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Integral as Net Change

Integral as Net Change. Section 7.1a. DO Now – p.370, #1 and 2. Find all values of x (if any) at which the function changes sign on the given interval. Sketch a number line graph of the interval, and indicate the sign of the function on each subinterval. 1. on. at.

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Integral as Net Change

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  1. Integral as Net Change Section 7.1a

  2. DO Now – p.370, #1 and 2 Find all values of x (if any) at which the function changes sign on the given interval. Sketch a number line graph of the interval, and indicate the sign of the function on each subinterval. 1. on at The function changes sign at each of these values. + – + –

  3. DO Now – p.370, #1 and 2 Find all values of x (if any) at which the function changes sign on the given interval. Sketch a number line graph of the interval, and indicate the sign of the function on each subinterval. 2. on at The function changes sign at each of these values. + – +

  4. Linear Motion Revisited A particle moves along a horizontal s-axis for , and the particle’s velocity is given by the following function. Describe the motion of the particle. Graph the velocity function in [0, 5] by [–10, 30] Zero: Initial Velocity: Final Velocity: The particle has an initial velocity of 8 cm/sec to the left. It slows to a halt at about 1.255 sec, after which it moves to the right (v > 0) with increasing speed, reaching a velocity of about 24.778 cm/sec at the end.

  5. Linear Motion Revisited Suppose the initial position of the particle from our last ex. is s(0) = 9. What is the particle’s position at (a) t = 1 sec? (b) t = 5 sec? Here we need displacement, the change in a particular body’s position… Displacement =

  6. Linear Motion Revisited Suppose the initial position of the particle from our last ex. is s(0) = 9. What is the particle’s position at (a) t = 1 sec? (b) t = 5 sec? Here we need displacement, the change in a particular body’s position… During the first second of motion, the particle moves 11/3 cm to the left. It starts at s(0) = 9, so its position at t = 1 is: New position = Initial position + displacement

  7. Linear Motion Revisited Suppose the initial position of the particle from our last ex. is s(0) = 9. What is the particle’s position at (a) t = 1 sec? (b) t = 5 sec? We model part (b) in a similar way: Displacement = The full motion has the net effect of displacing the particle 35 cm to the right of its starting point: Final position = Initial position + displacement

  8. Linear Motion Revisited We now know that our particle was at s(0) = 9 at the beginning of the motion and at s(5) = 44 at the end. Did the particle move directly between these two points? NO!!! It began its trip by moving to the left  so how far did the particle actually travel??? Total distance traveled: Let’s evaluate this one numerically:

  9. Linear Motion Revisited In general: Integrating velocity gives displacement (netarea between the velocity curve and the time axis). Integrating the absolute value of velocity gives total distance traveled (total area between the velocity curve and the time axis). Now let’s try some more practice problems!!!

  10. The function v(t) is the velocity in m/sec of a particle moving along the x-axis. Use analytic methods to do each of the following: (a) Determine when the particle is moving to the right, to left, and stopped. (b) Find the particle’s displacement for the given time interval. (c) Find the total distance traveled by the particle. (a) Right when velocity is positive: Left when velocity is negative: Stopped when velocity is zero:

  11. The function v(t) is the velocity in m/sec of a particle moving along the x-axis. Use analytic methods to do each of the following: (a) Determine when the particle is moving to the right, to left, and stopped. (b) Find the particle’s displacement for the given time interval. (c) Find the total distance traveled by the particle. (b) Displacement =

  12. The function v(t) is the velocity in m/sec of a particle moving along the x-axis. Use analytic methods to do each of the following: (a) Determine when the particle is moving to the right, to left, and stopped. (b) Find the particle’s displacement for the given time interval. (c) Find the total distance traveled by the particle. (c) Distance =

  13. The function v(t) is the velocity in m/sec of a particle moving along the x-axis. Use analytic methods to do each of the following: (a) Determine when the particle is moving to the right, to left, and stopped. (b) Find the particle’s displacement for the given time interval. (c) Find the total distance traveled by the particle. (a) Right when: Left when: Stopped when:

  14. The function v(t) is the velocity in m/sec of a particle moving along the x-axis. Use analytic methods to do each of the following: (a) Determine when the particle is moving to the right, to left, and stopped. (b) Find the particle’s displacement for the given time interval. (c) Find the total distance traveled by the particle. (b) Displacement =

  15. The function v(t) is the velocity in m/sec of a particle moving along the x-axis. Use analytic methods to do each of the following: (a) Determine when the particle is moving to the right, to left, and stopped. (b) Find the particle’s displacement for the given time interval. (c) Find the total distance traveled by the particle. (c) Distance =

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