PK¯]lXñB–Hmimetypetext/x-wxmathmlPK¯]lXQdBV55 format.txt This file contains a wxMaxima session in the .wxmx format. .wxmx files are .xml-based files contained in a .zip container like .odt or .docx files. After changing their name to end in .zip the .xml and eventual bitmap files inside them can be extracted using any .zip file viewer. The reason why part of a .wxmx file still might still seem to make sense in a ordinary text viewer is that the text portion of .wxmx by default isn't compressed: The text is typically small and compressing it would mean that changing a single character would (with a high probability) change big parts of the whole contents of the compressed .zip archive. Even if version control tools like git and svn that remember all changes that were ever made to a file can handle binary files compression would make the changed part of the file bigger and therefore seriously reduce the efficiency of version control wxMaxima can be downloaded from https://github.com/wxMaxima-developers/wxmaxima. It also is part of the windows installer for maxima (https://wxmaxima-developers.github.io/wxmaxima/). If a .wxmx file is broken but the content.xml portion of the file can still be viewed using an text editor just save the xml's text as "content.xml" and try to open it using a recent version of wxMaxima. If it is valid XML (the XML header is intact, all opened tags are closed again, the text is saved with the text encoding "UTF8 without BOM" and the few special characters XML requires this for are properly escaped) chances are high that wxMaxima will be able to recover all code and text from the XML file. PK¯]lXHOtÍ:: content.xml Kedvenc Matematikai Kísérleteim Alapok Halmazok Számok Határérték Differenciálszámítás Derivált Függvényvizsgálat Tétel Rolle tétele Cauchy középérték tétele Következmény: Lagrange középérték tétele Megjegyzés Tétel Megjegyzés Tétel Példák kill(all); (%o0) done plot2d(x*sin(1/x),[x,-2,2])$ plot2d: expression evaluates to non-numeric value somewhere in plotting range. plot2d(x^2*sin(1/x),[x,-2,2])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range. plot2d(x^2*sin(1/x^2),[x,-4,4])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range. A trigonometrikus és a hiperbolikus függvények inverzei kill(all); (%o0) done plot2d(asin,[x,-1,1])$ plot2d(acos,[x,-1,1])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range. plot2d(atan,[x,-4,4])$ plot2d(acot,[x,-4,4])$ plot2d(asec,[x,-4,4])$ asec: argument 0.0 isn't in the domain of asec.asec: argument 1.0 isn't in the domain of asec.plot2d: expression evaluates to non−numeric value somewhere in plotting range. plot2d(acsc,[x,-4,4])$ acsc: argument 0.0 isn't in the domain of acsc.plot2d: expression evaluates to non−numeric value somewhere in plotting range. plot2d(asinh,[x,-4,4])$ plot2d(acosh,[x,-4,4])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range. plot2d(atanh,[x,-4,4])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range. plot2d(acoth,[x,-4,4])$ acoth: argument 0.0 isn't in the domain of acoth.acoth: argument −1.0 isn't in the domain of acoth.acoth: argument 1.0 isn't in the domain of acoth.plot2d: expression evaluates to non−numeric value somewhere in plotting range. plot2d(asech,[x,-4,4])$ asech: argument 0.0 isn't in the domain of asech.plot2d: expression evaluates to non−numeric value somewhere in plotting range. plot2d(acsch,[x,-4,4])$ acsch: argument 0.0 isn't in the domain of acsch.plot2d: expression evaluates to non−numeric value somewhere in plotting range. Összefüggések a trigonometrikus és a hiperbolikus függvények inverzei között kill(all); (%o0) done asin(y)+acos(y); (%o1) asin

y

+acos

y

trigsimp(%); (%o2) asin

y

+acos

y

load("hypergeometric")$ atanh(y); (%o4) atanh

y

rectform(%); (%o5) log

y+1

−log

y−1

2
+%i*

atan2

0,y+1

−atan2

0,1−y

2
Darboux közbenső érték tétele kill(all); (%o0) done f:diff(x^2*sin(1/x^2),x); (%o1) 2*sin

1x2

*x−2*cos

1x2

x
plot2d(%,[x,-2,2])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range. L'Hospital-szabály Megjegyzés Példák Feladat kill(all); (%o0) done (1/1) limit((b^x-1)/x,x,0); (%o1) log

b

limit(log(b)*b^x,x,0); (%o2) log

b

(1/2) limit(log(1+x)/x,x,0); (%o3) 1 limit(1/(1+x),x,0); (%o4) 1 (1/3) limit((1+x)^(1/x),x,0); (%o5) %e limit(log(1+x)/x,x,0); (%o6) 1 (2/1) limit((1+b/n)^n,n,inf); (%o7) %eb limit(n*log(1+b/n),n,inf); (%o8) b f1:diff(log(1+b/n),n); (%o9) −b

bn+1

*n2
g1:diff(1/n,n); (%o10) −1n2 h1:f1/g1; (%o11) bbn+1 limit(%,n,inf); (%o12) b (2/2 limit(n^2*log(1+1/n)-n,n,inf); (%o13) −12 f1:diff(log(1+1/n)-1/n,n); (%o14) 1n2−1

1n+1

*n2
g1:diff(1/n^2,n); (%o15) −2n3 h1:f1/g1; (%o16) −

1n2−1

1n+1

*n2

*n32
ratsimp(%); (%o17) −n2*n+2 limit(%,n,inf); (%o18) −12 Feladat kill(all); (%o0) done (1/1) limit((x+exp(x))/x,x,0); (%o1) infinity (1/2 limit(cos(x)^(1/sin(x)),x,0); (%o2) 1 limit(log(cos(x))/sin(x),x,0); (%o3) 0 limit(-sin(x)/cos(x)^2,x,0); (%o4) 0 (1/3) limit(((1+exp(x))/2)^(1/sinh(x)),x,0); (%o5) %e limit(log((1+exp(x))/2)/sinh(x),x,0); (%o6) 12 limit(exp(x)/(1+exp(x))/cosh(x),x,0); (%o7) 12 (2/1) limit(sinh(x)^2/log(cos(3*x)),x,0); (%o8) −29 limit(2*sinh(x)*cosh(x)/(-3*sin(3*x)/cos(3*x)),x,0); (%o9) −29 limit(sinh(x)/sin(3*x),x,0); (%o10) 13 limit(cosh(x)/3*cos(3*x),x,0); (%o11) 13 (2/2 limit((2-x)^(1/cos(%pi/2/x)),x,1); (%o12) %e−2%pi limit(log(2-x)/cos(%pi/2/x),x,1); (%o13) −2%pi f1:diff(log(2-x),x); (%o14) −12−x g1:diff(cos(%pi/2/x),x); (%o15) %pi*sin

