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7.1 Integral as Net Change

7.1 Integral as Net Change. Photo by Vickie Kelly, 2006. Greg Kelly, Hanford High School, Richland, Washington. Door. My chair. My desk. If I start at my chair and walk to my desk and then back to my chair, what is my displacement? Total distance traveled?. Displacement: 8+(-8)=0

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7.1 Integral as Net Change

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  1. 7.1 Integral as Net Change Photo by Vickie Kelly, 2006 Greg Kelly, Hanford High School, Richland, Washington

  2. Door My chair My desk If I start at my chair and walk to my desk and then back to my chair, what is my displacement? Total distance traveled? Displacement: 8+(-8)=0 Total distance: 8+8=16

  3. Door My chair My desk If I start at my chair and walk to my desk and then to the door, what is my displacement? Total distance traveled? Displacement: 8+(-10)=-2 Total distance: 8+10=18 What is the significance of a positive or negative displacement? Is “total distance” ever negative?

  4. Given that f(x) is a position function relating distance in terms of time. Speed is the absolute value of velocity.

  5. v (m/sec) 4 t (sec) 5 10 The velocity of a particle moving along the x-axis is given. Describe the motion of the particle. -4

  6. A honey bee makes several trips from the hive to a flower garden. The velocity graph is shown below. What is the total distance traveled by the bee? 700 feet 200ft 200ft 200ft 100ft

  7. What is the displacement of the bee? 100 feet towards the hive 200ft 200ft -200ft -100ft

  8. Displacement: Distance Traveled: velocity graph

  9. v (m/sec) 4 t (sec) 5 10 The velocity of a particle moving along the x-axis is given. -4 What is the displacement of the particle? What is the total distance traveled?

  10. To find the displacement (position shift) from the velocity function, we just integrate the function. The negative areas below the x-axis subtract from the total displacement. To find distance traveled we have to use absolute value. Find the roots of the velocity equation and integrate in pieces, just like when we found the area between a curve and the x-axis. (Take the absolute value of each integral.) Or you can use your calculator to integrate the absolute value of the velocity function.

  11. Example 1 Page 363 represents the velocity of a particle moving along the x-axis for Describe the motion of the particle. The particle has an initial velocity of 8 cm/sec to the left. It slows to a halt at about 1.25 sec, after which it moves to the right with increasing speed, reaching a velocity of 24.8 cm/sec at the end.

  12. Example 2 Suppose the initial condition is What is the particle’s position at Method 1. Solve initial value problem.

  13. Example 2 Suppose the initial condition is What is the particle’s position at Method 2. New position=initial position + displacement.

  14. v(t) is the velocity of a particle in m/sec along the x-axis. Determine when the particle is moving to the right, to the left and stopped. Find the particle’s displacement for the given time interval. Find the total distance traveled by the particle.

  15. Support all answers graphically.

  16. v(t) is the velocity of a particle in m/sec along the x-axis. Determine when the particle is moving to the right, to the left and stopped. Find the particle’s displacement for the given time interval. Find the total distance traveled by the particle.

  17. The rate of potato consumption for a particular country was: where t is the number of years since 1970 and C is in millions of bushels per year. Example 5: National Potato Consumption

  18. million bushels Example 5: National Potato Consumption From the beginning of 1972 to the end of 1973:

  19. Homework Page 371 #1-3,4,5-11odds, 12-21all

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