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Mathematics LibreTexts

5.7E: Net Change Exercises

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5.7: Net Change Exercises

Use basic integration formulas to compute the following antiderivatives.

207) (x1x)dx

Answer:
(x1x)dx=x1/2dxx1/2dx=23x3/2+C12x1/2+C2=23x3/22x1/2+C

208) (e2x12ex/2)dx

209) dx2x

Answer:
dx2x=12ln|x|+C

210) x1x2dx

211) ​​​​​​​ π0(sinxcosx)dx

Answer:
π0sinxdxπ0cosxdx=cosx|π0(sinx)|π0=((1)+1)(00)=2

212) π/20(xsinx)dx

NET CHANGE

223) Suppose that a particle moves along a straight line with velocity v(t)=42t, where 0t2 (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=4tt2. The total distance is d(2)=4m.

224) Suppose that a particle moves along a straight line with velocity defined by v(t)=t23t18, where 0t6 (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)=|2t6|, where 0t6 (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(62t)dt=6tt2. For t>3,d(t)=d(3)+t3(2t6)dt=9+(t26t). The total distance is d(6)=9m.

226) Suppose that a particle moves along a straight line with acceleration defined by a(t)=t3, where 0t6 (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)=409.8t;h(t)=1.5+40t4.9t2m/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.


5.7E: Net Change Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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