3.4 Rates of Change Motion along a line What it means… Positive Negative Zero x(t) Position function Gives location at time t v(t) = x’(t) Velocity function Tells how position is changing a(t) = v’(t) = x’’(t) Object is on the positive side Position is increasing i.e. moving Right Velocity is Acceleration function increasing Tells how Velcoity is changing Object is on the Object is at the origin Negative side Position is decreasing i.e. moving left Velocity is decreasing Object stops moving Velocity is constant ON YOUR COVER PAGE OF YOUR NOTE PACKET… MOVING RIGHT MOVING LEFT Example1: Looking at a position function on our graphing calculator. A particle is moving along the x-axis. It’s x-coordinate at time t is given by 3 2 x ( t ) 4 t 16 t 15 t for t 0 Let’s look at the position of this object over various t values: . t 0 1 2 3 4 x y Look at the graph of this function. a) When is the object moving right? b) When is the object moving left? c) When does the change directions? d) Find the velocity function: e) Find the acceleration function: f) When does the acceleration equal zero? Example2: The position of a particle is given by the function x (t ) t 6t 9t 7 3 a) 2 Describe the motion of the object. b) what is the velocity function? c) What is the acceleration function? Example2: The position of a particle is given by the function x (t ) t 6t 9t 7 3 2 d) When does the velocity = 0? e) When does the acceleration = 0? f) What is the acceleration of a particle when its velocity = 0? Example3: The graph of the position function is given below for an object moving along the x-axis. a) Describe the motion of the object. b) When does the velocity = 0? c) When does the acceleration = 0? Vocab Word Speed Definition how fast object is going Always positive Formula Speed = |v(t)| Rate of Change of f(t) *Instantaneous Slope of tangent line f’(c) *Average Slope of secant line ARC(f) = Mean Value Theorem 𝑓 𝑏 −𝑓(𝑎) 𝑏−𝑎 More Vocab…. Initially At Rest At time 0 Not moving Initial velocity = v(0) Initial position = x(0) V=0 Total Distance Sum of all positive and negative distances Displacement Change in position from time a to time b ∆x = x(b) – x(a) Average velocity: Slope of secant line on the position function 𝑣= 𝑥 𝑏 −𝑥(𝑎) 𝑏−𝑎 𝑎= 𝑣 𝑏 −𝑣(𝑎) 𝑏−𝑎 Average Acceleration Slope of secant line on the velocity function INCREASING An object is “Speeding up” when the speed is ______________________. This will occur when a) Speed graph increasing Or b) a and v have the same signs DECREASING An object is “Slowing down” when the speed is ____________________. This will occur when a) Speed graph decreasing b) a and v have opposite signs Or Example4: Bugs Bunny has been captured by Yosemite Sam and forced to “walk the plank”. Instead of waiting for Yosemite Sam to finish cutting the board from underneath him, Bugs finally decides just to jump. Bugs’ height off the ground, h, is given by h(t) = -16t2 – 16t + 320, where x is measured in feet and t is measured in seconds. a) What is Bugs’ displacement from t = 1 to t = 2 seconds? b) When will Bugs hit the ground? c) What is Bugs’ velocity at impact? (What are the units of this value?) d) What is Bugs’ speed at impact? e) Find Bugs’ acceleration as a function of time. (What are the units of this value?) f) What is the average velocity over the first 3seconds. Example5: Suppose the graph below shows the velocity of a particle moving along the x-axis. Justify your responses. Example5: Suppose the graph below shows the velocity of a particle moving along the x-axis. Justify your responses. Example5: SPEED GRAPH Example5: ACCELERATION GRAPH Example6: The following table gives the positive velocity v of an object at various times t. t 0 1 3 5 8 12 v 4 10 8 17 20 15 a) Use the table to approximate the average acceleration over the interval 5 ≤ t ≤ 8. b) Use the table to approximate the acceleration at time 4.