1.6

Optimization, Related Rates and

Exponential Growth or Decay

REVIEW

OPTIMIZATION

: process of solving for a

maximum or a minimum,

in general, “efficient” values

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.

Projectile

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

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

Solution:

d

dR22

2

cosg

v

2

2

d

Rd42

2

sing

v

4

2

2

d

Rd

g

v24 0

(at a fixed

initial velocity).

Solution:

Hence, the range of the projectile

attains a maximum if the angle of

the projectile is 4

Blood pressure

Example 2. Blood pressure monitoring

P : blood pressure

t : time

Problem: maximum and minimum

pressure and the time

these values occur

tcosP 22590

Solution:

dt

dP2225 tsin

0dt

dP 0250 tsin

02 tsin

3202 ,,,t

tcosP 22590

420 twithin

2

3

20

,,,t

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

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

REVIEW

RELATED RATES

: how one variable changes

through time depending on

how another variable varies

through time

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

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?

800 meters

x

x : height at time t : intercepted angle

at the dish of the shuttle to the pad

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

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

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

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?

200

fee

t

x : distance of the boat from the cliff

x

: angle of depression

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

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

REVIEW

Exponential growth or decay

: rate of growth (or decay) is

proportional to the present

population or the present

quantity.

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, .

Modeling

kydt

dy dtk

y

dy

dtky

dy

Ckt

Cktey

Cktey Ckt ee

kteBy

yln

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)

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

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.

Conclusion

With the assumption of an

exponential growth at a

rate of 2%, the population

doubles in every 35 years.

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

Solution (continued)

At t = 5730, y = B/2.

kteBy keBB 5730

2

ke573021

kelnln 573021

elnkln 573021

000120970.k

At t = 0, y = B.

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

END