3.4 Velocity, Speed, and Rates of Change Created by Greg Kelly, Hanford High School, Richland,...
-
Upload
anna-mcallister -
Category
Documents
-
view
213 -
download
0
Transcript of 3.4 Velocity, Speed, and Rates of Change Created by Greg Kelly, Hanford High School, Richland,...
![Page 1: 3.4 Velocity, Speed, and Rates of Change Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover,](https://reader035.fdocuments.us/reader035/viewer/2022062618/55145d3e550346414e8b570c/html5/thumbnails/1.jpg)
3.4 Velocity, Speed, and Rates of Change
Created by Greg Kelly, Hanford High School, Richland, WashingtonRevised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts
![Page 2: 3.4 Velocity, Speed, and Rates of Change Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover,](https://reader035.fdocuments.us/reader035/viewer/2022062618/55145d3e550346414e8b570c/html5/thumbnails/2.jpg)
Consider a graph of position vs. time.
time (hours)
distance from
starting place
(miles)
Average velocity can be found by comparing:
change in position
change in time
s
t
t
sA
B
2 1avg
2 1( ) ( )
f t f tfV
t t t
The speedometer in your car measures instantaneous velocity (but without direction information!)
0
limt
f t t f tdfv t
dt t
![Page 3: 3.4 Velocity, Speed, and Rates of Change Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover,](https://reader035.fdocuments.us/reader035/viewer/2022062618/55145d3e550346414e8b570c/html5/thumbnails/3.jpg)
Velocity, the change in position as time goes by, is the first derivative of position.
![Page 4: 3.4 Velocity, Speed, and Rates of Change Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover,](https://reader035.fdocuments.us/reader035/viewer/2022062618/55145d3e550346414e8b570c/html5/thumbnails/4.jpg)
Example:Free Fall Position Equation
21
2s g t
GravitationalConstants:
2
ft2
c3
seg
2
m.8
c9
seg
2
cm980
secg
Speed is the absolute value of velocity.
s = - 4.9 t2 m
velocity =
s = 2 m9.1
( 8)2
t
speed = ds
dtm/s9.8t
ds
dt4.9 2t m/s9.8t
![Page 5: 3.4 Velocity, Speed, and Rates of Change Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover,](https://reader035.fdocuments.us/reader035/viewer/2022062618/55145d3e550346414e8b570c/html5/thumbnails/5.jpg)
Acceleration is the derivative of velocity, and the second derivative of position.
dva
dt
2
2
d s
dt
example: 32ft
v ts
232ft
as
If distance is in:
velocity would be in:
acceleration would be in:
meters
meters
second
msecsec 2
mor
sec
![Page 6: 3.4 Velocity, Speed, and Rates of Change Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover,](https://reader035.fdocuments.us/reader035/viewer/2022062618/55145d3e550346414e8b570c/html5/thumbnails/6.jpg)
Jerk, the change in acceleration as time goes by, is the third derivative of position.
Snap, Crackle and Pop are the fourth, fifth and sixth derivatives of position. (Honestly!)
![Page 7: 3.4 Velocity, Speed, and Rates of Change Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover,](https://reader035.fdocuments.us/reader035/viewer/2022062618/55145d3e550346414e8b570c/html5/thumbnails/7.jpg)
time
positionp increases => p′ pos
It is important to understand the relationship between a position graph, velocity (p′) and acceleration (p′′):
p horizontal => p´ zero
p′ constant => p′′ zero
p decreases => p′ neg
p horizontal => p′ zero
p increases => p′ pos
p increases => p′ pos p decreases=> p′ neg
p decreases
=> p′ neg
p′ constant => p′′ zero
p′ increases => p′′ pos
p′ decreases => p′′ negp′ decreases => p′′ neg
p′ increases
=> p′′ pos
p′ constant => p′′ zero
p′ decreases => p′′ neg
![Page 8: 3.4 Velocity, Speed, and Rates of Change Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover,](https://reader035.fdocuments.us/reader035/viewer/2022062618/55145d3e550346414e8b570c/html5/thumbnails/8.jpg)
Rates of Change:
Average rate of change in f f x h f x
h
Instantaneous rate of change in f 0
limh
f x h f xf x
h
These definitions are true for any function
( and x does not have to represent time! )
![Page 9: 3.4 Velocity, Speed, and Rates of Change Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover,](https://reader035.fdocuments.us/reader035/viewer/2022062618/55145d3e550346414e8b570c/html5/thumbnails/9.jpg)
Example 1:
For a circle:
2A r
2dA dr
dr dr
2dA
rdr
Instantaneous rate of change in surface area as the radius changes.
For tree ring growth, if the change in area is constant, then dr must get smaller as r gets larger.
2 dA r dr
![Page 10: 3.4 Velocity, Speed, and Rates of Change Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover,](https://reader035.fdocuments.us/reader035/viewer/2022062618/55145d3e550346414e8b570c/html5/thumbnails/10.jpg)
From economics:
Marginal cost is the first derivative of the cost function, and represents the change in cost as the number of manufactured items changes.
The marginal cost is also thought of as the increase in cost for manufacturing one additional item.
![Page 11: 3.4 Velocity, Speed, and Rates of Change Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover,](https://reader035.fdocuments.us/reader035/viewer/2022062618/55145d3e550346414e8b570c/html5/thumbnails/11.jpg)
Marginal cost is a linear approximation of a curved function. For large values of x, it gives a good approximation of the cost of producing “the next item.”
![Page 12: 3.4 Velocity, Speed, and Rates of Change Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover,](https://reader035.fdocuments.us/reader035/viewer/2022062618/55145d3e550346414e8b570c/html5/thumbnails/12.jpg)
Example 13: Suppose it costs: 3 26 15c x x x x to produce x stoves.
If you are currently producing 10 stoves, the next stove will cost roughly:
23 10 20 01 1 1 15c 300 120 15
$195
marginal cost after the 10th stove
The actual cost is: 11 10C C 3 2 3 211 6 11 15 11 10 6 10 15 10
770 550 $220
Note that for small values of x, this is not a perfect approximation– it’s much better as x grows large!
then c′ (x) = 3x2 - 12x + 15