| (%i1) | load(draw)$ |
nekoliko zajedničkih globalnih postavki za crtanje da
ih ne moramo kod svake slike posebno pisati
| (%i2) |
set_draw_defaults( grid=true, xaxis=true, yaxis=true, xaxis_width=2, yaxis_width=2, xaxis_type=solid, yaxis_type=solid, xlabel="x", ylabel="f(x)")$ |
1 Svojstva funkcija
a)
| (%i3) | f(x):=2*sqrt(x)/(3-x^2); |
| (%i4) |
wxdraw2d(xrange=[-1,5], yrange=[-5,5], line_width=3, color=blue, explicit(f(x),x,0,5), line_width=2, color=black, line_type=dots, parametric(sqrt(3),t,t,-5,5)), wxplot_size=[600,400]$ |
| (%i5) |
draw2d(terminal=wxt,xrange=[-1,5], yrange=[-5,5], line_width=3, color=blue, explicit(f(x),x,0,5), line_width=2, color=black, line_type=dots, parametric(sqrt(3),t,t,-100,100))$ |
b)
| (%i6) | f(x):=sqrt(x+5); |
| (%i7) |
wxdraw2d(xrange=[-10,50], yrange=[-1,8], line_width=3, color=blue, explicit(f(x),x,-5,50)), wxplot_size=[600,400]$ |
| (%i8) |
draw2d(terminal=wxt,xrange=[-10,50], yrange=[-1,8], line_width=3, color=blue, explicit(f(x),x,-5,50))$ |
c)
| (%i9) | f(x):=4/(x^2+3); |
| (%i10) |
wxdraw2d(yrange=[-0.3,1.5], line_width=3, color=blue, explicit(f(x),x,-10,10)), wxplot_size=[600,400]$ |
| (%i11) |
draw2d(terminal=wxt,yrange=[-0.3,1.5], line_width=3, color=blue, explicit(f(x),x,-10,10))$ |
d)
| (%i12) | f(x):=%e^(5-x); |
| (%i13) |
wxdraw2d(yrange=[-50,680],line_width=3, color=blue, explicit(f(x),x,-1.5,4)), wxplot_size=[600,400]$ |
| (%i14) |
draw2d(terminal=wxt,yrange=[-50,680],line_width=3, color=blue, explicit(f(x),x,-1.5,4))$ |
e)
| (%i15) | f(x):=2*x^2/(3-x^2); |
| (%i16) |
wxdraw2d(yrange=[-10,10], line_width=3, ytics=[-10,2,10],xtics=[-4,1,4], color=blue, explicit(f(x),x,-5,5), color=black, line_type=dots, line_width=2, parametric(t,-2,t,-5,5), parametric(-sqrt(3),t,t,-10,10), parametric(sqrt(3),t,t,-10,10)), wxplot_size=[600,400]$ |
| (%i17) |
draw2d(terminal=wxt,yrange=[-10,10], line_width=3, ytics=[-100,2,100],xtics=[-4,1,4], color=blue, explicit(f(x),x,-5,5), color=black, line_type=dots, line_width=2, parametric(t,-2,t,-5,5), parametric(-sqrt(3),t,t,-100,100), parametric(sqrt(3),t,t,-100,100))$ |
f)
| (%i18) | f(x):=(x+5*x^5)^(1/3); |
| (%i19) |
wxdraw2d(line_width=3, color=blue, explicit(f(x),x,-3,3)), wxplot_size=[600,400]$ |
| (%i20) |
draw2d(terminal=wxt,line_width=3, color=blue, explicit(f(x),x,-3,3))$ |
g) trebamo biti oprezni
| (%i21) | f(x):=log((3+2*x)/(3-2*x))/log(4); |
mogli bismo dobiti krivi dojam da je funkcija omeđena
| (%i22) |
wxdraw2d(xrange=[-3,3], yrange=[-5,5], xtics=[-3,0.5,3], ytics=[-5,1,5], line_width=3, color=blue, explicit(f(x),x,-3/2,3/2)), wxplot_size=[600,400]$ |
stavimo li decimalne brojeve -1.499 i 1.499 da se čim više približimo rubovima -3/2 i 3/2
| (%i23) |
wxdraw2d(xrange=[-3,3], yrange=[-7,7], xtics=[-3,1,3], ytics=[-7,1,7], line_width=3, color=blue, explicit(f(x),x,-1.499,1.499), line_width=2, color=black, line_type=dots, parametric(-3/2,t,t,-7,7), parametric(3/2,t,t,-7,7)), wxplot_size=[600,400]$ |
| (%i24) |
