applications for unit 1
DESCRIPTION
unitTRANSCRIPT
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1.6
Optimization, Related Rates and
Exponential Growth or Decay
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REVIEW
OPTIMIZATION
: process of solving for a
maximum or a minimum,
in general, “efficient” values
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REVIEW
Problem: value/s of x which OPTIMIZE
xfy
1. Determine critical points of f. (value/s of x where ) 0x'f
2. If there are several critical points,
compare function values at the
critical points.
3. If possible, use second-derivative
test on the critical points.
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Projectile
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Example 1. Range of a projectile
g
sinvR
22
R feet : range of the projectile
v ft/sec : initial velocity
g ft/sec2 : a constant
rad : angle of the projectile
Problem: value of that maximizes
the range of the projectile
20
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Solution:
g
sinvR
22
d
dR 2
2
sinDg
v
d
dR22
2
cosg
v
0d
dR 0222
cosg
v
02 cos2
0
22
4
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Solution:
d
dR22
2
cosg
v
2
2
d
Rd42
2
sing
v
4
2
2
d
Rd
g
v24 0
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(at a fixed
initial velocity).
Solution:
Hence, the range of the projectile
attains a maximum if the angle of
the projectile is 4
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Blood pressure
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Example 2. Blood pressure monitoring
P : blood pressure
t : time
Problem: maximum and minimum
pressure and the time
these values occur
tcosP 22590
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Solution:
dt
dP2225 tsin
0dt
dP 0250 tsin
02 tsin
3202 ,,,t
tcosP 22590
420 twithin
2
3
20
,,,t
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Solution (continued)
2
3
20
,,,ttcosP 22590
1150 tP
652tP
115tP
6523 tP
Within , 20 t
P attains a maximum at
,t 0
P attains a minimum at
2
3
2
,t
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Solution (continued)
In general, using the model
P attains a maximum at
is an integer, and k,kt
P attains a minimum at
is an odd integer. k,tk2
tcosP 22590
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REVIEW
RELATED RATES
: how one variable changes
through time depending on
how another variable varies
through time
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REVIEW
Assume x and y are functions of
time t such that . xfy
Problem: solve for given ,
or vice-versa. dt
dy
dt
dx
HOW: differentiate both sides of
with respect to t xfy
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Example 3. Radar tracking
After blast-off, a space shuttle
climbs vertically and a radar-
tracking dish, located 800 m from
the launch pad, follows the shuttle.
How fast is the radar dish revolving
10 sec after blast-off if the velocity
at that time is 100 m/sec and the
shuttle is 500 m above the ground?
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800 meters
x
x : height at time t : intercepted angle
at the dish of the shuttle to the pad
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Solution:
Solve for when x = 500 and . dt
d100
dt
dx
800
xtan
800
xDtanD tt
dt
dx
dt
dsec
800
12
dt
dx
secdt
d
2
1
800
1
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Solution (continued)
dt
dxcos
dt
d
800
2
At , the shuttle is
meters away from the dish.
(by Pythagorean theorem)
500x 89100
89
8cos
89
642 cos
100dt
dx
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Solution (continued)
dt
dxcos
dt
d
800
2
where and . 89
642 cos 100dt
dx
898870. radian per second
100800
1
89
64
dt
d
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Example 4. Speed tracking
A woman standing on top of a
vertical cliff is 200 feet above a sea.
As she watches, the angle of
depression of a motorboat (moving
directly away from the foot of the
cliff) is decreasing at a rate of 0.08
rad/sec. How fast is the motorboat
departing from the cliff when the
angle of depression is /4 rad?
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200
fee
t
x : distance of the boat from the cliff
x
: angle of depression
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Solution:
Solve for when =/4 and . dt
dx080.
dt
d
xtan
200
xDtanD tt
200
dt
dx
xdt
dsec
22 200
dt
dxsec
dt
dx
200
22
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Solution (continued)
At =/4, x
tan200
4 200x
dt
dxsec
dt
dx
200
22
080.dt
d
2
4
2
sec
32dt
dxfeet per second
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REVIEW
Exponential growth or decay
: rate of growth (or decay) is
proportional to the present
population or the present
quantity.
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Modeling
Suppose an organism (or an element)
grows (or decays) such that rate of
growth is proportional to the present
quantity (or population).
y : quantity at time t
dt
dy: rate of growth (or decay)
kydt
dyHence, .
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Modeling
kydt
dy dtk
y
dy
dtky
dy
Ckt
Cktey
Cktey Ckt ee
kteBy
yln
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Modeling
kteBy
Exponential model of growth or decay
where B is the quantity at t=0
k is the rate of change
(per one unit change of t)
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Example 5. Population growth
Suppose that the world population
grows exponentially at a rate of 2%
in a year. In how many years will
the world population double?
Solution:
kteBy
where k = 0.02 and t is in years
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Solution (continued)
t.eBy 020 At t = 0, y = B.
At what t will y = 2B ?
t.eBB 0202 t.e 0202
t.elnln 0202
elnt.ln 0202
020
2
.
lnt 35734.
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Conclusion
With the assumption of an
exponential growth at a
rate of 2%, the population
doubles in every 35 years.
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Example 6. Radioactive decay
Carbon-14 is radioactive and
decays exponentially. It takes 5730
years for a given amount of C-14 to
decay one-half its original size.
Construct a function which shows
the amount of C-14 after t years.
Solution:
kteBy
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Solution (continued)
At t = 5730, y = B/2.
kteBy keBB 5730
2
ke573021
kelnln 573021
elnkln 573021
000120970.k
At t = 0, y = B.
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Conclusion
With the assumption of an
exponential decay and a
half-life of 5730 years,
models the amount of C-14
at time t.
t.eBy 000120970
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END