1 Zadavanje niza
ispisivanje članova niza zadanih formulom
| (%i1) | b[n]:=(n+1)/(n^2+1); |
![(%o1) b[n]:=(1+n)/(1+n^2)](nizovi_htmlimg/nizovi_0.png)
| (%i2) | makelist(b[n],n,1,6); |
![(%o2) [1,3/5,2/5,5/17,3/13,7/37]](nizovi_htmlimg/nizovi_1.png)
rekurzivno zadavanje niza
| (%i4) |
a[1]:1; a[n]:=a[n-1]+n-1; |
![(%o4) a[n]:=-1+n+a[n-1]](nizovi_htmlimg/nizovi_3.png)
| (%i5) | makelist(a[n],n,1,6); |
![(%o5) [1,2,4,7,11,16]](nizovi_htmlimg/nizovi_4.png)
2 Omeđenost i monotonost niza
ako želimo ponovo koristiti varijablu "a" za spremanje članova novog niza, moramo prethodni niz ukloniti pomoću naredbe "kill"
| (%i6) | kill(a); |
1. zadatak
| (%i7) | a[n]:=(n-1)/(n+1); |
![(%o7) a[n]:=(n-1)/(1+n)](nizovi_htmlimg/nizovi_6.png)
| (%i8) | makelist(a[n],n,1,10); |
![(%o8) [0,1/3,1/2,3/5,2/3,5/7,3/4,7/9,4/5,9/11]](nizovi_htmlimg/nizovi_7.png)
| (%i9) | makelist([n,a[n]],n,1,10); |
![(%o9) [[1,0],[2,1/3],[3,1/2],[4,3/5],[5,2/3],[6,5/7],[7,3/4],[8,7/9],[9,4/5],[10,9/11]]](nizovi_htmlimg/nizovi_8.png)
| (%i10) | wxplot2d([discrete,makelist([n,a[n]],n,1,20)],[style,points],grid2d); |
| (%i11) | kill(a); |
2. zadatak
| (%i12) | a[n]:=log(n+5); |
![(%o12) a[n]:=log(5+n)](nizovi_htmlimg/nizovi_13.png)
| (%i13) | wxplot2d([discrete,makelist([n,a[n]],n,1,20)],[y,0,4],[style,points],grid2d); |
| (%i14) | kill(a)$ |
3. zadatak
| (%i15) | a[n]:=1/n!; |
![(%o15) a[n]:=1/n!](nizovi_htmlimg/nizovi_17.png)
| (%i16) | wxplot2d([discrete,makelist([n,a[n]],n,1,8)],[x,-1,8.5],[y,-1,1.5],[style,points],grid2d); |
| (%i17) | kill(a)$ |
4. zadatak
| (%i18) | a[n]:=(-1)^n/n!; |
![(%o18) a[n]:=(-1)^n/n!](nizovi_htmlimg/nizovi_21.png)
| (%i19) | wxplot2d([discrete,makelist([n,a[n]],n,1,8)],[x,-1,8.5],[y,-1.5,1],[style,points],grid2d); |
| (%i20) | kill(a)$ |
5. zadatak
| (%i21) | a[n]:=(3*n-18)/(3*n-19); |
![(%o21) a[n]:=(3*n-18)/(3*n-19)](nizovi_htmlimg/nizovi_25.png)
| (%i22) | makelist(a[n],n,1,10); |
![(%o22) [15/16,12/13,9/10,6/7,3/4,0,3/2,6/5,9/8,12/11]](nizovi_htmlimg/nizovi_26.png)
| (%i23) | makelist(float(a[n]),n,1,10); |
![(%o23) [0.9375,0.9230769230769231,0.9,0.8571428571428571,0.75,0.0,1.5,1.2,1.125,1.09090909090909]](nizovi_htmlimg/nizovi_27.png)
| (%i24) | wxplot2d([discrete,makelist([n,a[n]],n,1,15)],[x,-1,16],[y,-1,2],[style,[points,2]],grid2d); |
3 Limes niza
1. zadatak
| (%i25) | limit((n+3)/(2*n+5),n,inf); |
| (%i26) |
wxplot2d([[discrete,makelist([n,(n+3)/(2*n+5)],n,1,24)],1/2],[x,-1,25],[y,0,1], [style,[points,2],[lines,1]],grid2d); |
2. zadatak
| (%i27) | limit((5*n-2)/(n^2+n+1),n,inf); |
3. zadatak
| (%i28) | limit((n-1)^2/(3*n^2),n,inf); |
4. zadatak
| (%i29) | limit(sqrt(n^2+3*n)-n,n,inf); |
5. zadatak
| (%i30) | limit(n^3/(n^2+1)-n,n,inf); |
6. zadatak
| (%i31) | limit(n*(sqrt(n^2+1)-n),n,inf); |
7. zadatak
| (%i32) | limit((8*n^3-7*n+4)^(1/3)/(3*n+4),n,inf); |
8. zadatak
| (%i33) | limit(n!/((n+1)!+(n-1)!),n,inf); |
9. zadatak
| (%i34) | limit(sqrt(2*n-3)/(sqrt(n)+1),n,inf); |
| (%i35) |
