> restart; with(plots): > alpha0:=Pi/3; x0:=50; y0:=70; L:=80; alpha0 := 1/3 Pi x0 := 50 y0 := 70 L := 80 > #átmeneti ív: > kappa:=0.04*cos(2*Pi/320*l); kappa := 0.04 cos(1/160 Pi l) > plot(kappa, l=0..L); > alpha:=s->alpha0+int(kappa, l=0..s); s / | alpha := s -> alpha0 + | kappa dl | / 0 > x:=x0+int(cos(alpha(s)), s=0..l); l / | x := 50 + | cos(1/3 Pi + 2.037183271 sin(0.01963495409 s)) ds | / 0 > y:=y0+int(sin(alpha(s)), s=0..l); l / | y := 70 + | sin(1/3 Pi + 2.037183271 sin(0.01963495409 s)) ds | / 0 > plot([x,y,l=0..L]); > p_atm:=plot([x,y,l=0..L]): > xL:=subs(l=L, x); > yL:=subs(l=L, y); > alphaL:=alpha(L); 80 / | xL := 50 + | cos(1/3 Pi + 2.037183271 sin(0.01963495409 s)) ds | / 0 80 / | yL := 70 + | sin(1/3 Pi + 2.037183271 sin(0.01963495409 s)) ds | / 0 alphaL := 1/3 Pi + 2.037183271 > p_atm_kezd:=pointplot({[x0,y0]}, color=blue): > p_atm_veg:=pointplot({[xL,yL]}, color=green): > display(p_atm, p_atm_kezd, p_atm_veg); > #bevezető szakasz: ez egy körív > kappa:=0.04; kappa := 0.04 > alpha:=s->alpha0+int(kappa, l=0..s); s / | alpha := s -> alpha0 + | kappa dl | / 0 > x:=x0+int(cos(alpha(s)), s=0..l); x := 28.34936490 + 21.65063510 cos(0.04000000000 l) + 12.50000000 sin(0.04000000000 l) > y:=y0+int(sin(alpha(s)), s=0..l); y := 82.50000000 - 12.50000000 cos(0.04000000000 l) + 21.65063510 sin(0.04000000000 l) > p_bev:=plot([x,y,l=0..-40], color=blue): display(p_bev, p_atm); > #kivezető szakasz: > kappa:=0; kappa := 0 > alpha:=s->alphaL+int(kappa, l=L..s); s / | alpha := s -> alphaL + | kappa dl | / L > x:=xL+int(cos(alpha(s)), s=L..l); x := 129.8691080 80 / | + | cos(1/3 Pi + 2.037183271 sin(0.01963495409 s)) ds | / 0 - 0.9983638495 l > y:=yL+int(sin(alpha(s)), s=L..l); y := 65.42554994 80 / | + | sin(1/3 Pi + 2.037183271 sin(0.01963495409 s)) ds | / 0 + 0.05718062580 l > p_kiv:=plot([x,y,l=80..120], color=green): display(p_bev, p_atm, > p_kiv); > display(p_bev, p_atm, p_kiv, p_atm_kezd, p_atm_veg); > #érintőhöz: még egyszer az átmeneti ív: > kappa:=0.04*cos(2*Pi/320*l); > alpha:=s->alpha0+int(kappa, l=0..s); > x:=x0+int(cos(alpha(s)), s=0..l); > y:=y0+int(sin(alpha(s)), s=0..l); kappa := 0.04 cos(1/160 Pi l) s / | alpha := s -> alpha0 + | kappa dl | / 0 l / | x := 50 + | cos(1/3 Pi + 2.037183271 sin(0.01963495409 s)) ds | / 0 l / | y := 70 + | sin(1/3 Pi + 2.037183271 sin(0.01963495409 s)) ds | / 0 > x_f:=evalf(subs(l=L/2, x)); > y_f:=evalf(subs(l=L/2, y)); x_f := 41.39715035 y_f := 105.5280057 > m:=evalf(tan(alpha(L/2))); m := -0.7663591172 > p_erinto:=plot(y_f+m*(xx-x_f), xx=x_f-20..x_f+20, color=black): > display(p_bev, p_atm, p_kiv, p_atm_kezd, p_atm_veg, p_erinto); > #x közelítése: > plot(x, l=0..L); > x_pol:=convert(taylor(x, l=0, 23), polynom); 2 x_pol := 50 + 0.5000000002 l - 0.01732050808 l 3 -5 4 - 0.0001333333334 l + 0.2865867753 10 l -7 5 -9 6 + 0.2094750460 10 l - 0.4271014643 10 l -11 7 -13 8 - 0.2742088439 10 l + 0.5183960935 10 l -15 9 -17 10 + 0.3151214476 10 l - 0.5555585155 10 l -19 11 -21 12 - 0.3197489225 10 l + 0.5384346243 10 l -23 13 -25 14 + 0.2953247012 10 l - 0.4771427229 10 l -27 15 -29 16 - 0.2520547879 10 l + 0.3924905043 10 l -31 17 -33 18 + 0.2005747200 10 l - 0.3027465047 10 l -35 19 -37 20 - 0.1501616364 10 l + 0.2204799857 10 l -39 21 -41 22 + 0.1065172897 10 l - 0.1525269509 10 l > plot({x_pol,x}, l=0..L); > plot(x_pol-x, l=0..L); > > #2. feladat > restart; > f:=cosh(x); f := cosh(x) > n:=7; T_n:=convert(taylor(f, x=0, n), polynom); n := 7 2 4 6 T_n := 1 + 1/2 x + 1/24 x + 1/720 x > plot({f-T_n}, x=-1..1); > # 2.a.) Tehát n=7 már jó. > g:=a*x^2+b*x+c; 2 g := a x + b x + c > hiba:=int((f-g)^2, x=-1..1); hiba := -12 a sinh(1) + 8 a cosh(1) + cosh(1) sinh(1) - 4 c sinh(1) 2 2 2 + 2/3 b + 2/5 a + 4/3 a c + 2 c + 1 > hiba_a:=diff(hiba, a); > hiba_b:=diff(hiba, b); > hiba_c:=diff(hiba, c); > hiba_a := -12 sinh(1) + 8 cosh(1) + 4/5 a + 4/3 c hiba_b := 4/3 b hiba_c := -4 sinh(1) + 4/3 a + 4 c > solve({hiba_a,hiba_b, hiba_c}, {a,b,c}); {a = 30 sinh(1) - 45/2 cosh(1), b = 0, c = -9 sinh(1) + 15/2 cosh(1)} > g_opt:=(30*sinh(1)-45/2*cosh(1))*x^2+0*x+(-9*sinh(1)+15/2*cosh(1)); 2 g_opt := (30 sinh(1) - 45/2 cosh(1)) x - 9 sinh(1) + 15/2 cosh(1) > plot([f,g_opt], x=-1..1, color=[red,green]); > > #3. feladat > restart; > de:=diff(y(x), x)=-4*y(x)^2+1; d 2 de := -- y(x) = -4 y(x) + 1 dx > kezd:=y(0)=5; kezd := y(0) = 5 > dsolve({de,kezd}, y(x)); 1/2 exp(4 x + ln(11/9)) + 1/2 y(x) = ----------------------------- -1 + exp(4 x + ln(11/9)) > evalf(subs(x=2, rhs(%))); 0.5002745448 > #Tehát y(2)=.5002745448 . > #4. feladat > restart; > h:=sin(3*x-y)/(1+x^2+y^4); sin(3 x - y) h := ------------ 2 4 1 + x + y > plot3d(h, x=-3..2, y=-2..4, axes=boxed, shading=zhue, > style=patchcontour); > plot3d(h, x=0.2..0.5, y=-0.6..-0.2, axes=boxed, shading=zhue, > style=patchcontour); > #Max. hely: kb. x=0.37, y=-0.4 , max. érték: 0.86 > > #Pontosabban: a max. hely: > fsolve({diff(h, x), diff(h,y)}, {x,y}, {x=0.2..0.4,y=-1..0}); {x = 0.3321388709, y = -0.3811150960} > #Max. érték: > evalf(subs(%, h)); 0.8673950951 >