1.  Differentiate the following: u (dv/dx) + v(du/dx)

4. Lesson 3.4

5.  Students will be able to  use derivatives to analyze straight line motion and solve other problems involving rates of change.

6.  The instantaneous rate of change of with respect to x at a is the derivative provided the limit exists.

7. a)Evaluate the rate of change of the area of a circle at r = 7. (Rate of Change = derivative)

8. b)Evaluate the rate of change of the area of a circle at r =12.

9. c)If r is measured in feet and A is measured in square feet, what units would be appropriate for the derivative? square feet per foot

10.  Write the volume of a cube as function of side length s  V = s 3  Find the instantaneous rate of change of the volume V with respect to side s.  V’ = 3s 2  Evaluate the rate of change of V at s = 1 and s = 5  V’ = 3(1) 2 = 3  V’ = 3(5) 2 =75

11.  Suppose that an object is moving a long a coordinate line so that we know its position on that is line is a function of time t: s = f(t)  The displacement of the object over the time interval from t to t + Δt is  Δs = f(t + Δt) – f(t)  And the average velocity of the object over that time interval is

12.  The number of gallons of water in a tank t minutes after the tank has started to drain is Q(t) = 200 ( 30 – t) 2  How fast is the water running out at the end of 10 minutes?  Since this at a specific point, you can use the derivative and substitute 10 for t at the end.  Q(t) = 200 ( 900 – 60t + t 2 ) = 180000 – 12000t + 200t 2  Q’(t) = -12,000 + 400t  Q’(10) = -12,000 + 400 (10)  = -8000 gallons/minute

13. Page 135: 2-4, Page 181: 1, 2, 4, 7 Page 181: 55-60

14.  What is the average rate at which the water flows out during the first 10 minutes?  You will need to use the slope formula between (10, Q(10)) and (0, Q(0))

15.  The accompanying figure shows the velocity v = f(t) of a particle moving on a coordinate line.  When does the particle move forward (when v(t) > 0)?backward? (v (t) <0)?  When is the particle’s acceleration positive? Negative? Zero?  When does the particle move at its greatest speed? (think max and min)  When does the particle stand still for more than an instant?

16. When does the particle move forward? [0, 1) ( 5, 7) Backward? (1, 5) When is the particle’s acceleration positive? (3, 6) Negative? [0, 2) (6, 7) Zero? (2, 3) ( 7, 9] When does the particle move at its greatest speed? At t = 0 and ( 2, 3) When does the particle stand still for more than an instant? (2, 3), (7, 9]

17.  Instantaneous Velocity  The instantaneous velocity is the derivative of the position function s =f(t) with respect to time. At time t, the velocity is:  Speed  The absolute value velocity

18.  The derivative of velocity with respect to time. If v(t) = ds/st, then the acceleration at time t is:  a(t) = dv/dt = d 2 s/dt 2  Free fall constants (earth)

19. A dynamite blast rock propels a heavy rock straight up with a launch velocity of 192 ft/sec (about 130.91 mph). It reaches a height of s = 192t – 16t 2 after t seconds. a)How high does the rock go? The maximum height of the rock is 576 ft.

20. b)What is the velocity and speed of the rock when it is 512 ft above the ground on the way up?

21. c)What is the velocity and speed of the rock when it is 512 ft above the ground on the way down?

22. d)What is the acceleration of the rock at any time t during its flight (after the blast)? The acceleration is always downward. When the rock is rising, it is slowing down; when it is falling, it is speeding up.

23. e)When does the rock hit the ground? The rock will hit the ground 12 seconds after is blasts off.

24. A water rocket blasts off from the ground straight up with an initial velocity of 32 m/sec. It reaches a height of s = 32t – 0.8t 2 meters after t seconds. a) How high does the rocket go? b) What is the velocity of the rocket when it is 240 meters above the ground? c) What is the acceleration of the rocket at any time t? d) When does the rocket hit the ground?

25.  Page 136-137: 11-16,19