SCRIBE EPISODE 4:
Return of the Jedi




                     Yoda
SCRIBE EPISODE 4:
Return of the Jedi

        So, Some people (those who are not in our class, or
      weren't there yest...
SCRIBE EPISODE 4:
Return of the Jedi
      Well, Mr. K. then shoed us a couple pictures (man I wish the
       slides were...
SCRIBE EPISODE 4:
Return of the Jedi
     So we moved on to a quick review of composite functions
      and the chain rule...
SCRIBE EPISODE 4:
Return of the Jedi
         Then we were asked to identify THIS function:
SCRIBE EPISODE 4:
Return of the Jedi

              It is, of course, a SEMI CIRCLE
SCRIBE EPISODE 4:
Return of the Jedi

          Now what's the equation of the full circle???
SCRIBE EPISODE 4:
Return of the Jedi
SCRIBE EPISODE 4:
Return of the Jedi
SCRIBE EPISODE 4:
Return of the Jedi




              And that can be drawn as...
SCRIBE EPISODE 4:
Return of the Jedi



                     2




                              2
               -2




 ...
SCRIBE EPISODE 4:
Return of the Jedi

        However, a circle is not a function. But, is there a
     function (besides ...
SCRIBE EPISODE 4:
Return of the Jedi
            Sure there is: There's this one...

                           2




    ...
SCRIBE EPISODE 4:
Return of the Jedi
                     And this one...


                           2




             ...
SCRIBE EPISODE 4:
Return of the Jedi
                     And this one...

                          2




               ...
SCRIBE EPISODE 4:
Return of the Jedi

        Do you understand that there are an
     infinite amount of functions with x...
SCRIBE EPISODE 4:
Return of the Jedi
        Now think of a relation such as the circle given
    previously, and think ho...
SCRIBE EPISODE 4:
Return of the Jedi

      The only way to find the derivative of a
     circle at any given point is by ...
SCRIBE EPISODE 4:
Return of the Jedi

        First of all, let's use the following
               equation of a circle:
SCRIBE EPISODE 4:
Return of the Jedi


           Now let's find the derivative:
SCRIBE EPISODE 4:
Return of the Jedi



             First rewrite it as such:
SCRIBE EPISODE 4:
Return of the Jedi


        Now we can recognize that the derivative
                  of 25 is always 0.
SCRIBE EPISODE 4:
Return of the Jedi


         Next, use the chain rule on the left side of
                       the eq...
SCRIBE EPISODE 4:
Return of the Jedi


       Identify ƒ(x), g(x), and their derivatives:

                               ...
SCRIBE EPISODE 4:
Return of the Jedi
       Did everyone see the red text on the
       previous slide? It is very importa...
SCRIBE EPISODE 4:
Return of the Jedi

        Anyway, now using the outer and inner
         functions, complete the chain...
SCRIBE EPISODE 4:
Return of the Jedi


      Now combine it with the derivative of x2
     (2x) and you have the derivativ...
SCRIBE EPISODE 4:
Return of the Jedi


                     therefore...
SCRIBE EPISODE 4:
Return of the Jedi


             Now solve for quot; y' quot;...
SCRIBE EPISODE 4:
Return of the Jedi
   So now we know that the derivative (slope
    of the tangent line) at any given po...
SCRIBE EPISODE 4:
Return of the Jedi
                     So let's test it out.

         Let's say we need to find the de...
SCRIBE EPISODE 4:
Return of the Jedi
           First, find the output of when x=3:
SCRIBE EPISODE 4:
Return of the Jedi
        The next step is simple. To find the derivative
         at that point, plug ...
SCRIBE EPISODE 4:
Return of the Jedi
         The derivative (slope of the tangent) at x = 3
              is (-3)/4... Le...
SCRIBE EPISODE 4:
Return of the Jedi
         So yes, the derivative (slope of the tangent) at
          any point of any ...
SCRIBE EPISODE 4:
Return of the Jedi

        From there, Mr. K. gave us some
          examples to try and find the
     ...
SCRIBE EPISODE 4:
Return of the Jedi

          Then one final point was made:

            When the numerator of the
    ...
SCRIBE EPISODE 4:
Return of the Jedi
SCRIBE EPISODE 4:
Return of the Jedi




     About it, that is...

  For reading, thank you...

     Grey-M, the next
   ...
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Jedi (implicit)

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Scribe #4 implicit functions

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Jedi (implicit)

