3.4 Rates of Change

Motion along a line What it means…PositiveNegativeZero x(t) v(t) a(t) Position function Gives location at time t Object is on the positive side Object is on the Negative side Object is at the origin = x’(t) Velocity function Tells how position is changing Position is increasing i.e. moving Right Position is decreasing i.e. moving left Object stops moving = v’(t) = x’’(t) Acceleration function Tells how Velcoity is changing Velocity is increasing Velocity is decreasing 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 for. Let ’ s look at the position of this object over various t values: txy

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 a)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 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 DefinitionFormula Speed Rate of Change of f(t) *Instantaneous *Average Mean Value Theorem how fast object is going Always positive Speed = |v(t)| Slope of tangent line Slope of secant line f’(c)

More Vocab…. Initially At Rest Total Distance Displacement Average velocity: Average Acceleration At time 0 Not moving V = 0 Change in position from time a to time b ∆x = x(b) – x(a) Sum of all positive and negative distances Slope of secant line on the velocity function Slope of secant line on the position function Initial velocity = v(0) Initial position = x(0)

An object is “Speeding up” when the speed is ______________________. This will occur when a) Or b) An object is “Slowing down” when the speed is ____________________. This will occur when a) Or b) INCREASING DECREASING Speed graph increasing a and v have the same signs Speed graph decreasing a and v have opposite signs

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) = -16t 2 – 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: SPEED GRAPH

Example5: ACCELERATION GRAPH

t v Example6: The following table gives the positive velocity v of an object at various times t. 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.