3.4 Velocity, Speed, and Rates of Change
No excuses,Just do the work!
Calculus Date: 11/18/14 Obj: SWBAT apply the basic rules of derivatives
Bell Ringer: pg 125 #47, 46Homework Check and Requests: SM selected problems pg 39-41pg 124-125 #31-51 odds
IN Class: Start lecture on rate of change Section 3.4
https://www.youtube.com/watch?v=OnoNITE-CLcHomework: Pg 135 #1-4Hw: pg 136 #12-21 exclude direction changeAnnouncements: Quiz Friday DerivativesMon. Mandatory Tutoring Periodic functionsMath Team Thursdays
Winners never quitQuitters never win!!
If at first you don’t succeed,Try and try again!!
Distance =Displacement =
Change in position is measured in both
displacement distanceand
If you were to drive 10 miles east and then 4 miles west
10 miles east
4 miles west
The difference between the starting and ending positions of an object
The total amount of “ground covered” by an object or the total length of its path
14 miles6 miles
x
y
position
time
Distance =Displacement =
And the position graph would look like this:
10 miles east 4 miles west
14 miles6 miles
Consider a graph of position (distance from a starting point) vs. time.
time (hours)
position(miles)
Average velocity can be found by taking:
change in position
change in time
s
t
t
sA
B
ave
f t t f tsV
t t
The speedometer in your car does not measure average velocity, but instantaneous velocity.
0
limt
f t t f tdsV t
dt t
(The velocity at one moment in time.)
Velocity is the first derivative of position.
Mr. Murphy’s fall from the 196 foot platform canbe expressed with the equation:
216196)( tts …where t is in seconds and s is measured in feet. Given the above statement, the equation for his velocity is: ttv 32)( …which is expressed in what units?
feet/second
Find
Which is easy enough except…32 what?
)(tv 32)( tv
t (seconds)
v(t)
(fe
et/s
econ
d)
First, let’s look at the graph of velocity. Note that like the position graph, s(t), the y-axis represents velocity while the x axis represents time
Accelerationsec
sec/feet
2sec/32)( feetta feet/second2
Since the slope of a line is based on:x
y
…or in this case:
st
feet
sec
…we are now talking about:
vt
feet /sec
sec
Is an expression of…
which is a rate of change of velocity.
Acceleration
Acceleration is the derivative of velocity.
dva
dt
2
2
d s
dt
a(t) 32 feet /sec2 feet/second2
Remember: If you ever get lost, what can always save you?
UNITS!
Example: Free Fall Equation
21
2s g t
Speed is the absolute value of velocity.
GravitationalConstants:
2sec32
ftg
2sec8.9m
g
2sec980
cmg
2)32(2
1ts
216 ts
tdt
dsV 32
Wait! So what is the difference between speed and velocity?
Speed is the absolute value of velocity.
x
y
0v
If the object is moving upward…
…or to the right
0v
If the object is moving downward…
…or to the left
But speed does not indicate direction so speed will always be positive…
Wait! So what is the difference between speed and velocity?
Speed is the absolute value of velocity.
x
y
v 40 ft /sec
If an object is moving upward at a speed of 40 ft/sec…
v 40 ft /sec
If an object is moving downward at a speed of 40 ft/sec
But regardless of the direction, the speed will always be positive…
time
position
acc posvel pos &increasing
acc zerovel pos &constant
acc negvel pos &decreasing
velocityzero
acc negvel neg &decreasing acc zero
vel neg &constant
acc posvel neg &increasing
acc zero,velocity zero
It is important to understand the relationship between a position graph, velocity and acceleration:
WAIT! How can you tell that acceleration is positive? The slope (velocity) is tilting upward so the slope (velocity) is increasing
time
acc posvel pos &increasing
acc zerovel pos &constant
acc negvel pos &decreasing
velocityzero
acc negvel neg &decreasing acc zero
vel neg &constant
acc posvel neg &increasing
acc zero,velocity zero
Based on this graph, how can you use velocity and acceleration to determine when an object is speeding up or slowing down?
time
position
time
acc posvel pos &increasing
acc zerovel pos &constant
acc negvel pos &decreasing
velocityzero
acc negvel neg &decreasing acc zero
vel neg &constant
acc posvel neg &increasing
acc zero,velocity zero
0
0
a
v
0
0
a
v
0
0
a
v
0
0
a
vSpeeding
up
SlowingDown
time
position
v and a have the same sign
v and a have opposite signs
So the moral of the story is…
0
0
a
v
0
0
a
v0
0
a
v
0
0
a
vSpeeding
up
SlowingDown