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Description
This programme shows how the definition of scale factor together with the idea of a limit of a sequence of values can be used to define the derivative at a point of a function.
Module code and title: MS283, An introduction to calculus MS283; 10 27-06-1979 1979 Rights owned or controlled by The Open University This material can be used in accordance with The Open University conditions of use. A link to the conditions can be found at the bottom of all OU Digital Archive web pages. 00:23:42 + Show more... John Jaworski Graham Read; Allan I.,1936-2013 Solomon BBC Open University Animation; Computer; Derivatives; Differentiation; Limits; Scaling Graham Read introduces the programme. Allan Solomon looks at the graph for the function f(x)1 -x + 10 and its derived function. He reviews the method for calculating the area under the curve by rectangular approximation and does so for the curve of the derived function. Solomon then plots a curve for the values of the area at several intervals by means of the area-so-far function. He compares this with the curve for the original function. Graham Read demonstrates that while it is fairly easy to work out areas under a curve for simple cases, it soon becomes very much more difficult for functions such as f(x) = cos x. Allan Solomon uses a graphics board to demonstrate that the area under a curve can be defined by either lower or upper rectangles. An animated diagram shows how the area under a curve between two points can be approximated to arrive at the area-so-far function. Allan Solomon, voice over an animated diagram, explains how the area under a curve between two points can be calculated without the addition of lower or upper rectangles. The method involves finding two values for the area-so-far function and subtracting them. Another animation demonstrates that the starting point of the area-so-far function has no effect on the calculations. Graham Read, with the aid of an animated diagram demonstrates that the slope of a chord between two points, a and b, on the area-so-far function is also the height of the corresponding rectangle representing the area under the graph of f between a and b. In the limit, the derivative of the area-so-far function at x is f(x). Hence, an area-so-far function for f is simply a primitive for f. The result is the Fundamental Theorem of Calculus. Graham Read goes on to solve a problem by using the Fundamental theorem rather than the limit of a sum process. He then uses an animated diagram to show that sine is an area-so-far function. Finally Graham Read shows that an area-so-far function for f is simply a primitive for f and that to work out integrals one needs to look for the primitive of the function. 6HT/73041 00525_4319 454 no