. Sub-intervals refers to how many points we are using. Basically, is our interval is [a, b], we can split it up evenly using the following x-values:
[x0, x1, x2, ..., xn – 1, xn], where
x0 = a (the starting point of our interval where we're integrating)
xn = b (our ending point).
Each 'sub-interval' refers to the following intervals: [x0, x1], [x1, x2], ..., [xn – 1, xn]. As you can see, there are n sub-intervals, because we have a total of n + 1 points being used.
Since we have equal spacing, the width of each sub-interval is h = (total width of intervals)/(number of intervals) = (b – a)/n.
Now for Simpson's rule, we actually apply each approximation by spanning over two sub-intervals. E.g. if we had our original interval of integration as [0, 1], we'd need to split it up into at least two sub-intervals, e.g. via the partition {0,½,1} (here, we're using 3 x-values, and the two sub-intervals are [0, ½], [½, 1]).
Then what Simpson's rule would do is it essentially finds the unique quadratic (or line if the function we're integrating is linear) that goes through the points (0, f(0)), (½, f(½)), and (1, f(1)) and computes the integral of this quadratic (or line), and gives us this value as an approximation to the integral of our original function. So you can see that for Simpson's rule to work, we'll need at least 3 x-values. If we want more x-values, we can add two more (so 5 x-values), but we can't add just one more, since Simpson's rule requires a multiple of 2 to be the number of sub-intervals. E.g. if we tried using the x-values {0, ⅓, ⅔, 1} (i.e. 4 x values, which is 3 sub-intervals), our Simpson's rule would first be applied over the interval [0, ⅔] (using the y-values at 0, ⅓, and ⅔), but for the next interval, we couldn't use Simpson's rule, since we don't have 3 remaining x values.
So in general, for Simpson's rule, n (the no. of sub-intervals) must be even, because we use two sub-intervals for each 'use' of Simpson's rule.
(Sorry if that's hard to understand without a picture. Here's a picture: http://www.mathwords.com/s/simpsons_rule.htm ; Simpson's rule would be used on the interval [x0, x2] (as 3 x-values are used), and then the next one would use the interval [x2, x4], so even number of sub-intervals required (a sub-interval is just between adjacent x values)
So if we had 5 x values to use {x0 = a, x1, x2, x3, x4 = b}, we'd have n = 4, and Simpson's rule would be used on [x0, x2] and [x2, x4], and we'd sum up the results. (By using Simpson's rule on these intervals, I mean the rule for using it when there's just 3 values.) If you do this, you see that the 'odd' x's (i.e. the x's with odd-numbered subscripts) get counted four times, and the ones with even subscripts get counted twice (and the end points of the entire interval get counted once).
What? I can only think of one possible circle which is tangent to all three lines and in the first quadrant...Find the equations of the two circles (by solving simultaneously or otherwise) in the first quadrant which are tangent to the y axis, the x axis, and the line x + y = 2.
(Sorry I didn't answer a question, last 15 or so comments were just banter and I couldn't see an unanswered one.)
There can be one inside the triangle formed by the line and the axes, and also one outside.What? I can only think of one possible circle which is tangent to all three lines and in the first quadrant...
Find the equations of the two circles (by solving simultaneously or otherwise) in the first quadrant which are tangent to the y axis, the x axis, and the line x + y = 2.
(Sorry I didn't answer a question, last 15 or so comments were just banter and I couldn't see an unanswered one.)
How does he come to this conclusion? .__.To find the point from M, move <- 6 units, ^ 3units. P(-3, 4)
One way to do this is by noticing what values of x give you the max and min value for y.Find the range of the function y= 2 [(square root) 25-x^2]