%pi2*x

2*x2
h1:f1/g1; (%o16) −2*x2%pi*sin

%pi2*x

*

2−x

limit(%,x,1); (%o17) −2%pi (2/3) limit((x^(1/x)-1)*x/log(x),x,inf); (%o18) 1 f1:diff(x^(1/x)-1,x); (%o19) x1x*

1x2−log

x

x2

g1:diff(log(x)/x,x); (%o20) 1x2−log

x

x2
h1:f1/g1; (%o21) x1x limit(x^(1/x),x,inf); (%o22) 1 limit(log(x)/x,x,inf); (%o23) 0 Feladat kill(all); (%o0) done (1/1) limit((%e-(1+x)^(1/x))/x,x,0); (%o1) %e2 f1:diff((1+x)^(1/x),x); (%o2)

x+1

1x
*

1x*

x+1

−log

x+1

x2

l:taylor(log(1+x),x,0,5); (%o3)/T/ x−x22+x33−x44+x55+... l:l/x; (%o4)/T/ 1−x2+x23−x34+x45+... h:taylor(1/(1+x),x,0,4); (%o5)/T/ 1−x+x2−x3+x4+... h-l; (%o6)/T/ −x2+2*x23−3*x34+4*x45+... (h-l)/x; (%o7)/T/ −12+2*x3−3*x24+4*x35+... (1/2) limit(n*(n^(1/n)-1)/log(n),n,inf); (%o8) 1 f1:diff(n^(1/n),n); (%o9) n1n*

1n2−log

n

n2

g1:diff(log(n)/n,n); (%o10) 1n2−log

n

n2
h1:f1/g1; (%o11) n1n limit(log(n)/n,n,inf); (%o12) 0 (1/3) limit((tan(x)-x)/(x*(1-cos(x))),x,0); (%o13) 23 f1:diff(tan(x)-x,x); (%o14) sec

x

2
−1
g1:diff(x*(1-cos(x)),x); (%o15) x*sin

x

−cos

x

+1
h1:f1/g1; (%o16) sec

x

2
−1
x*sin

x

−cos

x

+1
limit(h1,x,0); (%o17) 23 f2:diff(f1,x); (%o18) 2*sec

x

2
*tan

x

g2:diff(g1,x); (%o19) 2*sin

x

+x*cos

x

h2:f2/g2; (%o20) 2*sec

x

2
*tan

x

2*sin

x

+x*cos

x

trigsimp(%); (%o21) 2*sin

x

2*cos

x

3
*sin

x

+x*cos

x

4
f3:diff(sin(x),x); (%o22) cos

x

g3:diff(2*cos(x)^3*sin(x)+x*cos(x)^4,x); (%o23) −6*cos

x

2
*sin

x

2
−4*x*cos

x

3
*sin

x

+3*cos

x

4
h3:f3/g3; (%o24) cos

x

−6*cos

x

2
*sin

x

2
−4*x*cos

x

3
*sin

x

+3*cos

x

4
(2/1) limit((x-sin(x))/(tan(x)-x),x,0); (%o25) 12 f1:diff(x-sin(x),x); (%o26) 1−cos

x

g1:diff(tan(x)-x,x); (%o27) sec

x

2
−1
h1:f1/g1; (%o28) 1−cos

x

sec

x

2
−1
trigsimp(%); (%o29) cos

x

2
cos

x

+1
limit(%,x,0); (%o30) 12 (2/2) limit((1+1/x)^x,x,-1,minus); (%o31) inf limit(-x*log(1-1/x),x,1,plus); (%o32) inf f1:diff(log(1+1/x),x); (%o33) −1

1x+1

*x2
g1:diff(1/x,x); (%o34) −1x2 h1:f1/g1; (%o35) 11x+1 limit(%,x,-1,minus); (%o36) inf (2/3) limit((1+1/x)^x,x,0,plus); (%o37) 1 f1:diff(log(1+1/x),x); (%o38) −1

1x+1

*x2
g1:diff(1/x,x); (%o39) −1x2 h1:f1/g1; (%o40) 11x+1 limit(%,x,0,plus); (%o41) 0 (2/4 limit(x^x,x,0,plus); (%o42) 1 f1:diff(log(x),x); (%o43) 1x g1:diff(1/x,x); (%o44) −1x2 h1:f1/g1; (%o45) −x limit(-x,x,0,plus); (%o46) 0 (3/1) limit(x^(1/x),x,0,plus); (%o47) 0 limit(log(x)/x,x,0,plus); (%o48) −inf (3/2) assume(c>0); (%o49) [c>0] limit(log(x)/x^(-c),x,0,plus); Is c an integer? yes; (%o50) 0 limit(1/(-c)*x^(-c),x,0,plus); (%o51) −inf (3/3) forget(c>0); (%o52) [c>0] assume(c<0); (%o53) [c<0] limit(log(x)/x^c,x,0,plus); Is c an integer? yes; (%o54) 0 limit(1/c/x^c,x,0,plus); (%o55) 0 Feladat kill(all); (%o0) done (1/1) limit(sin(3*x)/tan(5*x),x,0); (%o1) 35 limit(3*cos(3*x)/5*sec(5*x)^2,x,0); (%o2) 35 (1/2) limit(log(cos(a*x))/log(cos(b*x)),x,0); (%o3) a2b2 limit(tan(a*x)/tan(b*x),x,0); (%o4) ab limit(sec(a*x)^2/sec(b*x)^2,x,0); (%o5) 1 (1/3 limit((exp(x)-exp(-x)-2*x)/(x-sin(x)),x,0); (%o6) 2 limit((exp(x)+exp(-x)-2)/(1-cos(x)),x,0); (%o7) 2 limit((exp(x)-exp(-x))/sin(x),x,0); (%o8) 2 limit(exp(x)+exp(-x)/cos(x),x,0); (%o9) 2 (2/1) limit(1/x^2-1/sin(x)^2,x,0); (%o10) −13 limit((sin(x)^2-x^2)/(x^2*sin(x)^2),x,0); (%o11) −13 f1:diff(sin(x)^2-x^2,x); (%o12) 2*cos

