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IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
IB Maths .Basic differentiation
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IB Maths .Basic differentiation

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  • 1. Calculate the following limits:
  • 2. By the end of the lesson you will be able to: • Understand the definition of derivative. • Find derivatives from first principles. Differentiation • Calculate the gradient of a curve without drawing  the tangent line. • Relate derivatives and slopes of curves.
  • 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 The gradient of a curve at a point  is the slope of  the tangent line at that point. http://www.math.umn.edu/%7Egarrett/qy/TraceTangent.html Tangent line to a curve.ggb
  • 8. Gradient of a curve The gradient of a curve varies at each point.
  • 9. We need to find a method to calculate the slope of  the tangent line at any point.
  • 10. x Let's consider a point P on the curve. P f(x) Coordinates of P :      ( x ,  f(x) ) f(x) Secant to tangent.ggb
  • 11. Coordinates of Q :      ( x+h ,  f(x+h) ) x Let's consider another point on the curve, 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 The line PQ is  secant  to the curve. If we find the  gradient of this line , it is not the tangent line but is a  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 Secant to tangent.ggb
  • 17. slope of the secant = x P f(x)Q x + h If Q gets closer to P... 
  • 18. Then we need h to be as small as possible. h 0When the slope of the secant tends to be the slope of the tangent. This is the difference quotient , the definition of  the derivative.
  • 19. This  is called the  first derivative of function f with respect to x. Notation: f '(x)    or   y'     or 
  • 20. Find the derivative of the function      f(x) = x2 Pp R S We will find the gradient of the tangent at any  point for the function  f(x) = x2
  • 21. f(x)  = x 2
  • 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. Calculate the derivative of  f(x) = 3 x2  +1 To practice this topic on-line : http://archives.math.utk.edu/visual.calculus/2/definition.7/index.html
  • 24. The process of finding the derivative is called  differentiation.
  • 25. Book page 357 Ex 12B http://www.calculus­help.com/funstuff/phobe.html At the end of the lesson:
  • 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 Tangent line to a curve.ggb Secant to tangent.ggb

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