Key things to remember for approximation of roots:
- the root is always between two x values, one of which is a positive y value and one which has a negative y value (makes sense if you draw a picture)
- the best approximation for a root is the x value closest to ZERO that you have found so far. This makes sense too if you think about it because the root, which is an x-intercept exists when y = ZERO. (In this question, you should have evaluated f(3), f(4), f(3.5) and f(3.25). The f(x) value closest to ZERO out of these is your answer.
- When halving an interval more than once, you need to make the appropriate evaluations. So for example, in this question, you should have evalued
f(3) = - 1
f(4) 5
We don't need to state why it is between x = 3 and x = 4 here. The question already implies that it does
f(3.5) = 1.75 *
This is the first halving that you apply. You probably already know how to do this
Since f(3) < 0 and f(3.5) > 0, then the root must lie between x = 3 and x = 3.5 *
This goes back to the first dot point. When there is a negative and positive y value, the root lies between these
We have now narrowed down the possibilities of where the root is. We're only required to halve the interval twice, so we halve the interval between x = 3 and x = 3.5 because we know this is where the root is
f(3.25) = 0.3125
Since f(3) <0 and f(3.25) >0, then the root must lie between x = 3 and x = 3.25 *
We've narrowed it down once again
f(3) = - 1
f(3.25) = 0.3125
THEREFORE THE APPROXIMATION FOR THE ROOT IS 3.25 *
0.3125 IS CLOSER TO ZERO THAN - 1
So that's how you would do this question. Remember those key points at the top, and all the italicised stuff, keep that running through your mind.
The
BEST thing you could do to help yourself is to draw a diagram. I've attached one. The red dots show where the value of the function would be.
Hope this helps