x

*sin

x

−2*x
f2:diff(%,x); (%o13) −2*sin

x

2
+2*cos

x

2
−2
f3:diff(%,x); (%o14) −8*cos

x

*sin

x

f4:diff(%,x); (%o15) 8*sin

x

2
−8*cos

x

2
g1:diff(x^2*sin(x)^2,x); (%o16) 2*x*sin

x

2
+2*x2*cos

x

*sin

x

g2:diff(%,x); (%o17) −2*x2*sin

x

2
+2*sin

x

2
+8*x*cos

x

*sin

x

+2*x2*cos

x

2
g3:diff(%,x); (%o18) −12*x*sin

x

2
−8*x2*cos

x

*sin

x

+12*cos

x

*sin

x

+12*x*cos

x

2
g4:diff(%,x); (%o19) 8*x2*sin

x

2
−24*sin

x

2
−64*x*cos

x

*sin

x

−8*x2*cos

x

2
+24*cos

x

2
h4:f4/g4; (%o20) 8*sin

x

2
−8*cos

x

2
8*x2*sin

x

2
−24*sin

x

2
−64*x*cos

x

*sin

x

−8*x2*cos

x

2
+24*cos

x

2
limit(%,x,0); (%o21) −13 (2/2) limit((tan(x)-x)/(x-sin(x)),x,0); (%o22) 2 limit((sec(x)^2-1)/(1-cos(x)),x,0); (%o24) 2 (2/3) limit((x*cot(x)-1)/x^2,x,0); (%o25) −13 f1:diff(x*cot(x)-1,x); (%o26) cot

x

−x*csc

x

2
f1:trigsimp(%); (%o27) cos

x

*sin

x

−x
sin

x

2
limit(f1,x,0); (%o28) 0 g1:diff(x^2,x); (%o29) 2*x h1:f1/g1; (%o30) cos

x

*sin

x

−x
2*x*sin

x

2
limit(h1,x,0); (%o31) −13 f2:diff(cos(x)*sin(x)-x,x); (%o32) −sin

x

2
+cos

x

2
−1
g2:diff(2*x*sin(x)^2,x); (%o33) 2*sin

x

2
+4*x*cos

x

*sin

x

h2:f2/g2; (%o34) −sin

x

2
+cos

x

2
−1
2*sin

x

2
+4*x*cos

x

*sin

x

limit(h2,x,0); (%o35) −13 f3:diff(f2,x); (%o36) −4*cos

x

*sin

x

g3:diff(g2,x); (%o37) −4*x*sin

x

2
+8*cos

x

*sin

x

+4*x*cos

x

2
h3:f3/g3; (%o38) −4*cos

x

*sin

x

−4*x*sin

x

2
+8*cos

x

*sin

x

+4*x*cos

x

2
limit(h3,x,0); (%o39) −13 f4:diff(f3,x); (%o40) 4*sin

x

2
−4*cos

x

2
g4:diff(g3,x); (%o41) −12*sin

x

2
−16*x*cos

x

*sin

x

+12*cos

x

2
h4:f4/g4; (%o42) 4*sin

x

2
−4*cos

x

2
−12*sin

x

2
−16*x*cos

x

*sin

x

+12*cos

x

2
(3/1) limit(cot(x)-1/x,x,0); (%o43) 0 f1:diff(x*cot(x)-1,x); (%o44) cot

x

−x*csc

x

2
trigsimp(%); (%o45) cos

x

*sin

x

−x
sin

x

2
g1:diff(x,x); (%o46) 1 h1:f1/g1; (%o47) cot

x

−x*csc

x

2
limit(h1,x,0); (%o48) 0 f2:diff(cos(x)*sin(x)-x,x); (%o49) −sin

x

2
+cos

x

2
−1
g2:diff(x^2,x); (%o50) 2*x h2:f2/g2; (%o51) −sin

x

2
+cos

x

2
−1
2*x
limit(h2,x,0); (%o52) 0 f3:diff(cos(x)^2-1,x); (%o53) −2*cos

x

*sin

x

g3:diff(2*x,x); (%o54) 2 h3:f3/g3; (%o55) −cos

x

*sin

x

(3/2) limit((sin(x)/x)^(-x^2),x,0); (%o56) 1 limit(-x^2*log(sin(x)/x),x,0); (%o57) 0 f1:diff(log(sin(x)/x),x); (%o58) x*

cos

x

x
−sin

x

x2

sin

x

g1:diff(1/x^2,x); (%o59) −2x3 h1:f1/g1; (%o60) −x4*

cos

x

x
−sin

x

x2

2*sin

x

limit(h1,x,0); (%o61) 0 (3/3 limit(((1+exp(x))/2)^cot(x),x,0); (%o62) %e limit(cot(x)*log((1+exp(x))/2),x,0); (%o63) 12 f1:diff(log((1+exp(x))/2),x); (%o64) %ex%ex+1 g1:diff(tan(x),x); (%o65) sec

x

2
(4/1) limit((1-x)*tan(%pi*x/2),x,1); (%o66) 2%pi f1:diff(tan(%pi*x/2),x); (%o67) %pi*sec

%pi*x2

2
2
g1:diff(1/(1-x),x); (%o68) 1

1−x

2
limit((1-x)/cos(%pi*x/2),x,1); (%o69) 2%pi f2:diff(1-x,x); (%o70) −1 g2:diff(cos(%pi*x/2),x); (%o71) −%pi*sin

%pi*x2

2
(4/2) limit(x^(1/(1-x)),x,1); (%o72) %e−1 limit(log(x)/(1-x),x,1); (%o73) −1 limit((1/x)/(-1),x,1); (%o74) −1 (4/3) limit((2-x)^tan(%pi*x/2),x,1); (%o75) %e2%pi limit(log(2-x)/cot(%pi*x/2),x,1); (%o76) 2%pi f1:diff(log(2-x),x); (%o77) −12−x g1:diff(cot(%pi*x/2),x); (%o78) −%pi*csc

