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AP Calculus
   Derivative f’ (a) for specific values of a
   Function f’ (x)
  Domain of f’ (x) is all values of x in domain off (x) for which the limit exists. F’ (x) is differentiable on (a, b) i...
   Prove that f (x) = x3 – 12x is differentiable.    Compute f ‘(x) and write the equation of the    tangent line at x = ...
   F ‘(x) = 3x2 – 12   Equation of tangent line at x = -3    y = 15x + 54
   Calculate the derivative of y = x-2. Find the    domain of y and y’
   Solution: y’ = -2x-3   Domain of y: {x| x ≠ 0}   Domain of y’ : {x| x ≠ 0}   The function is differentiable.
   Another notation for writing the derivative:   Read “dy dx”   For the last example y = x-2, the solution could    ha...
   For all exponents n,
   Calculate the derivative of the function below
   Solution:
Assume that f and g are differentiable functions. Sum Rule: the function f + g is differentiable                 (f + g)’...
   Find the points on the graph of    f(t) = t3 – 12t + 4 where the tangent line(s) is    horizontal.
   Solution:
   How is the graph of f(x) = x3 – 12x related to    the graph of f’(x) = 3x2 – 12 ?
f(x) = x3 – 12 x        Decreasing on (-2, 2)                                                     Increasing on (2, ∞) Inc...
   Differentiability Implies Continuity     If f is differentiable at x = c, then f is continuous at      x = c.
   Show that f(x) = |x| is continuous but not    differentiable at x = 0.
   The function is continuous at x = 0 because
   The one-sided limits are not equal:   The function is not differentiable at x = 0
   Local Linearity   f(x) = x3 – 12x
   g(x) = |x|
   Show that f(x) = x 1/3 is not differentiable at    x = 0.
The limit at x = 0 is infinitef’(0) =  The slope of the tangent line is  infinite – vertical tangent line
3.2 Derivative as a Function
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3.2 Derivative as a Function

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A PowerPoint presentation on the Derivative as a Function. Includes example problems on finding the derivative using the definition, Power Rule, examining graphs of f(x) and f'(x), and local linearity.

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Transcript of "3.2 Derivative as a Function"

  1. 1. AP Calculus
  2. 2.  Derivative f’ (a) for specific values of a
  3. 3.  Function f’ (x)
  4. 4.  Domain of f’ (x) is all values of x in domain off (x) for which the limit exists. F’ (x) is differentiable on (a, b) if f ‘(x) exists for all x in (a, b). If f’ (x) exists for all x, then f (x) is differentiable.
  5. 5.  Prove that f (x) = x3 – 12x is differentiable. Compute f ‘(x) and write the equation of the tangent line at x = -3.
  6. 6.  F ‘(x) = 3x2 – 12 Equation of tangent line at x = -3 y = 15x + 54
  7. 7.  Calculate the derivative of y = x-2. Find the domain of y and y’
  8. 8.  Solution: y’ = -2x-3 Domain of y: {x| x ≠ 0} Domain of y’ : {x| x ≠ 0} The function is differentiable.
  9. 9.  Another notation for writing the derivative: Read “dy dx” For the last example y = x-2, the solution could have been written this way:
  10. 10.  For all exponents n,
  11. 11.  Calculate the derivative of the function below
  12. 12.  Solution:
  13. 13. Assume that f and g are differentiable functions. Sum Rule: the function f + g is differentiable (f + g)’ = f’ + g’ Constant Multiple Rule: For any constant c, cf is differentiable and (cf)’ = cf’
  14. 14.  Find the points on the graph of f(t) = t3 – 12t + 4 where the tangent line(s) is horizontal.
  15. 15.  Solution:
  16. 16.  How is the graph of f(x) = x3 – 12x related to the graph of f’(x) = 3x2 – 12 ?
  17. 17. f(x) = x3 – 12 x Decreasing on (-2, 2) Increasing on (2, ∞) Increasing on (-∞, -2) What happens to f(x) at x = -2 f’(x) = 3x2 - 12 and x = 2??Graph of f’(x) positive f’(x) is negative f’(x) is positive on (2, ∞)on (-∞, -2) on (-2,2) Zeros at -2, 2
  18. 18.  Differentiability Implies Continuity  If f is differentiable at x = c, then f is continuous at x = c.
  19. 19.  Show that f(x) = |x| is continuous but not differentiable at x = 0.
  20. 20.  The function is continuous at x = 0 because
  21. 21.  The one-sided limits are not equal: The function is not differentiable at x = 0
  22. 22.  Local Linearity f(x) = x3 – 12x
  23. 23.  g(x) = |x|
  24. 24.  Show that f(x) = x 1/3 is not differentiable at x = 0.
  25. 25. The limit at x = 0 is infinitef’(0) = The slope of the tangent line is infinite – vertical tangent line
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