298 Appendix A Selected Answers

Transcript

1 A Selected Answers (/3)x +(1/3) y = x ( /3)x +(1/3) y = x+,, y = x+6, 6, y = x/+1/, 1/, y = 3/, y-intercept: 3/, no x-intercept y = ( /3)x,, yes y = 0, y = x+, y = x y = 75t (t in hours); 164 minutes y = (9/5)x+3, ( 40, 40) y = 0.15x x (a) y = { 0 0 x < 100 (x/10)0 100 x 1000 x < x y = { 0.15x 0 x x < x x < x (a) P = x+ (b) x = 0000P (/5)x (16/5) (a) x +y = 9 (b) (x 5) +(y 6) = 9 (c) (x+5) +(y +6) = (a) x =, y = 3, m = 3/, y = (3/)x 3, 13 (b) x =, y = 3, m = 3, y = 3x+, 10 (c) x =, y =, m = 1, y = x, (x+/7) +(y 41/7) = 1300/ {x x 3/} {x x } {x x 1 and x } {x x < 0} {x x R}, i.e., all x 97

2 98 Appendix A Selected Answers {x x 0} {x h r x h+r} {x x 1 or x < 0} {x /3 < x < 1/3} {x x 0 and x 1} {x x 0 and x 1} R {x x 3}, {x x 0} A = x(500 x), {x 0 x 50} V = r(50 πr ), {r 0 < r 50/π} A = πr +000/r, {r 0 < r < } , , , , /3, 4/7, 7/4, 3/ , , , 3(3+ x) x+ x x , , , 3+3 x+ x m , 5/, 0, 15, 5, , 4.1, 4.01, 4.001, 4+ t , 9.849, , t undefined / does not exist a (a) 8, (b) 6, (c) dne, (d), (e), (f) 8, (g) 7, (h) 6, (i) 3, (j) 3/, (k) 6, (l).4.1. x/ 169 x t.4.3. x+1/x.4.4. ax+b x.4.8. /(x+1) 3/ /(t+) y = 3x or 1.3 or x x x πx π (3/4)x / (9/7)x 6/ x +4x x 4 +6x+10/x x+5

3 Appendix A Selected Answers x +x x +6x x x/ 65 x y = 13x/ y = 4x 48 π t/5+5, 49/5 n ka k x k k= x 3 /16 3x/ x (x 3 5x+10)+x 3 (3x 5) (x +5x 3)(5x 4 8x +6x 7)+ (x+5)(x 5 6x 3 +3x 7x+1) 65 x x x x 65 x x x 0 65 x x f = 4(x 3), y = 4x x x 3 5x+10 x3 (3x 5) (x 3 5x+10) x+5 x 5 6x 3 +3x 7x+1 (x +5x 3)(5x 4 8x +6x 7) (x 5 6x 3 +3x 7x+1) 1 x 65 x + x 3/ (65 x ) 3/ x x y = 17x/4 41/ y = 11x/165/ y = 19/169 5x/ / x 3 9x +x x 4x+/ x (x +1) x 65 x x x x / 169 x (x 4) 5 x (x 4x+5)x/ 5 x x/ r x x 3 / 1+x x(5 x) 3/ x x+1 1 x + x +x+1 (1 x) / 5 x 5 x /x ( ) 1 69 / x x x x x+1/x x 3 x (1/x) 300x (100 x ) 5/ 1+3x 3(x+x 3 ) /3 ( ) 4x(x 4x 3 +4x / +1)+ 1+(x +1) (x +1) + 1+(x +1) (x+8) (4 x) x(x +5) x(6 x ) x (1 4x 3 ) /x (4x)(x x+3) /(x+1) (8x )/(4x x+1) x +5x

