Can some body clearly explain the graphical meaning of differentiation and integration?

Graphical analysis of functions is an important technique in science and engineering. For example, consider the function F(x) = x 2 e ?x/5 , (13-26) which might describe a cause-and-effect relationship between two variables, x and F, in a physical system. A scientist encountering this function, e.g., in a theoretical calculation, would immediately sketch its graph. the pictorial representation is easier to comprehend than the symbolic formula. The graph shows that as x increases from 0 to 30, the variable F starts at 0, then rises until F reaches a maximum at x = 10 and then decreases more gradually to 0. The derivative dF/dx, or F 0 (x), is an important quantity in the analysis of a function F(x). It is the rate of change of the variable F with respect to a change in x. The derivative function has a simple graphical interpretation, which we studied in Sec. 4.2: The derivative F 0 (x) is the slope of the graph of F(x) at x. For example, looking at Fig. 13.2 for the function in (13-26) we can immediately see that the slope starts at 0 at x = 0, becomes positive as x increases, but returns to 0 at the maximum (x = 10); as x increases beyond this point the slope becomes negative but approaches 0 (from below) as x tends to ?. All of the detailed properties of F(x) and F 0 (x) can be seen from the graph. The slope of a graph is a “geometric” concept, because it involves the shape of the curve. The physical application of the function F(x) may be completely unrelated to geometry. But we introduce this geometric concept— the slope of the graph—as an abstract mathematical construction that will help us to analyze the function. The definite integral R b a F(x)dx is another important quantity in the analysis of a function. It is the amount of a quantity whose “density” (amount per unit of x) is F(x). This concept has many physical applications. F(x) help us to analyze the integral? The integral does have a simple graphical interpretation. R b a F(x)dx is a certain area in the graph of F(x): It is the area bounded above by the curve F(x), bounded below by the x axis, and bounded on the sides by the vertical lines x = a and x = b. For example, the integral of the specific function F(x) in (13-26), from x = 5 to x = 20. We’ll prove that R b a F(x)dx is equal to the shaded area presently. It is important to understand that the “area of the graph” does not generally refer to a real physical area. In most applications, R F(x)dx is not a physical area, or any other geometric quantity. It could be a mass, or kinetic energy, or an electric field, etc. The integral R b a g(x)dx is the area bounded above by g(x) and below by the x axis, between x = a and x = b, on a graph of g(x). read less