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.