%pi*x2

2
2
trigsimp(%); (%o79) −%pi2*sin

%pi*x2

2
Feladat etc. Feladat Feladat kill(all); (%o0) done (1) limit((1+1/n)^n,n,inf); (%o1) %e (2) limit((log(n)^(log(n)/n))/2,n,inf); (%o2) 12 Feladat Feladat Konvex és konkáv függvények, inflexiós hely Segédtétel Tétel Feladat kill(all); (%o0) done diff(a^x,x,2); (%o1) ax*log

a

2
Feladat diff(x^b,x,2); (%o2)

b−1

*b*xb−2
Feladat kill(all); (%o0) done diff(x*log(x),x,2); (%o1) 1x Feladat diff(x^b,x,2); (%o3)

b−1

*b*xb−2
Feladat diff(log(x),x,2); (%o2) −1x2 Feladat diff(x^3,x,2); (%o4) 6*x diff(x^(1/3),x,2); (%o5) −29*x53 Számtani és mértani közép közti egyenlőtlenség Feladat: harmonikus és mértani közép Alkalmazzuk az előző tételt a reciprokokra. Feladat (1) x^n szigorúan konvex. etc. Taylor-polinom kill(all); (%o0) done taylor(f(x),x,a,5); (%o1)/T/ f

a

+

dd*x*f

x

x=a

*

x−a

+

d2d*x2*f

x

x=a

*

x−a

2
2
+

d3d*x3*f

x

x=a

*

x−a

3
6
+

d4d*x4*f

x

x=a

*

x−a

4
24
+

d5d*x5*f

x

x=a

*

x−a

5
120
+...
subst(h,x-a,%); (%o2) h5*

d5d*x5*f

x

x=a

120
+h4*

d4d*x4*f

x

x=a

24
+h3*

d3d*x3*f

x

x=a

6
+h2*

d2d*x2*f

x

x=a

2
+h*

dd*x*f

x

x=a

+f

a

Taylor-formula maradéktag nélkül Taylor-formula Lagrange-féle maradéktaggal Tétel Aszimtóták Függvényvizsgálat kill(all); (%o0) done f:exp(-x^2); (%o1) %e−x2 solve(f,x); (%o2) [] f1:diff(f,x); (%o3) −2*x*%e−x2 solve(f1,x); (%o4) [x=0] f2:diff(f1,x); (%o5) 4*x2*%e−x2−2*%e−x2 solve(f2,x); (%o6) [x=−12,x=12] plot2d([f,f1,f2],[x,-4,4])$ Feladat kill(all); (%o0) done f:1+3*x+5*x^2-2*x^3; (%o1) −2*x3+5*x2+3*x+1 f0:subst(x=-1,f); (%o3) 5 fp:diff(f,x); (%o4) −6*x2+10*x+3 f1:subst(x=-1,fp); (%o5) −13 etc. Feladat etc. Feladat (1) diff(sin(x),x,5); (%o6) cos

x

.5^5/5!; (%o7) 2.604166666666666*10−4 etc. Feladat kill(all); (%o0) done assume(a>0); (%o1) [a>0] f:(a^n+x)^(1/n); (%o2)

x+an

1n
f0:subst(x=0,f); (%o7) a fp:diff(f,x); (%o8)

x+an

1n−1
n
subst(x=0,fp); (%o9) a

1n−1

*n
n
ratsimp(%); (%o10) a1−nn fpp:diff(fp,x); (%o18)

1n−1

*

x+an

1n−2
n
A maradéktag abszolút értékében a szorzó: M:-subst(x=θ*x,%); (%o19) −

1n−1

*

x*θ+an

1n−2
n
Mivel a kitevő negatív, θ=0 esetén kapjuk a maximumot: M1:subst(θ=0,M); (%o20) −a

1n−2

*n
*

1n−1

n
Ezt még szorozni kell: M2:M1*(θ*x)^2/2!; (%o22) −a

1n−2

*n
*

1n−1

*x2*θ2
2*n
A felső becslés tehát: M3:subst(θ=1,M2); (%o23) −a

1n−2

*n
*

1n−1

*x2
2*n
ratsimp(%); (%o24)

a*n−a

*x2
2*a2*n*n2
Feladat (1) 3/12!; (%o34) 1159667200 float(%), numer; (%o35) 6.26302709636043*10−9 3/13!; (%o36) 12075673600 float(%), numer; (%o37) 4.817713151046484*10−10 etc. Feladat (1) Számtani és mértani közép. etc. Feladat (1) taylor(cos(x),x,0,5); (%o39)/T/ 1−x22+x424+... taylor(exp(-x^2/2),x,0,5); (%o41)/T/ 1−x22+x48+... etc. Feladat (1/1) f:x^3-3*x^2+3*x+2; (%o42) x3−3*x2+3*x+2 fp:diff(f,x); (%o43) 3*x2−6*x+3 solve(fp,x); (%o44) [x=1] fpp:diff(fp,x); (%o45) 6*x−6 subst(x=1,fpp); (%o46) 0 fppp:diff(fpp,x); (%o47) 6 etc. Feladat Legyen a gömb sugara 1, és m a henger fél magassága. kill(all); (%o0) done v:r^2*%pi*2*m; (%o1) 2*%pi*m*r2 m^2+r^2=1; (%o2) r2+m2=1 v:(1-m^2)*2*%pi*m; (%o3) 2*%pi*m*

1−m2

diff(%,m); (%o4) 2*%pi*

1−m2

−4*%pi*m2
solve(%,m); (%o5) [m=−13,m=13] Feladat Legyen a gömb sugara 1, a kúp magassága 1+d, sugara r. kill(all); (%o0) done v:r^2*%pi*m/3; r^2=(1+d)*(1-d); v:2*(1-d^2)*(1+d)*%pi/3; diff(v,d); (%o1) %pi*m*r23(%o2) r2=

1−d

*

d+1

(%o3) 2*%pi*

d+1

*

1−d2

3
(%o4) 2*%pi*

1−d2

3
−4*%pi*d*

d+1

3
solve(%,d); (%o5) [d=13,d=−1] Feladat Legyen a gömb sugara 1, a kúp magassága 1+d, alkotója a, sugara r. kill(all); (%o0) done f:r^2*%pi+2*r*%pi*a/2; (%o1) %pi*r2+%pi*a*r r^2=1-d^2; a^2=1-d^2+(1+d)^2; (%o2) r2=1−d2(%o3) a2=

d+1

2
−d2+1
ratsimp(%); (%o4) a2=2*d+2 f:%pi*(1-d^2)+%pi*(2*(1+d)**1-d^2)^(1/2); (%o5) %pi*2*