  1. 1. SCRIBE EPISODE 4: Return of the Jedi Yoda
  2. 2. SCRIBE EPISODE 4: Return of the Jedi So, Some people (those who are not in our class, or weren't there yesterday) reading this Scribe post may be a little confused by the title. The title is as such because Mr. K. actually started off the class with a couple quot;Jedi mind tricksquot;!!! Isn't that awesome?!?! I won't go into detail, but let's just say: THERE ARE NO GREY ELEPHANTS IN DENMARK!!! And so I begin the Calculus Related part of this Scribe post...
  3. 3. SCRIBE EPISODE 4: Return of the Jedi Well, Mr. K. then shoed us a couple pictures (man I wish the slides were up) with questions pertaining the existence of somethings that were not shown in the pictures, but were eluded to by certain hints in these pictures. i.e. You see a shadow of person on the ground, but have not turned around to see this person yet. Is there someone behind you??? When we gave an answer he asked us why? We said because it is implied. There aren't any visuals of the objects in question, but certain evidence allows us to imply that they exist. This leads to the topic of the day: IMPLICIT FUNCTIONS!!!
  4. 4. SCRIBE EPISODE 4: Return of the Jedi So we moved on to a quick review of composite functions and the chain rule, because we would have to use it later on in the lesson. In one example the answer was:
  5. 5. SCRIBE EPISODE 4: Return of the Jedi Then we were asked to identify THIS function:
  6. 6. SCRIBE EPISODE 4: Return of the Jedi It is, of course, a SEMI CIRCLE
  7. 7. SCRIBE EPISODE 4: Return of the Jedi Now what's the equation of the full circle???
  8. 8. SCRIBE EPISODE 4: Return of the Jedi
  9. 9. SCRIBE EPISODE 4: Return of the Jedi
  10. 10. SCRIBE EPISODE 4: Return of the Jedi And that can be drawn as...
  11. 11. SCRIBE EPISODE 4: Return of the Jedi 2 2 -2 -2
  12. 12. SCRIBE EPISODE 4: Return of the Jedi However, a circle is not a function. But, is there a function (besides √4-x2 and its opposite: -√4-x2) that is a part of the circle???
  13. 13. SCRIBE EPISODE 4: Return of the Jedi Sure there is: There's this one... 2 2 -2 -2
  14. 14. SCRIBE EPISODE 4: Return of the Jedi And this one... 2 2 -2 -2
  15. 15. SCRIBE EPISODE 4: Return of the Jedi And this one... 2 2 -2 -2
  16. 16. SCRIBE EPISODE 4: Return of the Jedi Do you understand that there are an infinite amount of functions with x as the variable? All of these functions hidden in the main graph of a circle, are IMPLICIT FUNCTIONS!!!
  17. 17. SCRIBE EPISODE 4: Return of the Jedi Now think of a relation such as the circle given previously, and think how you would find the derivative of any given point in the domain that relation? There are infinite implicit functions, so who knows which one you should find the derivative of because there are many of these functions that go through the same points!
  18. 18. SCRIBE EPISODE 4: Return of the Jedi The only way to find the derivative of a circle at any given point is by finding the derivative of the relation using the following derivation:
  19. 19. SCRIBE EPISODE 4: Return of the Jedi First of all, let's use the following equation of a circle:
  20. 20. SCRIBE EPISODE 4: Return of the Jedi Now let's find the derivative:
  21. 21. SCRIBE EPISODE 4: Return of the Jedi First rewrite it as such:
  22. 22. SCRIBE EPISODE 4: Return of the Jedi Now we can recognize that the derivative of 25 is always 0.
  23. 23. SCRIBE EPISODE 4: Return of the Jedi Next, use the chain rule on the left side of the equation... remember it is:
  24. 24. SCRIBE EPISODE 4: Return of the Jedi Identify ƒ(x), g(x), and their derivatives: The reason why the inner function quot;g(x)quot; is simply quot;yquot; is because the inner function of y^2 is an implicit function. We do not know exactly what it is, but we know it is there.
  25. 25. SCRIBE EPISODE 4: Return of the Jedi Did everyone see the red text on the previous slide? It is very important! The reason why the inner function quot;g(x)quot; is simply quot;yquot; is because the inner function of y^2 is an implicit function. We do not know exactly what it is, but we know it is there.
  26. 26. SCRIBE EPISODE 4: Return of the Jedi Anyway, now using the outer and inner functions, complete the chain rule:
  27. 27. SCRIBE EPISODE 4: Return of the Jedi Now combine it with the derivative of x2 (2x) and you have the derivative of the left side of the equation...
  28. 28. SCRIBE EPISODE 4: Return of the Jedi therefore...
  29. 29. SCRIBE EPISODE 4: Return of the Jedi Now solve for quot; y' quot;...
  30. 30. SCRIBE EPISODE 4: Return of the Jedi So now we know that the derivative (slope of the tangent line) at any given point on the circle x2 + y2 = 25 is given by the following formula
  31. 31. SCRIBE EPISODE 4: Return of the Jedi So let's test it out. Let's say we need to find the derivative of the previous function at x=3.
  32. 32. SCRIBE EPISODE 4: Return of the Jedi First, find the output of when x=3:
  33. 33. SCRIBE EPISODE 4: Return of the Jedi The next step is simple. To find the derivative at that point, plug the coordinates into the equation of the derivative of the circle:
  34. 34. SCRIBE EPISODE 4: Return of the Jedi The derivative (slope of the tangent) at x = 3 is (-3)/4... Let's see if that is true.
  35. 35. SCRIBE EPISODE 4: Return of the Jedi So yes, the derivative (slope of the tangent) at any point of any circle with its center at the origin (because the derivative of the constant will always be zero) is given by:
  36. 36. SCRIBE EPISODE 4: Return of the Jedi From there, Mr. K. gave us some examples to try and find the derivative functions of relations (which could have been seen on the slides if they were up).
  37. 37. SCRIBE EPISODE 4: Return of the Jedi Then one final point was made: When the numerator of the derivative function of the relation is 0, the tangent is a horizontal line. When the denominator is 0, then the tangent is a vertical line. (as seen on the next slide)
  38. 38. SCRIBE EPISODE 4: Return of the Jedi
  39. 39. SCRIBE EPISODE 4: Return of the Jedi About it, that is... For reading, thank you... Grey-M, the next scribe is... Yoda

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