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3.1 Accumulated Changes

3.1 Accumulated Changes. Example 1 : An objects travels with a velocity of 15 mph. What is the distance traveled after 4 hours. The velocity is constant and the distance is: S = v.t = (4 ) (15) = 60 miles. v. Distance = area = (4)(15) =60. 15. t. 1. 2. 3. 4.

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3.1 Accumulated Changes

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  1. 3.1 Accumulated Changes Example 1: An objects travels with a velocity of 15 mph. What is the distance traveled after 4 hours The velocity is constant and the distance is: S = v.t = (4 ) (15) = 60 miles v Distance = area = (4)(15) =60 15 t 1 2 3 4

  2. Example 2: An objects travels for 4 hours with the following velocities: First hour : 15 mph, Second hour: 20 mph , Last 2 hours: 40 mph What is the distance traveled First hour : (15 mph)(1 hour) = 15 miles Total = 115 miles Second hour: (20 mph)(1 hour) = 20 miles Last 2 hours: (40 mph)(2 hours) = 80 miles v A=80 40 A=20 Distance = area = 15 + 20 + 80 = 115 miles 20 A=15 15 t 1 2 3 4

  3. Example 3: An objects travels with the following velocity: v = 15t + 10 Find the distance traveled after 4 hours 0 1 2 3 4 t Try t = 0,1,2,3,4 hours 10 25 40 55 70 v Total From Left: v (10)(1) + (40)(1) + (25)(1) + (55)(1) =130 70 70 55 55 40 40 25 25 10 t 1 2 3 4  = 1 hour

  4. Example 3 (Cont.): Using the same data and graph: Total From Left from previous slide: (10)(1) + (40)(1) + (25)(1) + (55)(1) =130 v 70 70 Total From Right: 55 55 (70)(1) + (40)(1) + (55)(1) + (25)(1) =190 40 40 25 Average: (130+190)/2 =160 miles 25 10 t 1 2 3 4  = 1 hour

  5. Note: • is the interval size n is the number of intervals v 70 70 70 55 55 55 40 40 40 25 25 25 10 t t 1 2 3 4 1 1.5 2 3 4 • = 1 hour n = 4 intervals • = 0.5 hour n = 8 intervals

  6. Example 4: An objects travels with the following velocity: v = 2t2 + 5 Find the distance traveled between t = 0 and t = 6 using n = 4 intervals 0 1.5 3 4.5 6 t • = 6 hours / 4 intervals =1.5 hour 23 77 5 9.5 45.5 v Total From Left: v 77 (5)(1.5) + (9.5)(1.5) + 45.5 (23)(1.5) + 23 (45.5)(1.5) + 9.5 =124.5 5 t 1.5 3 4.5 6  = 1.5 hour

  7. Example 4 (Cont.): Using the same data and graph: Total From Left from previous slide: =124.5 Total From Right: v 77 (77)(1.5) + (45.5)(1.5) + 45.5 (23)(1.5) + (9.5)(1.5) + 23 9.5 =232.5 5 t 1.5 3 4.5 6 Average: (124.5+232.5)/2 =178.5 miles  = 1.5 hour

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