d+1

−d2
+%pi*

1−d2

diff(%,d); (%o6) %pi*

2−2*d

2*2*

d+1

−d2
−2*%pi*d
e:2*%pi*d=%pi*(2-2*d)/2/(2+2*d-d^2)^(1/2); (%o7) 2*%pi*d=%pi*

2−2*d

2*−d2+2*d+2
e:e/%pi; (%o8) 2*d=2−2*d2*−d2+2*d+2 e:e^2; (%o9) 4*d2=

2−2*d

2
4*

−d2+2*d+2

ratsimp(%); (%o10) 4*d2=−d2−2*d+1d2−2*d−2 solve(%); (%o11) [d=−144*

95324+22491728

23
+312*

95324+22491728

13
+49
24*

95324+22491728

16
−−

95324+22491728

13
−78*

95324+22491728

16
144*

95324+22491728

23
+312*

95324+22491728

13
+49
−49144*

95324+22491728

13
+133
2
+12,d=−144*

95324+22491728

23
+312*

95324+22491728

13
+49
24*

95324+22491728

16
+−

95324+22491728

13
−78*

95324+22491728

16
144*

95324+22491728

23
+312*

95324+22491728

13
+49
−49144*

95324+22491728

13
+133
2
+12,d=144*

95324+22491728

23
+312*

95324+22491728

13
+49
24*

95324+22491728

16
−−

95324+22491728

13
+78*

95324+22491728

16
144*

95324+22491728

23
+312*

95324+22491728

13
+49
−49144*

95324+22491728

13
+133
2
+12,d=144*

95324+22491728

23
+312*

95324+22491728

13
+49
24*

95324+22491728

16
+−

95324+22491728

13
+78*

95324+22491728

16
144*

95324+22491728

23
+312*

95324+22491728

13
+49
−49144*

95324+22491728

13
+133
2
+12]
float(%), numer; (%o12) [d=−0.3960391884318551*

−1

0.5
−0.4730659892077556,d=0.3960391884318551*

−1

0.5
−0.4730659892077556,d=0.2429716025576799,d=2.703160375857831]
Feladat etc. Feladat etc. Feladat etc. Feladat etc. Feladat etc. Feladat etc. Feladat kill(all); (%o0) done (1/1) f(x):=x^2-x^4; (%o1) f

x

:=x2−x4
f1:diff(f(x),x); (%o2) 2*x−4*x3 solve(%,x); (%o3) [x=−12,x=12,x=0] f(0); f(2^(-1/2)); f(2); (%o4) 0(%o5) 14(%o6) −12 (1/2) f(x):=x-atan(x); (%o8) f

x

:=x−atan

x

f1:diff(f(x),x); (%o9) 1−1x2+1 f(-1); f(1); (%o10) %pi4−1(%o11) 1−%pi4 (1/3) f(x):=x+exp(-x); (%o12) f

x

:=x+exp

−x

f1:diff(f(x),x); (%o13) 1−%e−x solve(f1); (%o14) [x=0] [f(-1),f(0),f(1)]; (%o15) [%e−1,1,%e−1+1] (2/1) f(x):=x+1/x^2; (%o16) f

x

:=x+1x2
f1:diff(f(x),x); (%o17) 1−2x3 solve(f1,x); (%o18) [x=213*3*%i−2132,x=−213*3*%i+2132,x=213] [f(2^(1/3)),f(1/10),f(10)]; (%o19) [3223,100110,1001100] float(%), numer; (%o20) [1.889881574842309,100.1,10.01] plot2d(f(x),[x,1/10,10])$ (2/2) f(x):=atan(1/x); (%o22) f

x

:=atan

1x

f1:diff(f(x),x); (%o23) −1

1x2+1

*x2
ratsimp(%); (%o24) −1x2+1 solve(%); (%o25) [] [f(1/10),f(10)]; (%o26) [atan

10

,atan

110

]
float(%), numer; (%o27) [1.471127674303734,0.09966865249116204] (2/3) f(x):=cos(x^2); (%o28) f

x

:=cos

x2

f1:diff(f(x),x); (%o29) −2*x*sin

x2

solve(f1,x); solve: using arc−trig functions to get a solution.Some solutions will be lost.(%o30) [x=0] [f(0),f((%pi)^(1/2)),f((2*%pi)^(1/2)), f((3*%pi)^(1/2)),f(%pi)]; (%o31) [1,−1,1,−1,cos

%pi2

]
float(%), numer; (%o32) [1.0,−1.0,1.0,−1.0,−0.9026853619330714] (3/1) f(x):=sin(sin(x)); (%o33) f

x

:=sin

sin

x

A sin szigorúan monoton -π/2 és π/2 között. [f(-%pi/2),f(%pi/2)]; (%o34) [−sin

1

,sin

1

]
float(%), numer; (%o35) [−0.8414709848078965,0.8414709848078965] (3/2) f(x):=x*exp(x); (%o36) f

x

:=x*exp

x

f1:diff(f(x),x); (%o37) x*%ex+%ex solve(f1,x); (%o38) [x=−1] [f(-2),f(-1),f(2)]; (%o39) [−2*%e−2,−%e−1,2*%e2] float(%), numer; (%o40) [−0.2706705664732254,−0.3678794411714423,14.7781121978613] (3/3) assume(n>0); (%o41) [n>0] f(x):=x^n*exp(-x); (%o42) f

x

:=xn*exp

−x

f1:diff(f(x),x); (%o43) n*xn−1*%e−x−xn*%e−x solve(f1,x); (%o44) [xn=n*xn−1] [f(-2*n),f(n),f(2*n)]; (%o45) [nn*

−1

n
*2n*%e2*n,nn*%e−n,nn*%e−2*n*2n]
(4/1) f(x):=x*log(x); (%o46) f

x

:=x*log

x

f1:diff(f(x),x); (%o47) log

x

+1
solve(f1,x); (%o48) [x=%e−1] [f(1/2),f(1/%e),f(2)]; (%o49) [−log

2

2
,−%e−1,2*log

2

]
float(%), numer; (%o50) [−0.3465735902799726,−0.3678794411714423,1.38629436111989] (4/2) f(x):=1/(1+sin(x)^2); (%o51) f

x

:=11+sin

x

2
Az 1/(1+y) szigorúan monoton, ha y pozitív. g:sin(x)^2; (%o52) sin

x

2
g1:diff(g,x); (%o53) 2*cos

x

*sin

x

solve(g1,x); solve: using arc−trig functions to get a solution.Some solutions will be lost.(%o54) [x=0,x=%pi2] [f(%pi/2),f(%pi),f(3*%pi/2)]; (%o55) [12,1,12] (4/3) f(x):=(1-exp(-x^2))^(1/2); (%o56) f

x

:=

1−exp

−x2

12
A négyzetgyök szigorúan monoton pozitív számokra. g:1-exp(-x^2); (%o57) 1−%e−x2 g1:diff(g,x); (%o58) 2*x*%e−x2 solve(%,x); (%o59) [x=0] [f(-2),f(0),f(2)]; (%o60) [1−%e−4,0,1−%e−4] float(%), numer; (%o61) [0.9907998592608226,0.0,0.9907998592608226] (5/1) f(x):=x*sin(log(x)); (%o62) f

