1. 15: The Gradient of the Tangent as a Limit © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

2. The Gradient of the Tangent as a Limit Module C1 AQA Edexcel OCR MEI/OCR Module C2 "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"

3. The Gradient of the Tangent as a Limit We found the rule for differentiating by noticing a pattern in results found by measuring gradients of tangents. The Gradient of a Tangent 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.

4. The Gradient of the Tangent as a Limit Tangent at A A(1,1) Consider the tangent at the point A ( 1, 1 ) on 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 ). 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. (2,4)

5. The Gradient of the Tangent as a Limit The gradient of the chord AB 1 is given by B1B1 Consider the tangent at the point A( 1, 1 ) on A(1,1) (2,4) Tangent at A We can see this gradient is larger than the gradient of the tangent. Chord AB 1

6. The Gradient of the Tangent as a Limit To get a better estimate we can take a point B 2 that is closer to A ( 1, 1 ), e.g. The gradient of the chord AB 2 is A(1,1) Tangent at A B1B1 Chord AB 2

7. The Gradient of the Tangent as a Limit We can get an even better estimate if we use the point. We need to zoom in to the curve to see more clearly. A(1,1) B1B1 Tangent at A Chord AB 3

8. The Gradient of the Tangent as a Limit The gradient of AB 3 is A(1,1) Tangent at A Chord AB 3 We can get an even better estimate if we use the point.

9. The Gradient of the Tangent as a Limit Continuing in this way, moving B closer and closer to A ( 1, 1 ), and collecting the results in a table, we get As B gets closer to A, the gradient approaches 2. This is the gradient of the tangent at A. Gradient of AB x y  1 x  1 Point

10. The Gradient of the Tangent as a Limit We write that the gradient of the tangent at A As B gets closer to A, the gradient of the chord AB approaches the gradient of the tangent. The gradient of the tangent at A is “ the limit of the gradient of the chord AB as B approaches A ”

11. The Gradient of the Tangent as a Limit We will generalize the result above to find a formula for the gradient at any point on a curve given by STUDENTS! If you are working through this on your own, ask your teacher if you need to do the next ( final ) section.

12. The Gradient of the Tangent as a Limit We need a general notation for the coordinates of B that suggests it is near to A. So, is the small difference in x as we move from A to B. is the Greek letter d so we can think of as standing for “difference”. Let A be the point ( x, ) on the curve We use can be used for the difference in y values.

13. The Gradient of the Tangent as a Limit We have and So, the gradient of the chord AB is m where So, since the gradient of the tangent  the gradient of the tangent  We can’t put x = 0 in this as we would get which is undefined.

14. The Gradient of the Tangent as a Limit the gradient of the tangent  But, the gradient of the tangent gives the gradient of the curve, so If we use for the difference in y -values, ( the letter h is sometimes used instead of ) the gradient of the tangent  We then get

15. The Gradient of the Tangent as a Limit Solution: The gradient of the curve is given by the gradient of the tangent, so So, as, e.g. Prove that the gradient function of the curve where is given by For, So, Since has cancelled, we will not be dividing by zero.

16. The Gradient of the Tangent as a Limit

17. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

18. The Gradient of the Tangent as a Limit The gradient of the chord AB 1 is given by B1B1 Consider the tangent at the point A( 1, 1 ) on A(1,1) (2,4) Tangent at A We can see this gradient is larger than the gradient of the tangent.

19. The Gradient of the Tangent as a Limit We write that the gradient of the tangent at A As B gets closer to A, the gradient of the chord AB approaches the gradient of the tangent. We will generalize the result above to find a formula for the gradient at any point on a curve given by The gradient of the tangent at A is “ the limit of the gradient of the chord AB as B approaches A ”

20. The Gradient of the Tangent as a Limit We need a general notation for the coordinates of B that suggests it is near to A. So, is the small difference in x as we move from A to B. is the Greek letter d so we can think of as standing for “difference”. Let A be the point ( x, ) on the curve We use can be used for the difference in y values.

21. The Gradient of the Tangent as a Limit We have and So, the gradient of the chord AB is m where So, since the gradient of the tangent  the gradient of the tangent  We can’t put x = 0 in this as we would get which is undefined.

22. The Gradient of the Tangent as a Limit the gradient of the tangent  But, the gradient of the tangent gives the gradient of the curve, so If we use for the difference in y -values, ( the letter h is sometimes used instead of ) the gradient of the tangent  We then get

23. The Gradient of the Tangent as a Limit Solution: The gradient of the curve is given by the gradient of the tangent, so So, as, e.g. Prove that the gradient function of the curve where is given by For, So, Since has cancelled, we will not be dividing by zero.