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3.2 Derivative as a Function

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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