Question

20 10. For the position function S make a table of average velocities and make a conjecture about the 1+1 instantaneous velocity at time t = 0.

Answer 1

= 20 Given Rosition functia SIA) = o *+1 We are interested in the instantanens velocity at t=0, so we compute the average velocity over smaller and smaller time intervals [ot] using the temula of average relocity Vay = S(+)-500) = S(A)-SCO) #-o in [ot] ) Time interval Average velocity [0,1] -10 unitisée Ce 0.5] -13.3333 unitsee [0,0.1) 48.1818 Griflsea [0, 0.00] Cult/sec [0, 0.00 -19.986019 Guitlsea [o 0.0001] -19, gg8000 unit/sec average velocity * 2 19.8019 for. [0, 1] - average velocity Vav = S(I) - SCO) I P:30110 To 20 9V S(O)-20 2-10 unitisec. I ter 19,05] Vav e S10.5) -S(O) = 13133 -20 = -13.33 0.5 0.5., 0:1 for [0,0.1) SC012-510 = 18.18-20 -181818 Nav 0:1 fer (0,0.01) - Vav = $10,012 -S(O) = 19.8019-20 = -19.8019 0.01 ft to d. e) - Jav : S(6.be) See) = -19.48e1998 dua le 0,01 0.001

Dete Pege. for Co, ooool - Vav = S(0,0001)-510) 0,0001 = -19.998000 see the Notice that these average velocities are also Slopes of secant lines, several of which are shown in The table Because this time interval is so short, the average velocity gives instantaneous. Velocity at tao hadi approximation to we see that as to the average ve focifies appear to approach =20 uut lisse. In fact, we could make the average velocity to unit lsea as we like by taking it sufficiently close to o. Therefore -20 uit /see is a reasonable estimate of the Trostantinesis vel t=o as close velocity at