NET AND TOTAL CHANGESect. 6-4

Remember• Rate of change = derivative• F’(x) represents the rate of change of y = F(x) with respect

to x• F(b) – F(a) is the change in y from a to b

• This is the net change

Net Change

•Net or Total Change = ∫rate of change

b

a

aFbFdxxF )()()('

•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 the mass of a rod measured from the right end to a point x is M(x), then the linear density is given by such that the integral of p(x) is the mass of the segment of the rod that lies between points a and b

Examples

)(')( xMxp

b

a

b

a

aMbMdxxMxp )()()(')(

•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) 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

2) 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

456

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?

Net change b(t) d(t)0

10

30cos t

6

45 45 15cos

t

3

0

10

dt

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

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

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?

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?

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

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

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

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

Assignment

Worksheet: Net Change