Calculus From First Principles

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  • + kappuraj kappuraj 2 weeks ago
    Clear and precise. Excellent. Look forward for some more presentation on Calculus.
  • + guest2c85f9 guest2c85f9 8 months ago
    pure and utter...............................boring
  • + guestfd8b0b guestfd8b0b 2 years ago
    hi, really love it!but, can i downoad it? i need it badly..please.....
  • + MHennessy MHennessy 2 years ago
    You switched on the light! for me Bless you for that.

    I have waded through several books till I got out of my depth and sank.

    You explained it in one lesson. You showed me the kernel of the nut I could not crack

    And: it was sweet. Thank You. Keep it up.
  • + guest477fc9 guest477fc9 2 years ago
    Wow, this is really, really well done. Were you paid to do this? Because it sure seems very professional, yet remaining concise, easy to understand and well presented. As a student doing 3 Unit Mathematics, I can say that this slideshow has been more useful than my own text book. Do you have any more slides like this? Thanks very much : )
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Calculus From First Principles - Presentation Transcript

  1. Calculus From First Principles
  2. Finding the Gradient of a Straight Line
    • We know how to find the gradient of a straight line.
    • Gradient = rise
    • run
    ( x 1 , y 1 ) ( x 2 , y 2 ) rise run
  3. Finding the Gradient of a Curve
    • We have a problem if we would like to find the gradient of a curve.
    • If we look at which way each part of the curve slopes, we can see that the gradient is not the same at each part of the curve.
  4. Negative, Zero and Positive Gradients
    • Recall the following:
    Negative gradients slope downhill Positive gradients slope uphill Horizontal lines have a gradient of 0
  5. Finding the Gradient of a Curve
    • Going back to y = x 2 , we can see that the gradient changes from negative, to zero, to positive.
    - - - - - - - + + + + + + + 0
  6. Finding the Gradient of a Curve
    • Since finding the gradient of a curve is difficult, we can consider the following:
    • The gradient of a point on the curve is the same as the gradient of the tangent to the curve at that point.
  7. Finding the Gradient of a Tangent
    • This leads us to the question of how to find the gradient of a tangent.
    • Whilst a tangent is a straight line, we can only find the coordinates of one point on the tangent – where the line touches the curve.
  8. Finding the Gradient of a Secant
    • To find the gradient of a tangent, we will consider a similar problem – finding the gradient of a secant.
    • A secant is a line that crosses the curve in two places.
    • Now we can find two points – and hence the gradient of the secant.
  9. Finding the Gradient of a Secant
    • Let’s call the first x coordinate x .
    • The y coordinate that corresponds is f(x) .
    • Let’s make the second point some distance, h , away.
    • The x coordinate of the second point is x + h .
    • The y coordinate that corresponds is f(x + h) .
    x x + h f (x) f (x + h)
  10. Finding the Gradient of a Secant
    • Gradient = rise
    • run
    x x + h f (x) f (x + h) rise run
  11. Finding the Gradient of a Secant
    • The gradient of the secant is NOT the same as the gradient of the curve, which remember is what we would like.
    • If we make the distance h smaller, the gradient will be closer to the gradient of the tangent.
  12. Finding the Gradient of a Secant
    • Make the distance h shorter.
    • The gradient of the secant is now closer to the gradient of the tangent.
    • Make the distance shorter again.
    • As h -> 0 , the secant becomes a tangent and the gradient of the secant becomes the gradient of the tangent.
    x x + h f (x) f (x + h) x + h 2 f (x + h 2 ) x + h 3 f (x + h 3 )
  13. Finding the Gradient of a Tangent
    • We can now find the gradient of the tangent by finding:
    • Remember if we can find the gradient of the tangent to the curve at a point, we can find the gradient of the curve at that point.
  14. Terminology and Notation
    • The process of finding the gradient of the tangent to a curve is differentiation .
    • The gradient function is called the derivative .
    These are all different notations for the derivative

+ pprosserpprosser, 3 years ago

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