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Selected Answers

ASelected (2/3)x+ (1/3) ( 2/3)x+ (1/3) 2x+ 2, 2, x+ 6, 6, + 1/2, 1/2, 3/2,y-intercept: 3/2, ( 2/3)x 2, 2, 0,y= 2x+ 2,y= 2x+ 75t(tin hours); 164 (9/5)x+ 32, ( 40, 40) + + (a)y=(00 x <100(x/10) 10 100 x 1000x 9101000< ( x 19450< x 488147050< x (a)P= + 2(b)x= 10000P+ (2/25)x (16/5) (a)x2+y2= 9(b) (x 5)2+ (y 6)2= 9(c) (x+ 5)2+ (y+ 6)2= 9(d)x2+ (y 3)2= 9(e)x2+ (y+ 3)2= 9(f) (x 3)2+y2= (a) x= 2, y= 3,m= 3/2,y= (3/2)x 3, 13(b) x= 1, y= 3,m= 3,y= 3x+ 2, 10(c) x= 2, y= 2,m= 1,y=x, (x+ 2/7)2+ (y 41/7)2= 1300 {x|x 3/2} {x|x6= 1}297298 Appendix A Selected {x|x6= 1 andx6= 1} {x|x <0} {x|x R}, , {x|x 0} {x|h r x h+r} {x|x 1 orx <0} {x| 1/3< x <1/3} {x|x 0 andx6= 1} {x|x 0 andx6= 1} {x|x 3},{x|x 0} (500 2x),{x|0 x 250} (50 r2),{r|0< r p50/ } 2 r2+2000/r,{r|0< r < } 5, , , , 4/3, 24/7, 7/24, 3 , , , 13(3 + x)))

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1 ASelected (2/3)x+ (1/3) ( 2/3)x+ (1/3) 2x+ 2, 2, x+ 6, 6, + 1/2, 1/2, 3/2,y-intercept: 3/2, ( 2/3)x 2, 2, 0,y= 2x+ 2,y= 2x+ 75t(tin hours); 164 (9/5)x+ 32, ( 40, 40) + + (a)y=(00 x <100(x/10) 10 100 x 1000x 9101000< ( x 19450< x 488147050< x (a)P= + 2(b)x= 10000P+ (2/25)x (16/5) (a)x2+y2= 9(b) (x 5)2+ (y 6)2= 9(c) (x+ 5)2+ (y+ 6)2= 9(d)x2+ (y 3)2= 9(e)x2+ (y+ 3)2= 9(f) (x 3)2+y2= (a) x= 2, y= 3,m= 3/2,y= (3/2)x 3, 13(b) x= 1, y= 3,m= 3,y= 3x+ 2, 10(c) x= 2, y= 2,m= 1,y=x, (x+ 2/7)2+ (y 41/7)2= 1300 {x|x 3/2} {x|x6= 1}297298 Appendix A Selected {x|x6= 1 andx6= 1} {x|x <0} {x|x R}, , {x|x 0} {x|h r x h+r} {x|x 1 orx <0} {x| 1/3< x <1/3} {x|x 0 andx6= 1} {x|x 0 andx6= 1} {x|x 3},{x|x 0} (500 2x),{x|0 x 250} (50 r2),{r|0< r p50/ } 2 r2+2000/r,{r|0< r < } 5, , , , 4/3, 24/7, 7/24, 3 , , , 13(3 + x)))

2 + 3 x+ x21 + x , , ,3 + 3 x+ x2 , 25/2, 20, 15, 25, , , , , 4 + t , , , t not (a) 8, (b) 6, (c) dne, (d) 2, (e) 1,(f) 8, (g) 7, (h) 6, (i) 3, (j) 3/2,(k) 6, (l) x/p169 + 1 + 2/(2x+ 1)3 (t+ 2) 13x+ or or 100x 5x x (3/4)x 1 (9/7)x 16/7 Appendix A Selected + 20x4+ 6x+ 10 30x+ + 2x + 6x x/p625 13x/4 + 24x 48 49t/5 + 5, 49 3x/4 + (x3 5x+ 10) +x3(3x2 5) (x2+5x 3)(5x4 18x2+6x 7) +(2x+ 5)(x5 6x3+ 3x2 7x+ 1) 625 x22 x x x 625 1x19 625 x2 20 625 = 4(2x 3),y= 4x 5x+ 10 x3(3x2 5)(x3 5x+ 10) + 5x5 6x3+ 3x2 7x+ 1 (x2+ 5x 3)(5x4 18x2+ 6x 7)(x5 6x3+ 3x2 7x+ 1) x 625 x2+x3/2(625 x2)3 1x19 625 x2 20 625 17x/4 41 11x/16 15 19/169 5 9x2+x+ 4x+ 2/ (x2+ 1) x2 x2/p169 (2x 4)p25 x2 (x2 4x+ 5)x/ 25 x/pr2 + x(5 x)3 + + 11 x+x2+x+ 1(1 x) 1/p25 x2 p25 x2 169x2 1.