4 300 Appendix A Selected Answers x(x 4) 3 +6(3x +1)(x 4) /(x) x/(x +1) (x 6x+7)/(x 3) /(3x 4) x 4 +7x 3 +18x +18x (5 4x)/((x+1) (x 3) ) /((+3x) ) x 6 + 7x x x x +8x y = 3x/96 9/ y = 3 x/ y = 13x/ 3/ y = x y = x nπ π/, any integer n nπ ±π/6, any integer n ( + 6)/ (1+ 3)/(1 3) = t = π/ / / / sin( x)cos( x)/ x sinx x + xcosx cosx sin x (x+1)sinx (x +x)cosx sin x sinxcosx sin x cos x sin x sinxcos(cosx) tanx+xsec x xtanx csc x sec x(1+sinx) tanxcosx (1+sinx) cscxcotx x sin(3x )+46x 4 cos(3x ) cos(cos(6x)) sin(6x) secθtanθ (1+secθ) = sinθ (cosθ +1) t 4 cos(6t) 6t 5 sin(6t) t (sin(3t) + tcos(3t))/cos(t) + t 3 sin(3t)sin(t)/cos (t) nπ/, any integer n π/+nπ, any integer n y = 3x/+3/4 3π/ y = 8 3x+4 8 3π/ y = 3 3x/ 3π/ π/6+nπ, 5π/6+nπ, any integer n ln(3)x3 x e x cosx sinx e x e x cos(e x ) cos(x)e sinx

5 ( x sinx cosxlnx+ sinx ) x x e x +x 3 e x x ln() xln(3)(1/3) x e 4x (4x)/x (3x +3)/(x 3 +3x) tan(x) (1 ln(x ))/(x ln(x )) sec(x) x cos(x) (cos(x)/x sin(x)ln(x)) e x/y (x+y)/(x+y) (xy 3x y )/(xy 3y x ) sin(x) sin(y)/(cos(x) cos(y)) y/ x (ysec (x/y) y )/(xsec (x/y)+y ) (y cos(x+y))/(cos(x+y) x) y /x y = x± y = x/± ( 3, 3), ( 3, 3), ( 3, 3), ( 3, 3) y = 7x/ 3 8/ y = ( y 1/3 1 x+y 1/3 1 x 1 +x 1/3 1 y 1 )/x 1/ (y y 1 ) = (x3 1 +x 1 y 1 x 1 ) (y 3 1 +y 1x 1 +y 1) (x x 1 ) 1+x Appendix A Selected Answers 301 x 1 x 4 e x e x x cos(x 3 )/ 1 sin (x 3 ) (arcsinx) 1 x e x / 1 e x (1+lnx)x x ln5(1+x x )arctan(x x ) / / / / / / / /7

6 30 Appendix A Selected Answers / / / / / / does not exist y = 1 and y = min at x = 1/ min at x =, max at x = max at x =, min at x = min at x = ±1, max at x = min at x = none none min at x = 7π/1 + kπ, max at x = π/1+kπ, for integer k none local max at x = local min at x = local min at x = one min at x = 1/ 5... min at x =, max at x = max at x =, min at x = min at x = ±1, max at x = min at x = none none min at x = 7π/1 + kπ, max at x = π/1+kπ, for integer k none max at x = 0, min at x = ± min at x = 3/, neither at x = min at nπ, max at π/+nπ min at nπ, max at (n+1)π min at π/+nπ, max at 3π/+nπ min at x = 1/ min at x =, max at x = max at x =, min at x = min at x = ±1, max at x = min at x = none none

