1. Calculate the following limits:

2. Bytheendofthelessonyouwillbeableto: Understandthedefinitionofderivative. Findderivativesfromfirstprinciples. Differentiation Calculatethegradientofacurvewithoutdrawing thetangentline. Relatederivativesandslopesofcurves.

3. Introduction to Differentiation http://www.youtube.com/watch?v=QC5ITOflh3k&feature=related

4. Gradient of a line B A

5. Gradient of a line B A y2 y1 x1 x2

6. Gradient of a line B A y2 y1 x1 x2 The gradient of a line is constant.

7. Gradient of a curve Thegradientofacurveatapointistheslopeof thetangentlineatthatpoint. http://www.math.umn.edu/%7Egarrett/qy/TraceTangent.html Tangentlinetoacurve.ggb

8. Gradient of a curve The gradient of a curve varies at each point.

9. Weneedtofindamethodtocalculatetheslopeof thetangentlineatanypoint.

10. x Let'sconsiderapointPonthecurve. P f(x) CoordinatesofP:(x,f(x)) f(x) Secanttotangent.ggb

11. CoordinatesofQ:(x+h,f(x+h)) x Let'sconsideranotherpointonthecurve,Q. P f(x) Q x+h f(x)

12. Coordinates of Q :(x+h,f(x+h)) x Let's take another point on the curve , Q. P Q x+h f(x+h) f(x)

13. x ThelinePQissecanttothecurve.Ifwefindthe gradientofthisline,itisnotthetangentlinebutisa starter. P f(x)Q x+h

14. x P f(x)Q x+h f(x) f(x+h) m=

15. x P f(x)Q x+h f(x) f(x+h) Gradient of secant line PQ : We can rewrite this gradient in a different way : Gradient of secant PQ :

16. Q x+h Veamos que sucede si Q se acerca a P... P x Secanttotangent.ggb

17. slopeofthesecant= x P f(x)Q x+h IfQgetsclosertoP...

18. Thenweneedhtobeassmallaspossible. h 0When theslopeofthesecanttendsto betheslopeofthetangent. Thisisthedifferencequotient,thedefinitionof thederivative.

19. This iscalled thefirstderivativeoffunctionfwithrespecttox. Notation: f'(x)ory'or

20. Findthederivativeofthefunctionf(x)=x2 Pp R S Wewillfindthegradientofthetangentatany pointforthefunction f(x)=x2

21. f(x)=x2

22. Pp R S m =4 m =-6 We found for f(x)=x2 which is a new function that gives the value of the gradient of the curve at each point. Para its derivative:

23. Calculatethederivativeoff(x)=3x2 +1 To practice this topic on-line : http://archives.math.utk.edu/visual.calculus/2/definition.7/index.html

24. Theprocessoffindingthederivativeiscalled differentiation.

25. Book page 357 Ex 12B http://www.calculushelp.com/funstuff/phobe.html Attheendofthelesson:

26. http://www.calculushelp.com/funstuff/phobe.html To revise this lesson at home:

27. At the end of the lesson: http://animoto.com/play/hVIw4sOG1tEL3hYONxJyJQ

28. Attachments Tangentlinetoacurve.ggb Secanttotangent.ggb