1. L ESSON 57 – R ATES OF C HANGE IN P HYSICS October 17, 2013 Fernando Morales, Human Being

2. S ECOND D ERIVATIVE Differentiation of the first derivative of the function Notations: y ’’ f ’’( x ).

3. E XAMPLE #1 Determine f ’’(3) f (x) = (3 x 2 + 2) -1/2 (1- x ) Step 1: Find f ’( x ) Step 2: Differentiate f ’( x ) to get f ’’( x ) Step 3: Replace x with 3 to get f ’’( 3 )

4. D ISPLACEMENT, V ELOCITY, AND A CCELERATION Think of a position function/graph (displacement vs. time), s(t) Derivative = Instantaneous rate of change = The gradient of a tangent line Gradient: Rise/Run

5. D ISPLACEMENT, V ELOCITY, AND A CCELERATION 1 st Derivative : Rise/Run -> Displacement/time-> units of m/s -> Velocity, s’(t) = v(t) 2 nd Derivative : Rise/Run -> Velocity/time -> units of m/s 2 -> Acceleration s’’(t) = a(t)

6. D ISPLACEMENT, V ELOCITY, AND A CCELERATION

12. V ELOCITY & A CCELERATION

13. Velocit y Accelerati on Slope of GraphMotion ++ Positive & Increasing Speeding Up & Forward +- Positive & Decreasing Slowing Down & Forward -+ Negative & Increasing Slowing Down & Reverse -- Negative & Decreasing Speeding Up & Reverse 000 Stationary

14. F REEDOM Q UESTION

15. R EQUIRED B EFORE N EXT C LASS McGraw Hill Section 2.3 # 7, 8, 9, 10, 16, 17