3 R169x 2x+ 1/x22px3 x2 (1/x) (100 x2)5 + 3x23(x+x3)2 4x(x2+ 1) +4x3+ 4x2p1 + (x2+ 1)2!.2q(x2+ 1)2+p1 + (x2+ 1) (x+ 8) 3(4 x) (x2+ 5) 12x(6 2x2) (1 4x3) + 5 8(4x 1)(2x2 x+ 3) 3300 Appendix A Selected (x+ 1) (8x 2)/(4x2 2x+ 1) 3x2+ 5x (2x 4)3+ 6(3x2+ 1)(2x 4) 2/(x 1) (x2+ 1) (x2 6x+ 7)/(x 3) 5/(3x 4) + 72x3+ 18x2+ 18x (5 4x)/((2x+ 1)2(x 3)2) (2(2 + 3x)2) + 72x5+ 110x4+ 100x3+60x2+ 28x+ 23x/96 29 3 2 13x/2 23 2x + 2 55p4 + 5x+3 55p4 + /2, any /6, any ( 2 + 6) (1 + 3)/(1 3) = 2 + 2 ( x) cos( x)/ x+ (2x+ 1) sinx (x2+x) sinxcosxp1 sinxcos(cosx) +xsec2x2 (1 + sinx) tanxcosx(1 + sinx) (23x2) + 46x4cos(23x2) 6 cos(cos(6x)) sin(6x) tan (1 + sec )2=sin (cos + 1) (6t) 6t5sin(6t) (sin(3t) +tcos(3t))/cos(2t) +2t3sin(3t) sin(2t)/cos2(2t)

4 /2, any /2 +n , any 3x/2 + 3/4 3 8 3x+ 4 8 3 3 3x/2 3 /6 + 2n , 5 /6 + 2n , any ln(3) sinxexAppendix A Selected (ex) (x) cosxlnx+sinxx + + 2xln(2) 2xln(3)(1/3) (4x 1) (3x2+ 3)/(x3+ 3x) tan(x) (1 ln(x2))/(x2pln(x2)) (x) (x)(cos(x)/x sin(x) ln(x)) (2x+y)/(x+ 2y) (2xy 3x2 y2)/(2xy 3y2 x2) (x) sin(y)/(cos(x) cos(y)) y/ (ysec2(x/y) y2)/(xsec2(x/y)+y2) (y cos(x+y))/(cos(x+y) x) y2 2x ( 3,2 3), ( 3, 2 3),(2 3, 3), ( 2 3, 3) 7x/ 3 8/ ( y1/31x+y1/31x1+x1/31y1)/x1 (y y1) =(2x31+ 2x1y21 x1)(2y31+ 2y1x21+y1)(x x1) 11 + 1 + 3x2cos(x3)/q1 sin2(x3) (arcsinx) 1 ex/p1 (1 + lnx)xxln 5(1 +x2x) arctan(xx) 1 1 A Selected 1 1 not 1 andy= atx= 1 atx= 1, max atx= atx= 2, min atx= atx= 1, max atx= atx= atx= 7 /12 +k , max atx= /12 +k , for max atx= min atx= min atx= atx= 1 atx= 1, max atx= atx= 2, min atx= atx= 1, max atx= atx= atx= 7 /12 +k , max atx= /12 +k , for atx= 0, min atx= atx= 3/2, neither atx= atn , max at /2 +n at 2n , max at (2n+ 1)

