net and total change sect. 6-b. remember rate of change = derivative f(b) – f(a) is the change in...

NET AND TOTAL CHANGESect. 6-B

Remember• Rate of change = derivative• f(b) – f(a) is the change in y from a to b

• This is the net change

Net Change

Net or Total Change = integral of rate of change

b

a

b

a

dxxfafbfafbfdxxf )(')()( that followsit )()()(' since

•If V(t) is the volume of water in a reservoir at time t, then its derivative V’(t) is the rate at which water flows into the reservoir. So the integral of V’(t) is the change in the amount of water in the reservoir between an initial time and final time

Examples

b

a

aVbVdxxV )()()('

•If C(t) is the concentration of a chemical reaction at time t, then the rate of the reaction is the derivative C’(t). Such that the integral of C’(t) is the change in concentration of C from an initial time and final time

Examples

b

a

aCbCdxxC )()()('

•If an object moves along a straight line with a position x(t), then its velocity is such that the integral of v(t) is the net change of position, or DIPLACEMENT of the particle during the same time period. Distance is the integral of

Examples

b

a

b

a

axbxtxdxtv )()()(')( nt displaceme

)(')( txtv

)(tv

1.) If

find )2(f

4)0( and 33)(' 2 fxxf

2.) If

find )4(f

3)2( and )('2

fxx

xxf

3.) The rate at which water is dripping into a tub of water

is given by (gal/hour). Find how much water

entered the tub from t = 1 to t = 3 hours?1

2)(

t

ttr

4) The birth rate for a population of animals is given by

and the death rate is given by

a) About how many total births occurred in the years t = 0 to t = 6?

b) What was the net change in the population from t = 0 to t = 10?

3

cos1545)(

ttd

45

6

cos30)(

t

tb

a) About how many total births occurred in the years t = 0 to t = 6?

dttb6

0

)( Birth Total 456

cos30)(

t

tb

b) What was the net change in the population from t = 0 to t = 10?

10

0

)()(changeNet tdtb

dttt

10

0 3

cos154545

6

cos30

The population is 62 members less after ten years, than when it began

5) A tank contains 30 gallons of water. Water is pumped into the tank at the rate of 8 gal/min. Water leaks out of the tank at a rate of gallons per minute for

minutes.

Figure 6.21

1t1200 t

a) How many gallons of water leak out of the tank from time t = 0 to t = 3 minutes?

3) A tank contains 30 gallons of water. Water is pumped into the tank at the rate of 8 gal/min. Water leaks out of the tank at a rate of gallons per minute for

minutes.

Figure 6.21

1t1200 t

b) How many gallons of water are in the tank at time t = 3 minutes?

5) A tank contains 30 gallons of water. Water is pumped into the tank at the rate of 8 gal/min. Water leaks out of the tank at a rate of gallons per minute for

minutes.

Figure 6.21

1t1200 t

c) Write an expression for A(t), the total amount (number of gallons) in the tank at time t.

5) A tank contains 30 gallons of water. Water is pumped into the tank at the rate of 8 gal/min. Water leaks out of the tank at a rate of gallons per minute for

minutes.

Figure 6.21

1t1200 t

d) At what time t is the amount of water in the tank a maximum?

Assignment

Workshete 6-B: Net Change