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MAT 1235 Calculus II

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  1. MAT 1235Calculus II 4.4 Part II Indefinite Integrals and the Net Change Theorem http://myhome.spu.edu/lauw

  2. Homework • WebAssign HW 4.4 II

  3. Preview • We will discuss the Net Change Theorem • The NCT explains the formulas for computing displacement and distance traveled you have learned in physics.

  4. Recall: Fundamental Theorem of Calculus Part 2

  5. Recall: Fundamental Theorem of Calculus Part 2

  6. Recall: Fundamental Theorem of Calculus Part 2

  7. The Net Change Theorem The integral of rate of change is the net change Final value –Initial Value = Net change of from to Rate of change of

  8. Example 1: Solution leaking from a tank

  9. Example 1 = volume of the solution in the tank at time t = rate of change of the volume

  10. Example 2: Linear motion of a particle Reference Frame

  11. Example 2:

  12. Formulas

  13. Recall: Section 2.7 • displacement ≠ distance traveled

  14. displacement ≠ distance traveled displacement = distance traveled =

  15. Observations distance traveled = displacement =

  16. Observations distance traveled = -displacement =

  17. Observations: A Unified Formula distance traveled = displacement = distance traveled = -displacement =

  18. Formulas

  19. Example 3 The velocity function (in meters per second) is given for a particle moving along a line. (a) Find the displacement of the particle. (b) Find the distance traveled. Step 1: Find the interval where v(t)>0 and <0

  20. Example 3 Step 2: Use the distance formula

  21. Expectations • Please pay attention to the solutions steps. • Do not skip steps. • Do not do multiple things at one step. • It is very helpful to have a diagram.