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This programme uses the ideas of scaling and nests of intervals to demonstrate that the area, A(r), between the x-axis and the graph of x-1/x from x=1 to r, equals loge(r).
Metadata describing this Open University video programme
Module code and title: MS283, An introduction to calculus
Item code: MS283; 07
First transmission date: 06-11-1978
Published: 1978
Rights Statement: Rights owned or controlled by The Open University
Restrictions on use: 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.
Duration: 00:24:14
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Producer: Jack Koumi
Contributors: Mike Crampin; Allan I.,1936-2013 Solomon
Publisher: BBC Open University
Keyword(s): Nest of intervals; Scaling
Footage description: Allan Solomon introduces the programme. Using three cardboard pieces which have been shaped to fit under the graph of the function x-1/x he demonstrates, by weighing the pieces, that there is a connection between the addition of areas under a curve and the multiplication of their end points. Solomon, with the aid of still graphics and animation, formulates a general rule for the above cases and relates this to the rule for the logarithm of a product rs. Mike Crampin explains how the area under a curve can be calculated by dividing it into a number of squares and rectangles for which areas can be easily calculated. He uses still graphs and animations to illustrate his points. By taking more and more rectangles, Crampin shows that the area under the curve becomes trapped inside smaller and smaller intervals. Taking the area under the curve calculated by Mike Crampin above, Allan Solomon relates this figure to a table of logarithms to the base e (Naperian Logarithms). He shows that A of r is precisely log of r to the base e. Mike Crampin goes on to provide evidence that the logarithmic rule provides for the addition of areas. He manipulates graphs and areas and uses animations to demonstrate that by scaling it can be shown that the areas under x-1/x obey the logarithmic addition rule A(r)+A(s)=A(rs). Allan Solomon sums up the programme. He points out, particularly, the importance of scaling and nests of intervals for showing that the area under the curve, where x maps to 1/x from 1 to r is in fact the logarithm of r.
Master spool number: 6HT/73038
Production number: 00525_4326
Videofinder number: 483
Available to public: no