x

:=x*sin

log

x

f1:diff(f(x),x); (%o63) sin

log

x

+cos

log

x

solve(f1,x); (%o64) [sin

log

x

=−cos

log

x

]
tan(y)=-1; (%o65) tan

y

=−1
solve(%,y); solve: using arc−trig functions to get a solution.Some solutions will be lost.(%o66) [y=−%pi4] tan(3*%pi/4); (%o67) −1 [exp(3*%pi/4),exp(7*%pi/4)]; (%o68) [%e3*%pi4,%e7*%pi4] float(%), numer; (%o69) [10.55072407419776,244.1510628542749] [f(1),f(exp(3*%pi/4)),f(100)]; (%o70) [0,%e3*%pi42,100*sin

log

100

]
float(%), numer; (%o71) [0.0,7.460488539293395,−99.42575694137898] (5/2) f(x):=x^x; (%o72) f

x

:=xx
Az exp szigorúan monoton. g:x*log(x); (%o73) x*log

x

g1:diff(g,x); (%o74) log

x

+1
solve(g1,x); (%o75) [x=%e−1] f(1/%e); (%o76) %e−%e−1 float(%), numer; (%o77) 0.6922006275553464 (5/3) f(x):=x^(1/x); (%o78) f

x

:=x1x
Az exp szigorúan monoton. g:log(x)/x; (%o79) log

x

x
g1:diff(g,x); (%o80) 1x2−log

x

x2
solve(%,x); (%o81) [x=%e] f(%e); (%o82) %e%e−1 float(%), numer; (%o83) 1.444667861009766 (6/1) f(x):=log(x)/x; (%o84) f

x

:=log

x

x
f1:diff(f(x),x); (%o85) 1x2−log

x

x2
solve(%,x); (%o86) [x=%e] f(%e); (%o87) %e−1 float(%), numer; (%o88) 0.3678794411714423 (6/2) f(x):=x*log(x); (%o89) f

x

:=x*log

x

f1:diff(f(x),x); (%o90) log

x

+1
solve(%,x); (%o91) [x=%e−1] f(1/%e); (%o92) −%e−1 float(%), numer; (%o93) −0.3678794411714423 (6/3) f(x):=x^x*(1-x)^(1-x); (%o94) f

x

:=xx*

1−x

1−x
Az exp szigorúan monoton. g:x*log(x)+(1-x)*log(1-x); (%o95) x*log

x

+log

1−x

*

1−x

g1:diff(%,x); (%o96) log

x

−log

1−x

solve(g1,x); (%o97) [log

x

=log

1−x

]
f(1/2); (%o98) 12 (7) etc. (8) etc. Feladat kill(all); (%o0) done assume(n>0); (%o1) [n>0] f:x^(2*n)*exp(-x^2); (%o2) x2*n*%e−x2 solve(f,x); (%o3) [x=0] f1:diff(f,x); (%o4) 2*n*x2*n−1*%e−x2−2*x2*n+1*%e−x2 solve(f1,x); (%o5) [x2*n+1=n*x2*n−1] f2:diff(f1,x); (%o6) 4*x2*n+2*%e−x2+2*n*

2*n−1

*x2*n−2*%e−x2−2*

2*n+1

*x2*n*%e−x2−4*n*x2*n*%e−x2
solve(f2,x); (%o7) [xn=−2*x2*n+2+

2*n2−n

*x2*n−2
4*n+1
,xn=2*x2*n+2+

2*n2−n

*x2*n−2
4*n+1
]
n:2; (%o8) 2 plot2d([f,f1,f2],[x,-4,4])$ Feladat kill(all); (%o0) done (1/1) f:x^3-3*x; (%o1) x3−3*x f1:diff(f,x); (%o2) 3*x2−3 solve(f1,x); (%o3) [x=−1,x=1] f2:diff(f1,x); (%o4) 6*x solve(f2,x); (%o5) [x=0] plot2d([f,f1,f2],[x,-4,4])$ (1/2) f:x^2-x^4; (%o7) x2−x4 solve(f,x); (%o8) [x=−1,x=1,x=0] f1:diff(f,x); (%o9) 2*x−4*x3 solve(f1,x); (%o10) [x=−12,x=12,x=0] f2:diff(f1,x); (%o11) 2−12*x2 solve(f2,x); (%o12) [x=−16,x=16] plot2d([f,f1,f2],[x,-2,2])$ (1/3) f:x-atan(x); (%o14) x−atan

x

solve(f,x); (%o15) [x=atan

x

]
f1:diff(f,x); (%o16) 1−1x2+1 solve(f1,x); (%o17) [x=0] f2:diff(f1,x); (%o18) 2*x

x2+1

2
solve(f2,x); (%o19) [x=0] plot2d([f,f1,f2],[x,-2,2])$ (2/1) f:x+exp(-x); (%o21) %e−x+x solve(f,x); (%o22) [x=−%e−x] f1:diff(f,x); (%o23) 1−%e−x solve(f1,x); (%o24) [x=0] f2:diff(f1,x); (%o25) %e−x solve(f2,x); (%o26) [] plot2d([f,f1,f2],[x,-2,2])$ (2/2) f:x+1/x^2; (%o28) x+1x2 solve(f,x); (%o29) [x=−3*%i−12,x=3*%i+12,x=−1] f1:diff(f,x); (%o30) 1−2x3 solve(f1,x); (%o31) [x=213*3*%i−2132,x=−213*3*%i+2132,x=213] f2:diff(f1,x); (%o32) 6x4 solve(f2,x); (%o33) [] plot2d([f,f1,f2],[x,-4,4],[y,-10,10])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped. (2/3) f:atan(1/x); (%o35) atan

1x

solve(f,x); (%o36) [] f1:diff(f,x); (%o37) −1

1x2+1

*x2
f1:ratsimp(f1); (%o38) −1x2+1 solve(f1,x); (%o39) [] f2:diff(f1,x); (%o40) 2*x

x2+1

2
solve(f2,x); (%o41) [x=0] plot2d([f,f1,f2],[x,-4,4])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range. (3/1) f:cos(x^2); (%o43) cos

x2

solve(f,x); solve: using arc−trig functions to get a solution.Some solutions will be lost.(%o44) [x=−%pi2,x=%pi2] f1:diff(f,x); (%o45) −2*x*sin

x2

solve(f1,x); solve: using arc−trig functions to get a solution.Some solutions will be lost.(%o46) [x=0] f2:diff(f1,x); (%o47) −2*sin

x2

−4*x2*cos

x2

solve(f2,x); (%o48) [x2=−sin

x2

2*cos

x2

]
plot2d([f,f1,f2],[x,-4,4])$ (3/2) f:sin(sin(x)); (%o50) sin

sin

x

solve(f,x); solve: using arc−trig functions to get a solution.Some solutions will be lost.(%o51) [x=0] f1:diff(f,x); (%o52) cos

x

*cos

sin

x

solve(f1,x); solve: using arc−trig functions to get a solution.Some solutions will be lost.(%o53) [x=asin

