Rule of False?
Hello, I'm in 9th grade at a high school. Since the regular math material is too boring for me, my math teacher copied a page on the Rule of Falsity from my 11th grade textbook. He told me to try to understand it. I've tried it, but somehow I have the feeling it requires some prior knowledge, which I don't have (Newton's method, bisection method).
Can someone please try to explain it in such a way that with the knowledge from the 9th grade you have a chance of understanding it?
That would be very nice
Thanks
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Imagine you’re looking for a place on a line (the graph of a function), where the value is exactly zero, and you know two points on the line:
You know that the graph in between is about straight (so does not make big curves). Now you want to determine where the value is exactly zero. Regula Falsi helps you gradually approach the solution.
You can see what the graphic looks like in this animation:
This is the way:
1) Guess where the function could have a zero.
2) Select two points. One is located to the left of the assumed zero, the other is located to the right of the assumed zero. Both points must then have different signs at their y value.
2) Between these two points, you expect the sequencing equation (required pre-knowledge: Just determine from two points). There, where this secant snows the x axis (y=0), is the x value of the new point.
3) With this new x value you calculate the new point on the graph of the function that replaces the old one (depending on the sign the left or right).
4) Now you start with the procedure again at 1).
With each passage of this loop, you are always approaching the actual zero point of the function. You can then break the procedure when the two punks are so tightly squeezed to the left and right that you can get the accuracy of the approximation solution. For example, at the beginning, you can specify that you want to have the zero point, i.e. the intersection of the graph with the x axis, on 2 behind the comma. Once this place no longer changes, you can stop using the procedure.
Regula Falsi (“starting from the wrong”) is an iterative approximation method for the zero point search of a function that is difficult to calculate exactly. This method uses the 1st derivation at a point which is assumed to be near the zero point. Since, where the tangent itself (hopefully easier to calculate) has its zero position, the new x value is now closer to the zero point of the function. From here, the R.F. is used again until the zero point of the function has been determined in the desired accuracy.
This is the Newton method.
This requires differential calculation to calculate the derivation.
We only had that in the 11th grade. However, it is quite possible to show a pupil in the 9th grade how to do this at least for all-rational functions. It’s really not difficult.
It becomes more difficult to explain why this works. But that comes later 😉.
Yeah, you’re right.
For example, two values x1 and x2 can be searched, for which the associated function values have different signs.
Then (with continuous functions) between x1 and x2 is a zero point. The plug from (x1, y1) to (x2, y2) intersects the x axis at x3. The function value there is y3. Now one of the points (x1, y1) or (x2, y2) is replaced by (x3, y3), so that the y values have different signs. This can be repeated as often as desired in order to work towards the zero point.
If you are too lazy to calculate the intersection x3, you can also simply take x3 = (x1 + x2)/2. This then takes longer overall.