7 Appendix A Selected Answers min at x = 7π/1 + nπ, max at x = π/1+nπ, for integer n max at x = 63/ max at x = max at 5 /4, min at 5 / none max at, min at min at / none min at nπ max at nπ, min at π/+nπ max at π/+nπ, min at 3π/+nπ concave up everywhere concave up when x < 0, concave down when x > concave down when x < 3, concave up when x > concave up when x < / 3 or x > 1/ 3, concave down when / 3 < x < 1/ concave up when x < 0 or x > /3, concave down when 0 < x < / concave up when x < 0, concave down when x > concave up when x < or x > 1, concave down when < x < 0 or 0 < x < concave down on ((8n)π/4,(8n+ 3)π/4), concave up on ((8n + 3)π/4,(8n+7)π/4), for integer n concave down everywhere concave up on (,(1 497)/4) and (1+ 497)/4, ) concave up on (0, ) concave down on (nπ/3,(n + 1)π/3) concave up on (0, ) concave up on (,) and (0, ) concave down everywhere concave up everywhere concave up on (π/4+nπ, 3π/4+nπ) inflection points at nπ, ±arcsin( /3)+nπ up/incr: (3, ), up/decr: (, 0), (, 3), down/decr: (0, ) max at (,5), min at (0,1) P/4 P/ w = l = 5 /3, h = 5 /3, h/w = 1/ , h/s = w = l = 1/3 V 1/3, h = V 1/3 / /3, h/w = 1/ square feet l /8 square feet $ r h/r = h/r = r = 5 cm, h = 40/π cm, h/r = 8/π /π / Go direct from A to D (a), (b) 7/

8 304 Appendix A Selected Answers (a) a/6, (b) (a + b a ab+b )/ meters wide by 1.5 meters tall If k /π the ratio is ( kπ)/4; if k /π, the ratio is zero: the window should be semicircular with no rectangular part a/b w = r/ 3, h = r/ / 3 58% r = 5/(π) 1/3.7 cm, h = 5 5/3 /π 1/3 = 4r 10.8 cm h = 750 ( ) π 1/3, r = π 750 ( ) 750 1/6 π h/r = The ratio of the volume of the sphere to the volume of the cone is 1033/ / , so the cone occupies approximately 8.54% of the sphere P should be at distance c 3 a/( 3 a+ 3 b) from charge A / $ There is a critical point when sinθ 1 /v 1 = sinθ /v, and the second derivative is positive, so there is a minimum at the critical point /(16π) cm/s /(1000π) meters/second /4 m/s /5 m/s π mi/min ft/s /(3π) cm/s /0 ft/s / m/s /64 m/min π/7 m/s π/144 m/min π /36 ft 3 /s tip: 6 ft/s, length: 5/ ft/s tip: 0/11 m/s, length: 9/11 m/s / mph / km/hr m/s / km/hr m/s m/s / km/hr / / km/hr /49 m/s (a) x = acosθ asinθcot(θ+β) = asinβ/sin(θ +β), (c) ẋ 3.79 cm/s x 3 = or y = 65/16, dy = y = 11/10, dy = y = sin(π/50), dy = π/50

9 Appendix A Selected Answers dv = 8π/ c = 1/ c = x 3 /3+47x / 5x+k arctanx+k x 4 /4 lnx+k cos(x)/+k / x x x b a rectangles: 41/4 = 10.5, 8 rectangles: 183/16 = / (16/3)x 3/ 7... t 3 +t x /z lns (5x+1) 3 / (x 6) 3 / x 5/ / / x t t, t < ; t 4t+8, t / ln(10) e / /6/ x 3x x(x 4 3x ) e x xe x tan(x ) 7... xtan(x 4 ) It rises until t = 100/49, then falls. The position of the object at time t is s(t) = 4.9t + 0t + k. The net distance traveled is 45/, that is, it ends up 45/ meters below where it started. The total distance traveled is 605/98 meters π 0 sintdt = net: π, total: π/ / A = 18, B = 44/3, C = 10/ (1 t) 10 / x 5 /5+x 3 /3+x (x +1) 101 / (1 5t) /3 / (sin 4 x)/ (100 x ) 3/ / x 3 / sin(sinπt)/π /(cos x) = (1/)sec x ln cosx tan (x)/

