Application of differentiation and integration in graph analysis.
See, let’s say some quantity y is the function of x, that means y varies with x. There are two ways of representing the variation of y with x, one is we write the function, and the other way is to give the graph between y and x. If function is given then how do we discuss that, we have learnt that with the help of integral and differentiation. So now we will discuss that if graph is given between y and x, then how can differentiation and integral used, okay. If y is function of x, then we had defined two things, average rate of change of y with respect to x, and that was defined by delta y by delta x. And another thing we had defined was instantaneous rate of change of y with respect to x, and that was defined by dy by dx. See if I know the function of yx then for average rate I will take change in y, meaning delta y divide dx, and for instantaneous change dy by dx, that means differentiation of y with respect to x. if I have to get these two information through the graph then how do we do that using a graph, average rate of change and how will we get instantaneous rate of change.
See what the graph is explaining. What does this point tell us? That if I make a vertical line then what is the value of x coordinate. And if I make a horizontal line then what is the value of y coordinate. When x is equal to x1 the y is equal to y1. Similarly when x is sum x2 to y will be sum y2. So this point represents x2y2 and this point represents x1y1, is this clear. Now listen carefully, then what is that distance. It will be x2 minus x1 and it will be delta x. And what is that distance? Delta y or y2 minus y1. That means if I will connect these two points, let’s say I connect these two points with a line. So what will this line connecting the curve be called? Secant or chord. What is chord or secant, a straight line with intersects the curve at two different points. Can you remember we defined two different things, secant of a curve and tangent of a curve. What is tangent? Line which touches the curve at one unique point. And when a line intersects the curve at two different points that is called secant. Now secant is a straight line, yes or no? Now class, let me ask you something? Do you know what is the meaning of slope? What is this slope? Slope is defined as tan theta. What is it? If I make a straight line between y and x, then the angle made by straight line with positive x axis is known as the slope. And then if we make two points here only then you will remember that this y2 minus y1 upon x2 minus x1, and this can also be written as delta y by delta x. So for a straight line delta y by delta x represent tan theta which is equal to the slope of a straight line. So, if we consider two points A and B and if we make a straight line passing through those two points. So then will there be a slope here of this straight line? Yes or no? So slope of secant which is tan theta is equal to delta y by delta x. So, average rate of change of y with respect to x is basically the slope of secant which passes through two different points of the curve. If I give you a graph and I ask you delta y by delta x, please hold and write it later. Let’s say, listen carefully….