5 At /2+2n , max at 3 /2+2n atx= 1 atx= 1, max atx= atx= 2, min atx= atx= 1, max atx= A Selected atx= atx= 7 /12 +n , max atx= /12 +n , for atx= 63 atx= at 5 1/4, min at 5 1 at 1, min at at 2 1 atn atn , min at /2 +n at /2+2n , min at 3 /2+2n up up whenx <0, concavedown whenx > down whenx <3, concaveup whenx > up whenx < 1/ 3 orx >1/ 3, concave down when 1/ 3< x <1/ up whenx <0 orx >2/3,concave down when 0< x <2 up whenx <0, concavedown whenx > up whenx < 1 orx >1,concave down when 1< x <0 or0< x < down on ((8n 1) /4,(8n+3) /4), concave up on ((8n+3) /4,(8n+ 7) /4), for down up on ( ,(21 497)/4)and (21 + 497)/4, ) up on (0, ) down on (2n /3,(2n+1) /3) up on (0, ) up on ( , 1) and (0, ) down up up on ( /4+n ,3 /4+n ) points atn , arcsin(p2/3) +n : (3, ), up/decr: ( ,0),(2,3), down/decr: (0,2) at (2,5), min at (0,1) 2 52/3,h= 52/3,h/w=1 100 3 100 23 100,h/s= 21/3V1/3,h=V1/3/22/3,h/w= 1 square square $ 5 cm,h= 40/ cm,h/r= 8/ A Selected direct (a) 2, (b) 7 36 36+12 14 (a)a/6, (b) (a+b pa2 ab+b2) meters wide by meters 2/ the ratio is (2 k )/4.

6 Ifk 2/ , the ratio is zero: thewindow should be semicircular withno rectangular 2r/ 3,h= 2 2r/ 3 58% 18 5/(2 )1/3 cm,h= 5 25/3/ 1/3= 4r 2 27502 1/3,r= 75022 2 1 ratio of the volume of thesphere to the volume of the cone is1033/4096 + 33/4096 17 ,so the cone occupies of the be at distancec3 a/(3 a+3 b) from $ is a critical point whensin 1/v1= sin 2/v2, and thesecond derivative is positive, sothere is a minimum at the (16 ) (1000 ) 5 (3 ) 10/2 /72 /144 2/36 ft3 : 6 ft/s, length: 5/2 : 20/11 m/s, length: 9/11 3 150 3 200 475/19 156 50 329 210 329 + 150 57/ 47 210 (a)x=acos asin cot( + ) =asin /sin( + ), (c) x or A Selected y= 65/16,dy= y=p11/10 1,dy= y= sin( /50),dy= 8 1 18 + 47x2/2 5x+ + lnx+ cos(2x)/2 + rectangles: 41/4 = ,8 rectangles: 183/16 = (16/3)x3/2+ +t+ x+ 2/z+ lns+ (5x+ 1)3/15 + (x 6)3/3 + + 4/ x+ t2+C,t <2;t2 4t+ 8 +C,t (10) 1 (x4 3x2) (x2) (x4) rises untilt= 100/49, then position of the object at timetiss(t) = + 20t+k.

7 Thenet distance traveled is 45/2, thatis, it ends up 45/2 meters belowwhere it started. The total distancetraveled is 6205/98 0sint dt= : 2 , total: 2 /3 + 4 18,B= 44/3,C= 10 (1 t)10/10 + + 2x3/3 +x+ (x2+ 1)101/202 + 3(1 5t)2/3/10 + (sin4x)/4 + (100 x2)3/2/3 + 2p1 x3/3 + (sin t)/ + (2 cos2x) = (1/2) sec2x+C306 Appendix A Selected ln|cosx|+ (x)/2 + cos(tanx) + 3 (27/8)(x2 7)8/9+ (37+ 1) (x)2 sin(2x)/4 + cosx+ (cos3x)/3 + (sin 2x)/4 + (sin 4x)/32 + (cos5x)/5 (cos3x)/3 + (sin3x)/3 + (sin 4x)/32 + (sin3x)/3 (sin5x)/5 + 2(cosx)5/2/5 + cotx+ (sec3x)/3 secx+ ln|cscx+ cotx|+ cscxcotx/2 (1/2) ln|cscx+cotx|+ 1/2 ln|x+px2 1|/2+ + 4x2/2 +(9/4) ln|2x+p9 + 4x2|+ (1 x2)3/2/3 + (x)/8 sin(4 arcsinx)/32 + |x+p1 +x2|+ (x+ 1)px2+ 2x/2 ln|x+ 1 +px2+ 2x|/2 + arctanx 1/x+ arcsin(x/2) xp4 x2/2 + ( x) x 1 x+ (2x2+ 1)

