2. Chapter 16B.5 Graphing Derivatives

3. The derivative is the slope of the original function. The derivative is defined at the end points of a function on a closed interval.

4. Example 2: Find the derivative f(x) = x 2 + 1 Since f’(x) = 2x therefore f’(-2) = -4 f’(-1) = -2 f’(0) = 0 f’(1) = 2 f’(2) = 4 F(x) F’(x) What can you conclude about the relationship between the graph of f(x) and the sign of f’(x)? Note that f(x) is increasing from 0 to ∞ and decreasing from -∞ to 0

5. 3/8/2016 Calculus - Santowski 4 Example #2

6. 3/8/2016 5 Example #2 (1) f(x) has a min at x = 2 and the derivative has an x-int at x = 2 (2) f(x) decr on (-∞,2) and the derivative has neg values on (-∞,2) (3) f(x) incr on (2,+∞) and the derivative has pos values on (2,+∞) (4) f(x) changes from decr to incr at the min while the derivative values change from neg to pos

7. Graphing f ʹ from f 3.Use the graph of f (x) to graph the derivative f ʹ (x).

8. 3/8/2016 7 Example #3

9. 3/8/2016 8 Example #3 f(x) has a max. at x = -3.1 & f `(x) has an x-int at x = -3.1 f(x) has a min. at x = -0.2 and f `(x) has a root at –0.2

10. 3/8/2016 9 Example #3 f(x) incr (- , -3.1) & (-0.2,  ) and on the same intervals, f `(x) has positive values f(x) decr on (-3.1, -0.2) and on the same interval, f `(x) has negative values

11. 3/8/2016 10 Example #3 At max (x = -3.1), the f(x) changes from being an incr to decr  the derivative changes from pos to neg values At the min (x = -0.2), f(x) changes from decr to incr  the derivative changes from neg to pos