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.
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