Derivatives Lesson Oct 15

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Derivatives Lesson Oct 15

  1. 1. LIMITS
  2. 2. We say that the limit of f(x), as x approaches a written lim f (x) = L x      a if we can take outputs f(x) as close to L as we wish by taking  inputs x sufficiently close to a but not equal to a.
  3. 3. x2 ­ 9 Graph the function f(x) =  x ­ 3
  4. 4. Now lets estimate  lim f(x)  x      3 On graphing calculator press 2nd [Tblset]
  5. 5. Then press 2nd [TABLE]
  6. 6. Graph f(x) =  sinx x
  7. 7. estimate  lim f(x) x      0
  8. 8. Graph f(x) =  x x
  9. 9. x lim x lim = ­1 x      0+ x = 1 x      0­ x Right ­ hand limit:  lim f(x) = L means that f(x) can be  x      a + made as close to L as we wish by taking x sufficiently close  to a but greater than a Left ­ hand limit: lim f(x) = L means that f(x) can be  x      a­ made as close to L as we wish by taking x sufficiently close  to a but less than a
  10. 10. lim sinx = lim sinx + = lim sinx = 1 x      0 x x      0 x ­ x      0 x lim x lim = x      0 x lim x therefore,  x      0 does not exist x      0+ x ­ x x
  11. 11. Rules for calculation limits 1. Sum Rule:  lim [ f(x) + g(x) ] = lim f(x) + lim g(x) = L + M x      a x      a x      a 2. Difference Rule: lim [ f(x) ­ g(x) ] = lim f(x) ­ lim g(x) x      a x      a x      a = L ­ M x      a . . 3. Product Rule: lim [ f(x)  g(x) ] = lim f(x) lim g(x) = LM x      a x      a
  12. 12. 4. Constant Multiple Rule: lim k f(x) = kL x      a f(x) lim f(x) lim x      a L = if M      0 5. Quotient Rule: x      a = = g(x) lim g(x) M x      a
  13. 13. Find lim x2 x      2 3x + 4 lim x2 lim x2 x      2 = x      2 3x + 4 lim (3x + 4) x      2 lim x lim x = x      2 x      2 3 lim x + lim 4 x      2 x      2 2 5
  14. 14. 2 Find  lim x  ­ 5x + 6 x      2 x ­ 2
  15. 15. Questions 1 ­ 19 odd only Exercise 2.7

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