Aim: What is the relationship between the graph of a derivative function and the graph of the original function?

1. Aim: What is the relationship between the graph of a derivative function and the graph of the original function? Do Now: Sketch the derivative of the function given by the following graph:

2. f(x) f’(x) Sketching the Derivative derivative is positive when original is increasing. derivative is negative when original is decreasing. derivative is zero when original is at relative maximum/minimum. pay attention to steep parts.

3. Derivative Graphs Sketch the graph of this function’s derivative

4. Derivative Graphs Sketch the graph of this function’s derivative

5. Derivative Graphs of Position Equation Sketch the graph of this function’s derivative

6. Derivative Graphs Use the following info to graph f over [-5, 6] 1) f is made of closed line segments end to end 2) graph starts at point (-5, 1) 3) derivative of f is step function shown here

7. Derivative Graphs of Position Equation Graph shows distance s, velocity v, and acceleration a as function of time. B = position, A = velocity, C = acceleration Which graph is which?

8. Derivative Graphs of Position Equation Graph shows distance s, velocity v, and acceleration a as function of time. A = position, C = velocity, B = acceleration Which graph is which?

9. Derivative Graphs of Position Equation Graph shows distance s, velocity v, and acceleration a as function of time. C = position, B = velocity, A = acceleration Which graph is which?

10. Derivative Graphs of Position Equation Graph shows velocity v as function of time t. 2 < t < 3, 6 < t < 7 When is the body's acceleration equal to zero?

11. Derivative Graphs of Position Equation Graph shows distance s as function of time t. 1 < t < 2 When is the body standing still?

12. Derivative Graphs of Position Equation y = h’(x) On what interval(s) is the function h positive? • -2 < x < -.6 • (-2, -.6)  (1, 1.6) • (-, -2) • All x values • Impossible to tell

13. Derivative Graphs of Position Equation y = h(x) What are the zeros of g’(x)? • x = -2, -0.6 • x = -2, -0.6, 1, 1.6 • x = -1.6, 1.3 • x = -1.6, 0.1, 1.3 • Impossible to tell

14. Derivative Graphs of Position Equation This is the graph of y = h(x). What can you expect concerning the function h’(x)? • h’ has no zeros • h’ is undefined at x = 1 • h’ is always negative on its domain • h’ has no smallest possible value • all of the above

15. Derivative Graphs of Position Equation This is the graph of y = h(x). What can you expect concerning the function h’(x)? • h’ has no zeros • h’ is undefined at x = 1 • h’ is always negative on its domain • h’ has no smallest possible value • all of the above

16. Derivative Graphs of Position Equation no relative max or min – no zero slope from l – r: slope is (-) and gets more (-) as x approaches 0 where f is undefined at x > 0 slope is very (-) and gets less negative (-); slope is never positive