draw2d(terminal=wxt,xrange=[-3,3], xtics=[-3,1,3], line_width=3, color=blue, explicit(f(x),x,-1.4999,1.4999), line_width=2, color=black, line_type=dots, parametric(-3/2,t,t,-7,7), parametric(3/2,t,t,-7,7))$ |
možemo izračunati i limese
| (%i25) | limit(f(x),x,3/2,minus); |
| (%i26) | limit(f(x),x,-3/2,plus); |
h)
| (%i27) | f(x):=%e^(x^2-4*x+3)-1; |
| (%i28) |
wxdraw2d(yrange=[-5,70], xrange=[-1,4.5], line_width=3, color=blue, explicit(f(x),x,-0.3,4.3)), wxplot_size=[400,600]$ |
| (%i29) |
draw2d(terminal=wxt,yrange=[-5,70], xrange=[-1,4.5], line_width=3, color=blue, explicit(f(x),x,-0.3,4.3))$ |
i)
| (%i30) | f(x):=sin(2*x)*cos(3*x); |
| (%i31) |
wxdraw2d(yrange=[-1,1], xrange=[-11,11], line_width=3, color=blue, explicit(f(x),x,-10,10)), wxplot_size=[600,400]$ |
| (%i32) |
draw2d(terminal=wxt,yrange=[-1,1], xrange=[-11,11], line_width=3, color=blue, explicit(f(x),x,-10,10))$ |
j)
| (%i33) | f(x):=sin(1/x); |
| (%i34) |
wxdraw2d(yrange=[-1.2,1.2], line_width=3, color=blue, explicit(f(x),x,-3,3)), wxplot_size=[600,400]$ |
| (%i35) |
draw2d(terminal=wxt,yrange=[-1.2,1.2], line_width=3, color=blue, explicit(f(x),x,-3,3))$ |
2 Injekcija, surjekcija, bijekcija
a)
| (%i36) | f(x):=log(4*x^2)/log(2); |
| (%i37) |
wxdraw2d(yrange=[-7,7], nticks=200, line_width=3, color=blue, explicit(f(x),x,-5,5)), wxplot_size=[600,400]$ |
| (%i38) |
draw2d(terminal=wxt,yrange=[-7,7], nticks=200, line_width=3, color=blue, explicit(f(x),x,-5,5))$ |
b)
| (%i39) | f(x):=2^(x+3); |
| (%i40) |
wxdraw2d(yrange=[-10,70],nticks=200, line_width=3, color=blue, explicit(f(x),x,-3,3)), wxplot_size=[600,400]$ |
| (%i41) |
draw2d(terminal=wxt,yrange=[-10,70],nticks=200, line_width=3, color=blue, explicit(f(x),x,-3,3))$ |
c)
| (%i42) | f(x):=3/(3*x^2+1); |
| (%i43) |
wxdraw2d(yrange=[-0.5,3.5], line_width=3, color=blue, explicit(f(x),x,-5,5)), wxplot_size=[600,400]$ |
| (%i44) |
draw2d(terminal=wxt,yrange=[-0.5,3.5], line_width=3, color=blue, explicit(f(x),x,-5,5))$ |
d)
| (%i45) | f(x):=x/(4-2*x); |
| (%i46) |
wxdraw2d(yrange=[-5,5], ytics=[-5,1,5], line_width=3, color=blue, explicit(f(x),x,-5,5), line_width=2, color=black, line_type=dots, parametric(t,-1/2,t,-5,5),parametric(2,t,t,-5,5)), wxplot_size=[600,400]$ |
| (%i47) |
draw2d(terminal=wxt,yrange=[-5,5], ytics=[-100,1,100], line_width=3, color=blue, explicit(f(x),x,-5,5), line_width=2, color=black, line_type=dots, parametric(t,-1/2,t,-5,5),parametric(2,t,t,-100,100))$ |
3 Inverzna funkcija
a)
| (%i48) | f(x):=(2*x-1)/(1-x); |
inverzna funkcija
| (%i49) | solve(y=(2*x-1)/(1-x),x); |
![(%o49) [x=(1+y)/(y+2)]](funkcije_slike_htmlimg/funkcije_slike_47.png)
| (%i50) | g(x):=(x+1)/(x+2); |
| (%i51) |
wxdraw2d(yrange=[-5,5], xtics=[-5,1,5], ytics=[-5,1,5], proportional_axes=xy, line_width=3, color=blue, explicit(f(x),x,-5,5), color=red, explicit(g(x),x,-5,5), color=green,line_width=2,explicit(x,x,-5,5)), wxplot_size=[500,500]$ |
| (%i52) |
draw2d(terminal=wxt,yrange=[-5,5], xtics=[-100,1,100], ytics=[-100,1,100], proportional_axes=xy, line_width=3, color=blue, explicit(f(x),x,-5,5), color=red, explicit(g(x),x,-5,5), color=green,line_width=2,explicit(x,x,-5,5))$ |