10 306 Appendix A Selected Answers / cos(tanx) / / (7/8)(x 7) 8/ (3 7 +1)/ f(x) / x/ sin(x)/ cosx+(cos 3 x)/ x/8 (sinx)/4+(sin4x)/ (cos 5 x)/5 (cos 3 x)/ sinx (sin 3 x)/ x/8 (sin4x)/ (sin 3 x)/3 (sin 5 x)/ (cosx) 5/ / tanx cotx (sec 3 x)/3 secx ln cscx+cotx cscxcotx/ (1/)ln cscx + cotx x x / ln x+ x / x 9+4x / + (9/4)ln x+ 9+4x (1 x ) 3/ / arcsin(x)/8 sin(4 arcsin x)/3 + C ln x+ 1+x (x + 1) x +x/ ln x+1+ x +x / arctanx/x arcsin(x/) x 4 x / arcsin( x) x 1 x (x +1) 4x / cosx+xsinx x sinx sinx+xcosx (x)e x (1/)e x (x/) sin(x)/4 = (x/) (sinxcosx)/ xlnx x (x arctanx+arctanx x)/ x 3 cosx + 3x sinx + 6xcosx 6sinx x 3 sinx + 3x cosx 6xsinx 6cosx x /4 (cos x)/4 (xsinxcosx)/+ C x/4 (xcos x)/+(cosxsinx)/4+ C xarctan( x)+arctan( x) x sin( x) xcos( x) secxcscx cotx ln x /4+ln x+ / x 3 /3 4x 4ln x + 4ln x /(x+5) x ln x +ln x x+x 3 /3+8arctan(x/) (1/)arctan(x/+5/) x / ln(4+x ) (1/4)ln x+3 (1/4)ln x (1/5)ln x 3 (1/5)ln 1+x (1/3)ln x (1/3)ln x T,S: 4±0

11 Appendix A Selected Answers T: 9.815±0.8115; S: 9± T: 60.75±1; S: 60± T: ±0.0833; S: ± T: ± 0.006; S: ± T: ± 0.005; S: ± T:.8833±0.0834; S:.9000± T: ±0.0077; S: ± T: 1.097±0.0147; S: 1.089± T: 3.63±0.087; S: 3.6± (t+4) 4 4 (t 9) 5/ 5 (e t +16) cost 3 cos3 t tan t ln t +t ln 1 4/t tan(arcsin(t/5)) + C = t 5 5 t sin3t ttant+ln cost e t +1 3t sint + sin4t 4 3 ln t ln t sinarctant = 1+t /t (1+tant) (t +1) 5/ (t +1) 3/ 5 3 e t sint e t cost (t 3/ +47) ( t ) 1 3/ ( t ) 1/ ln sin(arctan(t/3)) + C = 9 ln(4t ) ln(9+4t ) 18 (arctan(t)) 4 3ln t ln t 4 cos 7 t cos5 t t lnt t (lnt) t lnt + t (t 3 3t +6t 6)e t ln(t + 1 5) ln(t+1+ 5) / / / / /3 /π /π 3 3/(π)/8

12 308 Appendix A Selected Answers / / / / / /π, 5/π , , (3 π)/(π), (18 3+π)/(4π) /49 meters, 0/49 seconds /98 meters, 30/49 seconds /49 meters, 1000/49 seconds s(t) = cost, v(t) = sint, maximum distance is 1, maximum speed is s(t) = sin(πt)/π +t/π, v(t) = cos(πt)/π+1/π, maximum speed is /π s(t) = t / sin(πt)/π +t/π, v(t) = t cos(πt)/π+1/π s(t) = t /+sin(πt)/π t/π, v(t) = t+cos(πt)/π/π π/ π/ π(π/) (a) 114π/5 (b) 74π/5 (c) 0π (d) 4π π, 4π πh (3r h)/ π /π; /π; / /A π/ /3, ft/s; 8 14 ft/s ,305,08,516 N-m ,457,854,041 N-m , 500π N-m π /3 N-m π N-m N-m /5 N-m N-m N-m N-m / / x = 45/8, ȳ = 93/ x = 0, ȳ = 4/(3π) x = 1/, ȳ = / x = 0, ȳ = 8/ x = 4/7, ȳ = / x = ȳ = 1/ x = 0, ȳ = 8/(9π) x = ȳ = 8/(9π) x = 0, ȳ = 44/(7π) / diverges diverges diverges