wxplot2d([[discrete,makelist([n,sqrt(2*n-3)/(sqrt(n)+1)],n,2,100)],sqrt(2)],[x,-1,100],[y,-1,2], [style,[points,2],[lines,1]],grid2d); |
10. zadatak
| (%i36) | limit((16*4^(n-2)+4^n)/(1+4^n),n,inf); |
11. zadatak
| (%i37) | limit((n+1)^(1/3)-n^(1/3),n,inf); |
12. zadatak
| (%i38) | limit((2^n+3^n+4^n+5^(n-1))/(2^n+2*6^n),n,inf); |
13. zadatak
| (%i39) | limit(((n+6)/(n-1))^(2*n),n,inf); |
14. zadatak
| (%i40) | limit(((3*n)/(3*n+1))^(n+1),n,inf); |
4 Nizovi - wxdraw2d i draw2d
| (%i41) | load(draw)$ |
nekoliko zajedničkih globalnih postavki za crtanje da
ih ne moramo kod svake slike posebno pisati
| (%i42) |
set_draw_defaults( grid=true, xaxis=true, yaxis=true, xaxis_width=2, yaxis_width=2, xaxis_type=solid, yaxis_type=solid, xlabel="n", ylabel="a_n")$ |
a)
| (%i43) | a[n]:=(n-1)/(n+1); |
![(%o43) a[n]:=(n-1)/(1+n)](nizovi_htmlimg/nizovi_51.png)
| (%i44) | niz:points(makelist([n,a[n]],n,1,20))$ |
| (%i45) |
wxdraw2d(xrange=[0,20.2], yrange=[-0.2,1.5], xtics=[0,1,20], color=blue, point_size=2, point_type=7, niz), wxplot_size=[700,500]$ |
| (%i46) |
draw2d(terminal=wxt, xrange=[0,20.2], yrange=[-0.2,1.5], xtics=[0,1,20], color=blue, point_size=2, point_type=7, niz)$ |
b)
| (%i47) | kill(a)$ |
| (%i48) | a[n]:=log(n+5); |
![(%o48) a[n]:=log(5+n)](nizovi_htmlimg/nizovi_54.png)
| (%i49) | niz2:points(makelist([n,a[n]],n,1,20))$ |
| (%i50) |
wxdraw2d(xrange=[0,20.2], yrange=[-0.2,4], xtics=[0,1,20], color=blue, point_size=2, point_type=7, niz2), wxplot_size=[700,500]$ |
| (%i51) |
draw2d(terminal=wxt, xrange=[0,20.2], yrange=[-0.2,4], xtics=[0,1,20], color=blue, point_size=2, point_type=7, niz2)$ |
c)
| (%i52) | kill(a)$ |
| (%i53) | a[n]:=1/n!; |
![(%o53) a[n]:=1/n!](nizovi_htmlimg/nizovi_57.png)
| (%i54) | niz3:points(makelist([n,a[n]],n,1,8))$ |
| (%i55) |
wxdraw2d(xrange=[0,8.5], yrange=[-0.2,1.1], xtics=[0,1,9], color=blue, point_size=2, point_type=7, niz3), wxplot_size=[700,500]$ |
| (%i56) |
draw2d(terminal=wxt, xrange=[0,8.5], yrange=[-0.2,1.1], xtics=[0,1,9], color=blue, point_size=2, point_type=7, niz3)$ |
d)
| (%i57) | kill(a)$ |
| (%i58) | a[n]:=(-1)^n/n!; |
![(%o58) a[n]:=(-1)^n/n!](nizovi_htmlimg/nizovi_60.png)
| (%i59) | niz4:points(makelist([n,a[n]],n,1,8))$ |
| (%i60) |
wxdraw2d(xrange=[0,8.5], yrange=[-1.1,0.6], xtics=[0,1,9], color=blue, point_size=2, point_type=7, niz4), wxplot_size=[700,500]$ |
| (%i61) |
draw2d(terminal=wxt, xrange=[0,8.5], yrange=[-1.1,0.6], xtics=[0,1,9], color=blue, point_size=2, point_type=7, niz4)$ |
e)
| (%i62) | kill(a)$ |
| (%i63) | a[n]:=(3*n-18)/(3*n-19); |
![(%o63) a[n]:=(3*n-18)/(3*n-19)](nizovi_htmlimg/nizovi_63.png)
| (%i64) | niz5:points(makelist([n,a[n]],n,1,15))$ |
| (%i65) |
wxdraw2d(xrange=[0,15.5], yrange=[-1,2], xtics=[0,1,15], color=blue, point_size=2, point_type=7, niz5), wxplot_size=[700,500]$ |
| (%i66) |
draw2d(terminal=wxt,xrange=[0,15.5], yrange=[-1,2], xtics=[0,1,15], color=blue, point_size=2, point_type=7, niz5)$ |