5.7E: Net Change Exercises
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5.7: Net Change Exercises
Use basic integration formulas to compute the following antiderivatives.
207) ∫(√x−1√x)dx
- Answer:
- ∫(√x−1√x)dx=∫x1/2dx−∫x−1/2dx=23x3/2+C1−2x1/2+C2=23x3/2−2x1/2+C
208) ∫(e2x−12ex/2)dx
209) ∫dx2x
- Answer:
- ∫dx2x=12ln|x|+C
210) ∫x−1x2dx
211) ∫π0(sinx−cosx)dx
- Answer:
- ∫π0sinxdx−∫π0cosxdx=−cosx|π0−(sinx)|π0=(−(−1)+1)−(0−0)=2
212) ∫π/20(x−sinx)dx
NET CHANGE
223) Suppose that a particle moves along a straight line with velocity v(t)=4−2t, where 0≤t≤2 (in meters per second). Find the displacement at time t and the total distance traveled up to t=2.
- Answer:
- d(t)=∫t0v(s)ds=4t−t2. The total distance is d(2)=4m.
224) Suppose that a particle moves along a straight line with velocity defined by v(t)=t2−3t−18, where 0≤t≤6 (in meters per second). Find the displacement at time t and the total distance traveled up to t=6.
225) Suppose that a particle moves along a straight line with velocity defined by v(t)=|2t−6|, where 0≤t≤6 (in meters per second). Find the displacement at time t and the total distance traveled up to t=6.
- Answer:
- d(t)=∫t0v(s)ds. For t<3,d(t)=∫t0(6−2t)dt=6t−t2. For t>3,d(t)=d(3)+∫t3(2t−6)dt=9+(t2−6t). The total distance is d(6)=9m.
226) Suppose that a particle moves along a straight line with acceleration defined by a(t)=t−3, where 0≤t≤6 (in meters per second). Find the velocity and displacement at time t and the total distance traveled up to t=6 if v(0)=3 and d(0)=0.
227) A ball is thrown upward from a height of 1.5 m at an initial speed of 40 m/sec. Acceleration resulting from gravity is −9.8 m/sec2. Neglecting air resistance, solve for the velocity v(t) and the height h(t) of the ball t seconds after it is thrown and before it returns to the ground.
- Answer:
- v(t)=40−9.8t;h(t)=1.5+40t−4.9t2m/s
228) A ball is thrown upward from a height of 3 m at an initial speed of 60 m/sec. Acceleration resulting from gravity is −9.8m/sec2. Neglecting air resistance, solve for the velocity v(t) and the height h(t) of the ball t seconds after it is thrown and before it returns to the ground.