1. MAT 1221 Survey of Calculus Section 2.3 Rates of Change http://myhome.spu.edu/lauw

2. Expectations Use equal signs Show formula steps Show individual derivatives steps Double check the algebra

3. HW WebAssign HW 2.3 There is a hint on problem 1 at the end of your HO. Additional HW listed at the end of the handout (need to get done, but no need to turn in)

4. Fact: Slope of Tangent Line

5. What is “Rate of Change”? We are going to look at how to understand and how to find the “ rate of change ” in terms of functions. (The connection between derivatives, slope of tangent lines and the rates of change.)

6. Two Worlds and Two Problems ?

8. The Velocity Problem y = distance dropped (ft) t = time (s) Displacement Function (Positive Downward) Find the velocity of the ball at t=2.

9. The Velocity Problem Again, we are going to use a limiting process. Find the average velocity of the ball from t=2 to t=2+h by the formula

10. The Velocity Problem thAverage Velocity (ft/s) 2 to 31 2 to 2.10.1 2 to 2.010.01 2 to 2.0010.001

11. The Velocity Problem We “see” from the table that velocity of the ball at t=2 should be ____ft/s.

12. The Velocity Problem We “see” from the table that velocity of the ball at t=2 should be ____ft/s. The instantaneous velocity at t=2 is _____ ft/s. (The ball is traveling at____ ft/s 2 seconds after it dropped.)

13. Limit Notations When h is approaching 0, is approaching 64. We say as h  0, Or,

14. Definition For the displacement function, the instantaneous velocity at time t is if it exists.

15. Two Worlds and Two Problems

16. Remarks In the context of moving objects, the independent variable is time t. We use the following notations Distance function Velocity function Acceleration function : rate of change of the velocity function

17. Example 2 Given where s is in meters and t is in seconds, find (a) v(t) (b) a(t) (c) The velocity and acceleration at t=2s (d) The time when the velocity is 5m/s.

18. Remarks When units are given, you answers in (c) and (d) should have units. The wonderful design of the notations helps you to get the units easily.

19. Example 2 Suppose we model the amount of certain drug inside a patient’s body by mg after t hours of injection. (a) Find (b) Explain the meaning of the answer in (a)

20. Definition For y=f(t), the (instantaneous) rate of change at t is

21. Expectations Show the substitution step. Units are required for some of the answers. Use equal signs