8 P4x2 1/24 + +xsinx+ 2 sinx+ 2xcosx+ (x 1)ex+ (1/2)ex2+ (x/2) sin(2x)/4 +C=(x/2) (sinxcosx)/2 + x+ (x2arctanx+ arctanx x)/2 + x3cosx+ 3x2sinx+ 6xcosx 6 sinx+ + 3x2cosx 6xsinx 6 cosx+ (cos2x)/4 (xsinxcosx)/2+ (xcos2x)/2+(cosxsinx)/4+ ( x)+arctan( x) x+ sin( x) 2 xcos( x) + 2 cotx+ ln|x 2|/4 + ln|x+ 2|/4 + x3/3 4x 4 ln|x 2|+4 ln|x+ 2|+ 1/(x+ 5) + x ln|x 2|+ ln|x+ 2|+ 4x+x3/3 + 8 arctan(x/2) + (1/2) arctan(x/2 + 5/2) + 2 ln(4 +x2) + (1/4) ln|x+ 3| (1/4) ln|x+ 7|+CAppendix A Selected (1/5) ln|2x 3| (1/5) ln|1+x|+ (1/3) ln|x| (1/3) ln|x+ 3|+ ,S: 4 : ; S: 9 : 1; S: 60 : ; S: : ; S: : ; S: : ; S: : ; S: : ; S: : ; S: (t+ 4)44+ (t2 9)5/25+ (et2+ 16)24+ 23cos3t+ + |t2+t+ 3|+ |1 4/t2|+ (arcsin(t/5)) +C=t25 25 t2+ sin 3t+ + ln|cost|+ et+ 1 + +sin 2t4+sin 4t32+ |t|3 ln|t+ 3|3+ 1sin arctant+C= p1 +t2/t+ 12(1 + tant)2+ (t2+ 1)5/25 (t2+ 1)3/23+ etcost2+ (t3/2+ 47)46+ (2 t2)3/2 1(2 t2)1/2+ |sin(arctan(2t/3))|9+C=ln(4t2) ln(9 + 4t2)18+ (arctan(2t))24+ ln|t+ 3|4+ln|t 1|4+ cos5t5+ 1t 3+ 1lnt+ (lnt)22 t2lnt2+t24+ (t3 3t2+ 6t 6)et+ + 510ln(2t+ 1 5) +5 510ln(2t+ 1 + 5) + 2 A Selected 2/ 3 3/(2 ) 1 5/3 , 5/ , , (3 )/(2 ), (18 12 3 + )/(4 ) meters, 20/49 meters, 30/49 meters, 1000/49 (t) = cost,v(t) = sint,maximum distance is 1,maximum speed is (t) = sin( t)/ 2+t/ ,v(t) = cos( t)

9 / + 1/ ,maximum speed is 2/ (t) =t2/2 sin( t)/ 2+t/ ,v(t) =t cos( t)/ + 1/ (t) =t2/2 + sin( t)/ 2 t/ ,v(t) =t+ cos( t)/ 1/ ( /2 1) (a) 114 /5 (b) 74 /5 (c) 20 (d) 4 , 24 h2(3r h) ; 2/ ; 1/3, 4 1224 ft/s; 8 1224 5,305,028,516 4,457,854,041 ,500 + 196000/3 x= 45/28, y= 93 x= 0, y= 4/(3 ) x= 1/2, y= 2 x= 0, y= 8 x= 4/7, y= 2 x= y= 1 x= 0, y= 28/(9 ) x= y= 28/(9 ) x= 0, y= 244/(27 ) A Selected , , , no mph: to N-m90 mph: to mph: to = 1/c, = 1 = (a+b)/2, =(b a)2 (22 22 8) (2) + 3 +a3 (( 2 + 1)/ 3) +e2 2 + 1 ln(p1 +e2+1) + ln( 2 + 1) 3 16 73027 10 + 2 e+14 e2 4e2 2 +8 23 > b: 2 b2+2 a2b a2 b2arcsin(pa2 b2/a),a < b: 2 b2+2 a2b b2 a2ln ba+ b2 a2a!

10 = arctan(3) 4 csc sin2 = sin sec3 = sin(rcos ) 2/(sin 5 cos ) 2 sec =r2cos2 rsin + = 3r2cos2 2rcos rsin sin (x2+y2)2= 4x2y (x2+y2) (x2+y2)3/2= +y2= +x2y2= ( cos + sin )/( sin + cos ),( 2+ 2)/( sin + cos ) + 2 sin cos cos2 sin2 sin ,3(1 + sin )(cos2 sin2 sin ) (sin2 cos2 )/(2 sin cos ), 1/(4 sin3 cos3 ) sin cos cos2 sin2 ,2(cos2 sin2 )3310 Appendix A Selected sin 3 sin3 3 cos3 2 cos ,3 cos4 3 cos2 + 22 cos3 (3 cos2 2) 3 /12 + 3 /4 3 3 /2 + 3 3 /3 + 3 /3 3 3 /24 3 /12 15/2 7 arccos(1/4) sin(t)2,y= 1 cos(t) 4 cost cos(4t),y= 4 sint sin(4t) 2 cost+ cos(2t),y= 2 sint sin(2t) cost+tsint,y= sint is a horizontal tangent at allmultiples of.


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