Transcript EE005_fhs_lnt_005_Sep09

Derivatives: Part 4 (Application)
-Motion Along
a Line
-Velocity
-Acceleration
-Jerk
-Applications: Economics
and Genetics
Derivatives: Applications
Previously:
 To find gradient, m
 To find tangent line equations
 To enable easy sketching of graphs
(max/min points)
Today:
 Rate
of change along a line
Average and Instantaneous Rate of Change
Definitions
1. The average rate of change of a function f(x) with respect to x over the
interval from x0 to (x0 + h) is
Average rate of change =
f(x0  h)  f(x0 )
h
2. The (instantaneous) rate of change of f(x) with respect to x at x0 is the
f ( x0  h)  f ( x0)
derivative of f(x) = f ' ( x)  lim
provided the limit exists.
h
h0
dy
Instantaneous rate of change = f ‘(x0) =
dx
x  x0
Rate of Change: Example
How fast does the area change with respect to the diameter when the diameter is
10 m?

The area of a circle is related to its diameter by the equation A  D 2 ,
4
where A = area; D = diameter
A

4
D2
dA 
  2D
dD 4


2
D
When D=10m,
dA
m2
5
dD
m
Motion Along a Line
Suppose that object moving along a coordinate, we know its position s as function of time t
s  f (t )
t  t is
s  f (t  t )  f (t )
The displacement of the object over time interval
The average velocity of the object over that time
displacement
travel  time
s f (t  t )  f (t )

t
t
vavg 
Velocity (Instantaneous)
The instantaneous velocity is the derivative of the position function s = f(t) with
respect to time. At time t, the velocity is
ds
f (t  t )  f (t )
 lim
dt
t
t 0
Besides telling how fast an object is moving, its velocity tells the direction of
motion.
v(t ) 
Velocity is the functions derivative
s  f (t )
f ' (t )  velocity
Speed
Driving to point A from point B  60 km/h
But what if you are driving back? Will it show -60km/h?
Speedometer always shows speed – absolute value of velocity
Speed is the absolute value of velocity.
Speed  v(t ) 
ds
dt
Speed measures the rate of progress regardless of direction (unlike
velocity).
Acceleration
Acceleration commonly is used for an increase in
speed (the magnitude of velocity);
Acceleration  the derivative of velocity with respect
to time.
If a body’s position at time t is s = f(t), then the body’s
acceleration at time t is
2
dv d s
a(t ) 
 2
dt dt
How fast the velocity is changing with time
Jerk
Jerk is the rate of change of acceleration
Jerk  the derivative of acceleration with respect
to time.
If a body’s position at time t is s = f(t), then the body’s
acceleration at time t is
3
da d s
j (t ) 
 3
dt dt
Motion along a line: Summary
Velocity
ds
v(t ) 
dt
Acceleration
d 2s
a (t )  2
dt
Jerk
3
d s
j (t )  3
dt
Motion along a Line: Examples
1. Given that
s  4t  12t  10t  9
3
2
Find velocity, v, and speed, acceleration and
jerk as a function of time, t.
Motion along a Line: Examples
2. The position s in meters, of a particle
moving along a straight line is given by its
position function
s = 5t3 - 3t2 + t
where t is in seconds.
Find the velocity, acceleration and jerk of
the particle at t = 4 seconds.
Motion along a Line: Examples
3. The metric fall equation of the ball is
s = 4.9t2 , where s = m, t = sec.
(a) How many meters does the ball fall in
the first 2 seconds.
(b) What is the velocity, speed, acceleration
and jerk at t = 2 sec?
Motion along a Line: Examples
4. A dynamite blast blows a heavy rock
straight up with a launch velocity of 160
ft/sec. It reaches the height of
s = 160t-16t2, after t sec.
(a) How high does the rock go?
(b) What are the velocity and speed of the
rock when it is 256 ft above the ground on
the way up?
(c)What is the acceleration of the rock at
any time, t.
Application: Genetics
Gregor Meldel worked with peas and discovered
Medellian Law I and II and was henceforth known as
Father of Genetics. How he documented:-
y = 2p – p2
(p=frequency of smooth[dominant]; 1-p=frequency of
wrinkled[recessive])
Via graphical methods of y and y’, it is noted that a small
change in introducing dominant alleles into highly
recessive population will have a drastic effect
Application: Economics
Marginal costs and Marginal Revenues
Suppose it costs
c(x) = x3 - 6x2 + 15x (dollars)
To produce x radiators when 8 – 30 radiators are
produced and that
r(x) = x3 - 3x2 + 12x
Gives the dollar revenue from selling x radiators. Your
shop currently produces 10 radiators a day. About how
much extra will it cost to produce one more radiator a
day? What is your estimated increase in revenue for
selling 11 radiators a day?