%pi2

,x=%pi2]
f2:diff(f1,x); (%o54) −cos

x

2
*sin

sin

x

−sin

x

*cos

sin

x

solve(f2,x); (%o55) [sin

sin

x

=−sin

x

*cos

sin

x

cos

x

2
]
plot2d([f,f1,f2],[x,-4,4])$ (3/3) f:sin(1/x); (%o57) sin

1x

solve(f,x); solve: using arc−trig functions to get a solution.Some solutions will be lost.(%o58) [] f1:diff(f,x); (%o59) −cos

1x

x2
solve(f1,x); solve: using arc−trig functions to get a solution.Some solutions will be lost.(%o60) [x=2%pi] f2:diff(f1,x); (%o61) 2*cos

1x

x3
−sin

1x

x4
plot2d([f,f1,f2],[x,-4,4],[y,-4,4])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped. (4/1) f:x*exp(-x); (%o63) x*%e−x solve(f,x); (%o64) [x=0] f1:diff(f,x); (%o65) %e−x−x*%e−x solve(f1,x); (%o66) [x=1] f2:diff(f1,x); (%o67) x*%e−x−2*%e−x solve(f2,x); (%o68) [x=2] plot2d([f,f1,f2],[x,-4,4],[y,-4,4])$ plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped. (4/2) f:x^n*exp(-x); (%o70) xn*%e−x solve(f,x); (%o71) [x=0] f1:diff(f,x); (%o72) n*xn−1*%e−x−xn*%e−x solve(f1,x); (%o73) [xn=n*xn−1] f2:diff(f1,x); (%o74) xn*%e−x−2*n*xn−1*%e−x+

n−1

*n*xn−2*%e−x
solve(f2,x); (%o75) [xn=2*n*xn−1+

n−n2

*xn−2]
n:2; (%o76) 2 plot2d([f,f1,f2],[x,-4,4],[y,-4,4])$ plot2d: some values were clipped.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped. (4/3) f:x*log(x); (%o78) x*log

x

solve(f,x); (%o79) [x=1,x=0] f1:diff(f,x); (%o80) log

x

+1
solve(f1,x); (%o81) [x=%e−1] f2:diff(f1,x); (%o82) 1x f2:diff(f1,x); (%o83) 1x plot2d([f,f1,f2],[x,0,4],[y,-4,4])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped. (5/1) f:1/(1+sin(x)^2); (%o85) 1sin

x

2
+1
solve(f,x); (%o86) [] f1:diff(f,x); (%o87) −2*cos

x

*sin

x

sin

x

2
+1

2
solve(f1,x); solve: using arc−trig functions to get a solution.Some solutions will be lost.(%o88) [x=0,x=%pi2] f2:diff(f1,x); (%o89) 2*sin

x

2

sin

x

2
+1

2
−2*cos

x

2

sin

x

2
+1

2
+8*cos

x

2
*sin

x

2

sin

x

2
+1

3
solve(f2,x); (%o90) [cos

x

=−9*cos

x

4
+10*cos

x

2
+1
−1
3
,cos

x

=9*cos

x

4
+10*cos

x

2
+1
−1
3
,cos

x

=−%i*9*cos

x

4
+10*cos

x

2
+1
+1
3
,cos

x

=%i*9*cos

x

4
+10*cos

x

2
+1
+1
3
]
plot2d([f,f1,f2],[x,-4,4],[y,-4,4])$ (5/2) f:(1+1/x)^x; (%o92)

1x+1

x
solve(f,x); (%o93) [

x+1x

x
=0]
f1:diff(f,x); (%o94)

1x+1

x
*

log

1x+1

−1

1x+1

*x

solve(f1,x); (%o95) [x=−log

x+1x

−1
log

x+1x

,

x+1x

x
=0]
f2:diff(f1,x); (%o96)

1x+1

x
*

log

1x+1

−1

1x+1

*x

2
−

1x+1

x−2
x3
solve(f2,x); (%o97) [0=x4*

x+1x

x
*

2*log

x+1x

2−2*log

x+1x

+x3*

x+1x

x
*

log

x+1x

2−2*log

x+1x

+1

+x5*

x+1x

x
*log

x+1x

2
−x2*

x+1x

x−2
−2*x*

x+1x

x−2
−

x+1x

x−2
]
plot2d([f,f1,f2],[x,0,8],[y,-4,4])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped. (5/3) f:(1+1/x)^(x+1); (%o99)

1x+1

x+1
solve(f,x); (%o100) [

x+1x

x+1
=0]
f1:diff(f,x); (%o101)

1x+1

x+1
*

log

1x+1

−x+1

1x+1

*x2

ratsimp(%); (%o102)

x+1x

x
*

x2+x

*log

x+1x

+

−x−1

*

x+1x

x
x2
solve(f1,x); (%o103) [x=1log

x+1x

,

x+1x

x+1
=0]
f2:diff(f1,x); (%o104)

1x+1

x+1
*

log

1x+1

−x+1

1x+1

*x2

2
+

1x+1

x+1
*

2*

x+1

1x+1

*x3
−x+1

1x+1

2
*x4
−2

1x+1

*x2

ratsimp(%); (%o105)