13 Appendix A Selected Answers diverges π/ diverges, diverges, diverges, no CPV π mph: 90.8 to 95.3 N-m 90 mph: to 10.6 N-m mph: to N-m µ = 1/c, σ = 1/c µ = (a+b)/, σ = (b a) / / r = ( 8)/ ln()+3/ a+a 3 / ln(( +1)/ 3) / e +1 ln( 1+e + 1)+ln( +1) π 3 16π π π π +πe+ 1 4 πe π 4e π e π π + 8π a > b: πb + πa b a b arcsin( a b /a), a < b: πb + ( ) πa b b a ln b a + b a a θ = arctan(3) r = 4cscθ r 3 cosθsin θ = r = r = sinθsec 3 θ rsinθ = sin(rcosθ) r = /(sinθ 5cosθ) r = secθ = r cos θ rsinθ = 3r cos θ rcosθ rsinθ r = sinθ (x +y ) = 4x y (x +y )y (x +y ) 3/ = y x +y = x y x 4 +x y = y (θcosθ + sinθ)/( θsinθ + cosθ), (θ +)/( θsinθ +cosθ) cosθ +sinθcosθ cos θ sin θ sinθ, 3(1+sinθ) (cos θ sin θ sinθ) (sin θ cos θ)/(sinθcosθ), /(4sin 3 θcos 3 θ) sinθcosθ cos θ sin θ, (cos θ sin θ) undefined

14 310 Appendix A Selected Answers π/ /3 sinθ 3sin 3 θ 3cos 3 θ cosθ, 3cos 4 θ 3cos θ + cos 3 θ(3cos θ ) π/1+ 3/ πa π/ π/ π/ π/ π/4 3 3/ π/+3 3/ / / π/ π/3+ 3/ π/3 3/ π 3 / π π/4 3/ π/ π 15/ 7arccos(1/4) π x = t sin(t), y = 1 cos(t) x = 4cost cos(4t), y = 4sint sin(4t) x = cost+cos(t), y = sint sin(t) x = cost+tsint, y = sint tcost There is a horizontal tangent at all multiples of π π/ π cost dt Four points: 3 3 5,± ,± π/ / π / (π/, 1) π 3 / π 5 5, (π 4π +1 + ln(π + 4π +1))/ lim n n /(n +1) = 1/ lim n 5/(1/n +14) = 1/ n=1 1 n diverges, so 3 1 n diverges / n=1

15 Appendix A Selected Answers / / / diverges diverges converges converges converges converges diverges converges N = N = N = any integer greater than e converges converges diverges converges converges converges converges diverges diverges diverges converges diverges converges diverges converges absolutely diverges converges conditionally converges absolutely converges conditionally converges absolutely diverges converges conditionally converges converges converges diverges R = 1, I = (,1) R =, I = (, ) R = e, I = ( e,e) R = e, I = ( e,+e) R = 0, converges only when x = R = 1, I = [ 6, 4] the alternating harmonic series (n+1)x n (n+1)(n+)x n C (n+1)(n+) x n, R = 1 (n+1)(n+) xn+ () n x n /(n)!, R = x n /n!, R =

16 31 Appendix A Selected Answers () n(x 5)n, R = 5 5 n+1 () n(x)n, R = 1 n n= ln() converges () n(x )n n n, R = converges n=1 () n (n+1)(x) n, R = n=1 n= (n) n! n x n = (n)! n (n)!n! xn, R = x+x 3 / () n x 4n+1 /(n)! () n x n+1 /n! x + x4 4 x ; x1 1! x+ x3 3 + x5, error ± diverges converges converges diverges diverges diverges converges converges converges converges converges converges converges converges diverges (, ) ( 3, 3) ( 3, 3) (, 1) radius is 0 it converges only when x = ( 3, 3) (, ) (ln()) n n! x n () n n+1 xn+1 n+1 xn x/ + () n (n 3) n n! n= () n x n π = () n n+1 xn+1 () n 4 n+1 x n