CHAPTER 02.02: DIFFERENTIATION OF CONTINUOUS FUNCTIONS: Forward Divided Difference: Part 2 of 2
Let's go ahead and see that, if I change delta x, if I decrease delta x, what's going to happen to the true error?
So I drew a . . . I'm going to draw a table here, I'm going to show you what the values of delta x are, what the value of f prime of x is, and then I will put down what the true error is. So this is for the same function, f of x equal to 2 e to the power 1.5 x, and x is 3, that's where we are calculating this derivative of the function. So when delta x is 0.1, I just calculated what the value of the derivative of the function is, 291.35, and the true error is -21.30. If I choose 0.05, which is half of what I have now, I get 280.43, and the true error now in this case is -10.38. I choose 0.025, which is again half of what I have chosen previously, I get 275.18, and in this case the true error is turning out to be -5.127. So what I'm trying to show you here is that as you keep on making delta x to be smaller and smaller, the values which you are getting for the derivative of the function are getting closer and closer to the exact value, so let me write down the exact value right here, which is 270.05, and of course your true error is also decreasing as you go from a larger delta x to a smaller delta x.
One of the things which I want you to note here, so these are some of the questions which you do need to answer. One is why is the true error getting approximately halved as delta x is halved? What I mean by that is that, if you look at delta x is 0.1, you are making delta x here to be half of what it was before, but look at the true error, it's 21.30, and now it is approximately half of that. Then, when you are making this delta x to be half of this, which is 0.025, look at the true error, that's also getting approximately halved, it's not exactly halved, but approximately halved, so maybe you need to answer this question, why is the true error getting approximately halved as delta x is being halved? And keep in mind that this is not coincidental, this has happened in almost every case, that, as you keep on decreasing your delta x, halving your delta x, the true error is also going to get halved. The second one is that how, using relative approximate error can you get an accurate answer within a tolerance? And what I mean by here is that, if we don't have the privilege of knowing the exact value, the exact value is 270.05, but as we said that numerical methods are not used to calculate exact values, they are used to calculate approximate values. How can I use the concept of relative approximate error, coupled with the number of significant digits which we talked about? How can we calculate an accurate answer within a pre-specified tolerance? So somebody's telling me that I want to calculate my derivative of the function for this particular case that, at least I want two or three significant digits to be correct, how will I use the concept of relative approximate error to be able to do that? And these are concepts which we already covered in previous segments. So I would like you to answer this question and this question, right here. And that's the end of this segment.