chapter 2 51 - SECTION 2.7 TANGENTS VELOCITIES AND OTHER...

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Unformatted text preview: SECTION 2.7 TANGENTS, VELOCITIES, AND OTHER RATES OF CHANGE El 115 13. (a) Using (1). (w3—4m+1)—(a3—4a+1) . (ms—a3)—4(x—a) m: lim ——————— = 1m “— x—va m—a z—m 13—0' 2 =lim (m~a)(m2+ax+a)—4(m—a) A lim (x2+am+a2—4) 23a2~4 z—vu. {II—(l, (It—HI (b) At (1., —2): m = 3(1)2 — 4 2 ~15 so an equation of the tangent line is y — (—2) = —1(ac — 1) {I} y : —:z: — 1. At (2. 1): m = 3(2)2 — 4 = 8. so an equation of the tangent line is y — 1 = 8 (av ~ 2) 4:) y:8x—15. 14. (a) Using (1). 1 1 f — fl x/Ex/c—L : ‘ = 1' a — 1' m 3113111 x—a gel—13}; m~a ml—rgzi/am(:c~a)(\/E+\/§) 1' (1—0: lim _1 _1 1 or 1053/2 — 1m — ~ —— z—nz i/a$(m—a)(\/a+\/:c) z~>a Jamfi/a—lq/w) Va? (2\/5) 2113/2 2 (b) At (1.1):m: —%.so an equation ofthetangentlineisy— 1 : —%(ac— 1) (i) y=—%$+ % At (46):m:—1—16.soanequationofthetangentline isy— % 271—16(a:—4) 4:) yz—l—lfiw+%. (C) 2 l 15. (a) Since the slope of the tangent at t : 0 is 0, the car’s initial velocity was 0. O (b) The slope of the tangent is greater at C than at B, so the car was going faster at C'. (c) Near A. the tangent lines are becoming steeper as m increases, so the velocity was increasing, so the car was speeding up. Near B. the tangent lines are becoming less steep. so the car was slowing down. The steepest tangent near 0 is the one at C', so at C the car had just finished speeding up, and was about to start slowing (d) Between D and E. the slope of the tangent is 0, so the car did not move during that time. ...
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