total change theorem. the integral of a rate change is the total change

Total Change Theorem

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Total Change Theorem

The integral of a rate change is the total change

'( ) ( ) ( )b

af x dx F b F a

Page 3: Total Change Theorem. The integral of a rate change is the total change

If V’(t) is the rate that water flows into a lake at time t, then

shows the total volume of water that flows into the lake between t1 and t2

'( )t

tV t dt

Page 4: Total Change Theorem. The integral of a rate change is the total change

Example

If the rate of a reaction (concentration change/sec) is given by C’(t), then what does the integral below represent?

'( )t

tC t dt

Page 5: Total Change Theorem. The integral of a rate change is the total change

A particle moves along a line so that its velocity at time t is v(t) = t2 – t – 6. What is the displacement of the particle between 1 and 4 seconds?

Page 6: Total Change Theorem. The integral of a rate change is the total change

The rate of disbursement of a 2 million dollar federal grant is proportional to the square of 100 –t where t is measured in days.

A) Find the total disbursed between 20 and 30 days

B) Find the amount that remains to be disbursed after 50 days.

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