2Module C1 Module C2 AQA MEI/OCR Edexcel OCR "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"
3The Gradient of a Tangent We found the rule for differentiating by noticing a pattern in results found by measuring gradients of tangents.However, if we want to prove the rule or find a rule for some other functions we need a method based on algebra.This presentation shows you how this is done.The emphasis in this presentation is upon understanding ideas rather than doing calculations.
4Consider the tangent at the point A( 1, 1 ) on Tangent at AA(1,1)(2,4)As an approximation to the gradient of the tangent we can use the gradient of a chord from A to a point close to A.e.g. we can use the chord to the point ( 2, 4 ).( We are going to use several points, so we’ll call this point B1 ).
5We can see this gradient is larger than the gradient of the tangent. Consider the tangent at the point A( 1, 1 ) on(2,4)B1Chord AB1Tangent at AA(1,1)The gradient of the chord AB1 is given byWe can see this gradient is larger than the gradient of the tangent.
6The gradient of the chord AB2 is To get a better estimate we can take a point B2 that is closer to A( 1, 1 ), e.g.A(1,1)Tangent at AB1Chord AB2The gradient of the chord AB2 is
7We can get an even better estimate if we use the point . Tangent at AChord AB3We need to zoom in to the curve to see more clearly.
8We can get an even better estimate if we use the point . Tangent at AChord AB3The gradient of AB3 is
9Continuing in this way, moving B closer and closer to A( 1, 1 ), and collecting the results in a table, we getxy - 1x - 1PointGradient of ABAs B gets closer to A, the gradient approaches 2. This is the gradient of the tangent at A.
10As B gets closer to A, the gradient of the chord AB approaches the gradient of the tangent. We write that the gradient of the tangent at AThe gradient of the tangent at A is “ the limit of the gradient of the chord AB as B approaches A ”