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Net and Total Change

Net and Total Change. Sect. 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. Examples.

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Net and Total Change

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  1. Net and Total Change Sect. 6-B

  2. 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 ofrate of change

  3. Examples • 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

  4. Examples • 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

  5. Examples • 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

  6. 1.) If find

  7. 2.) If find

  8. 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?

  9. 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?

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

  11. b) What was the net change in the population from t = 0 to t = 10? The population is 62 members less after ten years, than when it began

  12. 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. a) How many gallons of water leak out of the tank from time t = 0 to t = 3 minutes? Figure 6.21

  13. 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. b) How many gallons of water are in the tank at time t = 3 minutes? Figure 6.21

  14. 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. c) Write an expression for A(t), the total amount (number of gallons) in the tank at time t. Figure 6.21

  15. 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. d) At what time t is the amount of water in the tank a maximum? Figure 6.21

  16. Assignment Workshete 6-B: Net Change

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