b)
| (%i53) | f(x):=5^(1-x/3); |
inverzna funkcija
| (%i54) | solve(y=5^(1-x/3),x); |
![(%o54) [x=-(3*log(y)-3*log(5))/log(5)]](funkcije_slike_htmlimg/funkcije_slike_52.png)
| (%i55) | expand(%); |
![(%o55) [x=3-(3*log(y))/log(5)]](funkcije_slike_htmlimg/funkcije_slike_53.png)
| (%i56) | g(x):=3-3*log(x)/log(5); |
| (%i57) |
wxdraw2d(yrange=[-2,10], xtics=[-5,1,10], ytics=[-1,1,10], proportional_axes=xy, line_width=3, color=blue, explicit(f(x),x,-5,10), color=red, explicit(g(x),x,0.01,10), color=green,line_width=2,explicit(x,x,-5,10)), wxplot_size=[600,500]$ |
| (%i58) |
draw2d(terminal=wxt,yrange=[-2,10], xtics=[-5,1,10], ytics=[-1,1,10], proportional_axes=xy, line_width=3, color=blue, explicit(f(x),x,-5,10), color=red, explicit(g(x),x,0.001,10), color=green,line_width=2,explicit(x,x,-5,10))$ |
c)
| (%i59) | f(x):=log(x)/log(3)+2; |
inverzna funkcija
| (%i60) | solve(y=f(x),x); |
![(%o60) [x=%e^(log(3)*y)/9]](funkcije_slike_htmlimg/funkcije_slike_58.png)
| (%i61) | radcan(%); |
![(%o61) [x=3^(y-2)]](funkcije_slike_htmlimg/funkcije_slike_59.png)
| (%i62) | g(x):=3^(x-2); |
| (%i63) |
wxdraw2d(yrange=[-2,10], xtics=[-3,1,10], ytics=[-1,1,10], proportional_axes=xy, line_width=3, color=blue, explicit(f(x),x,0.01,10), color=red, explicit(g(x),x,-2,10), color=green,line_width=2,explicit(x,x,-2,10)), wxplot_size=[500,500]$ |
| (%i64) |
draw2d(terminal=wxt,yrange=[-2,10], xtics=[-3,1,10], ytics=[-1,1,10], proportional_axes=xy, line_width=3, color=blue, explicit(f(x),x,0.01,10), color=red, explicit(g(x),x,-2,10), color=green,line_width=2,explicit(x,x,-2,10))$ |
d)
| (%i65) | f(x):=log(x+2)/log(3); |
inverzna funkcija
| (%i66) | solve(y=f(x),x); |
![(%o66) [x=%e^(log(3)*y)-2]](funkcije_slike_htmlimg/funkcije_slike_64.png)
| (%i67) | radcan(%); |
![(%o67) [x=3^y-2]](funkcije_slike_htmlimg/funkcije_slike_65.png)
| (%i68) | g(x):=3^x-2; |
| (%i69) |
wxdraw2d(yrange=[-4,10], xtics=[-3,1,10], ytics=[-4,1,10], proportional_axes=xy, line_width=3, color=blue, explicit(f(x),x,-1.99,10), color=red, explicit(g(x),x,-4,10), color=green,line_width=2,explicit(x,x,-4,10)), wxplot_size=[500,500]$ |
| (%i70) |
draw2d(terminal=wxt,yrange=[-4,10], xtics=[-3,1,10], ytics=[-4,1,10], proportional_axes=xy, line_width=3, color=blue, explicit(f(x),x,-1.99,10), color=red, explicit(g(x),x,-4,10), color=green,line_width=2,explicit(x,x,-4,10))$ |
e)
| (%i71) | f(x):=sqrt(x+1); |
inverzna funkcija
| (%i73) | solve(y=f(x),x);p; |
![(%o72) [x=y^2-1]](funkcije_slike_htmlimg/funkcije_slike_71.png)
| (%i74) | g(x):=x^2-1; |
| (%i75) |
wxdraw2d(yrange=[-4,10], xtics=[-3,1,10], ytics=[-4,1,10], proportional_axes=xy, line_width=3, color=blue, explicit(f(x),x,-1,10), color=red, explicit(g(x),x,0,10), color=green,line_width=2,explicit(x,x,-4,10)), wxplot_size=[500,500]$ |
| (%i76) |
draw2d(terminal=wxt,yrange=[-4,10], xtics=[-3,1,10], ytics=[-4,1,10], proportional_axes=xy, line_width=3, color=blue, explicit(f(x),x,-1,10), color=red, explicit(g(x),x,0,10), color=green,line_width=2,explicit(x,x,-4,10))$ |