Warmup: YES calculator 1) 2)

Warmup Find k such that the line is tangent to the graph of the function

3.4: Velocity, Speed, and Rates of Change

f(x) = position function f ’(x) = velocity function f ”(x) = acceleration function Speed is the absolute value of velocity. Acceleration is the derivative of velocity.

example: Its position is in: Velocity would be in: Acceleration would be in: if t is in seconds, and f(t) is in feet Write the position, velocity, and acceleration functions with appropriate units

time distance acc pos vel pos (speeding up) acc zero vel pos (constant speed) acc neg vel pos (slowing down) velocity zero acc neg vel neg (speeding up) acc zero vel neg (constant speed) acc pos vel neg (slowing down) acc zero, velocity zero (not moving) It is important to understand the relationship between a position graph, velocity and acceleration: This is a POSITION GRAPH f(x)

Rectilinear motion (motion of an object along a straight line): Position is the location of an object and is given as a function of time. Conventional notation uses s(t). Displacement is the difference between the final position and the initial position… displacement = s( final time ) – s( initial time ). Total distance traveled… Sum of each distance between turns. (turns may occur when velocity =0)

Velocity info: Advancing (moving right)….. when velocity is positive. v>0 Retreating (moving left) … when velocity is negative. v<0 Acceleration Info: Accelerating… when acceleration is positive. a>0 Decelerating… when acceleration is negative. a<0 Both: Speeding up (going faster)… when velocity and acceleration have the same signs. (+)(+) or (-)(-) Slowing down (going slower) … when velocity and acceleration have opposite signs. (+)(-) or (-)(+)

Example 2: Find the displacement of the object over the interval [0 sec, 10sec]. Example 3: What is the total distance traveled? Example 4: Describe the motion of the object in terms of advancing (forward) and/or retreating (backwards).

Example 5: Describe when the object is accelerating and/or decelerating. Example 6: is the object speeding up or slowing down at 1 second? justify answer. v(1)= negative a(1)= negative Since both v(1)<0 and a(1)<0, the object is speeding up at t = 1 second

In problems assume an object is moving rectilinearly in time according to s(t) = 4t 2 – 6t + 1 meters over the time interval [0, 4] seconds. 14. Find the velocity, speed, and acceleration as functions of time and give the appropriate units of each. 15. What is the velocity at t = 1 sec? 16. What is the acceleration at t = 1 sec? 17. On what time interval(s) is the particle advancing (moving to the right) and retreating? Justify your answers. 18. What is the total distance traveled?

Ex. A particle is moving along a line with its position at time t given by Find: a)Find the velocity function b)Find v(0) and v(2) c)When is the velocity 0? where is the particle at that time? d)Is the particle speeding up or slowing down at t = 5 seconds. Justify your answer.

Use your calculator

The Average Speed = The Average Velocity = The Average acceleration =

e) What is Bugs speeding up or slowing down at 3 seconds? Justify your answer.

d) Is the shot speeding up or slowing down at 4 seconds? Justify your answer.

f) At what value or values of t does the particle change directions?

the end