Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

3.2 Derivative as a Function

522 views

Published on

Published in: Technology, Education
  • Be the first to comment

  • Be the first to like this

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

×