x+1x

x
*

x3+x2

*log

x+1x

2
+

x+1x

x
*

−2*x2−2*x

*log

x+1x

+

x+1x

x
*

x+2

x3
solve(f2,x); (%o110) [x=

−12−3*%i2

*

4*log

x+1x

3
+12*log

x+1x

2
+39*log

x+1x

+4log

x+1x

2*332*log

x+1x

2
+−

log

x+1x

−2

*

2*log

x+1x

−1

log

x+1x

*log

x+1x

2
−6log

x+1x

2
6+

−1

*

log

x+1x

−2

327*log

x+1x

3

13−

3*%i2+−12

*

−1

*

log

x+1x

−2

29*log

x+1x

2
−2*log

x+1x

−1
3*log

x+1x

2

4*log

x+1x

3
+12*log

x+1x

2
+39*log

x+1x

+4
log

x+1x

2*332*log

x+1x

2
+−

log

x+1x

−2

*

2*log

x+1x

−1

log

x+1x

*log

x+1x

2
−6log

x+1x

2
6+

−1

*

log

x+1x

−2

327*log

x+1x

3

13+

−1

*

log

x+1x

−2

3*log

x+1x

,x=

3*%i2+−12

*

4*log

x+1x

3
+12*log

x+1x

2
+39*log

x+1x

+4log

x+1x

2*332*log

x+1x

2
+−

log

x+1x

−2

*

2*log

x+1x

−1

log

x+1x

*log

x+1x

2
−6log

x+1x

2
6+

−1

*

log

x+1x

−2

327*log

x+1x

3

13−

−12−3*%i2

*

−1

*

log

x+1x

−2

29*log

x+1x

2
−2*log

x+1x

−1
3*log

x+1x

2

4*log

x+1x

3
+12*log

x+1x

2
+39*log

x+1x

+4
log

x+1x

2*332*log

x+1x

2
+−

log

x+1x

−2

*

2*log

x+1x

−1

log

x+1x

*log

x+1x

2
−6log

x+1x

2
6+

−1

*

log

x+1x

−2

327*log

x+1x

3

13+

−1

*

log

x+1x

−2

3*log

x+1x

,x=

4*log

x+1x

3
+12*log

x+1x

2
+39*log

x+1x

+4log

x+1x

2*332*log

x+1x

2
+−

log

x+1x

−2

*

2*log

x+1x

−1

log

x+1x

*log

x+1x

2
−6log

x+1x

2
6+

−1

*

log

x+1x

−2

327*log

x+1x

3

13−

−1

*

log

x+1x

−2

2
9*log

x+1x

2
−2*log

x+1x

−1
3*log

x+1x

2

4*log

x+1x

3
+12*log

x+1x

2
+39*log

x+1x

+4
log

x+1x

2*332*log

x+1x

2
+−

log

x+1x

−2

*

2*log

x+1x

−1

log

x+1x

*log

x+1x

2
−6log

x+1x

2
6+

−1

*

log

x+1x

−2

327*log

x+1x

3

13+

−1

*

log

x+1x

−2

3*log

x+1x

,

x+1x

x+1
=0]
plot2d([f,f1,f2],[x,0,8],[y,-4,4])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped. (6/1) declare(x,real); (%o112) done f:(1-exp(-x^2))^(1/2); (%o113) 1−%e−x2 solve(f,x); (%o114) [%e−x22*%ex2−1=0] solve(exp(-x^2)=1,x); (%o115) [x=0] f1:diff(f,x); (%o116) x*%e−x21−%e−x2 limit(f/x,x,0); (%o117) und limit(f/x,x,0,plus); (%o118) 1 limit(f/x,x,0,minus); (%o119) −1 solve(f1,x); (%o120) [x=0] limit(f1,x,0); (%o121) und limit(f1,x,0,plus); (%o122) 1 limit(f1,x,0,minus); (%o123) −1 f2:diff(f1,x); (%o124) −2*x2*%e−x21−%e−x2+%e−x21−%e−x2−x2*%e−2*x2

1−%e−x2

32
solve(f2,x); (%o125) [x2=%ex2−12*%ex2−1] plot2d([f,f1,f2],[x,-4,4])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: expression evaluates to non−numeric value somewhere in plotting range. (6/2) assume(x>0); (%o127) [x>0] f:x*sin(log(x)); (%o128) x*sin

log

x

solve(f,x); solve: using arc−trig functions to get a solution.Some solutions will be lost.(%o129) [x=1,x=0] f1:diff(f,x); (%o130) sin

log

x

+cos

log

x

solve(f1,x); (%o131) [sin

log

x

=−cos

log

x

]
solve(tan(y)=-1); solve: using arc−trig functions to get a solution.Some solutions will be lost.(%o132) [y=−%pi4] f2:diff(f1,x); (%o133) cos

log

x

x
−sin

log

x

x
solve(f2,x); (%o134) [sin

log

x

=cos

log

x

]
solve(tan(y),y); solve: using arc−trig functions to get a solution.Some solutions will be lost.(%o135) [y=0] plot2d([f,f1,f2],[x,0,30],[y,-4,4])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped. (6/3) assume(x>0); (%o137) [redundant] f:x^x; (%o138) xx solve(f,x); (%o139) [xx=0] f1:diff(f,x); (%o140) xx*

log

x

+1

solve(f1,x); (%o141) [x=%e−1,xx=0] f2:diff(f1,x); (%o142) xx*

log

x

+1

2
+xx−1
solve(f2,x); (%o143) [log

x

=−xx+%i*xx−12xx,log

x

=−xx−%i*xx−12xx]
plot2d([f,f1,f2],[x,0,4],[y,-4,4])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped. (7/1) f:x^(1/x); (%o145) x1x solve(f,x); (%o146) [x1x=0] f1:diff(f,x); (%o147) x1x*

1x2−log

x

x2

solve(f1,x); (%o148) [x=%e,1x2*x−1x=0] f2:diff(f1,x); (%o149) x1x*

1x2−log

x

x2

2
+x1x*

2*log

x

x3−3x3

solve(f2,x); (%o150) [x=−log

x

2
−2*log

x

+1
2*log

x

−3
,1x4*x−1x=0]
plot2d([f,f1,f2],[x,0,4],[y,-4,4])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped. (7/2) f:log(x)/x; (%o152) log

x

x
solve(f,x); (%o153) [x=1] f1:diff(f,x); (%o154) 1x2−log

x

x2
solve(f1,x); (%o155) [x=%e] f2:diff(f1,x); (%o156) 2*log

x

x3
−3x3
solve(f2,x); (%o157) [x=%e32] plot2d([f,f1,f2],[x,0,8],[y,-4,4])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.Message from maxima's stderr stream: (gnuplot:31110): GLib−CRITICAL **: 09:42:03.482: Source ID 20264 was not found when attempting to remove it (7/3) f:x*log(x); (%o159) x*log

x

solve(f,x); (%o160) [x=1,x=0] f1:diff(f,x); (%o161) log

x

+1
solve(f1,x); (%o162) [x=%e−1] f2:diff(f1,x); (%o163) 1x solve(f2,x); (%o164) [] plot2d([f,f1,f2],[x,0,8],[y,-4,4])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped. (8/1) f:x^x*(1-x)^(1-x); (%o200)

1−x

1−x
*xx
solve(f,x); (%o201) [xx=0,

1−x

1−x
=0]
f1:diff(f,x); (%o207)

1−x

1−x
*xx*

log

x

+1

+

−log

1−x

−1

*

1−x

1−x
*xx
f1:ratsimp(%); (%o209) −

x−1

*xx*log

x

+xx*

log

1−x

−log

1−x

*x

1−x

x
solve(f1,x); (%o210) [x=1,log

x

=log

1−x

,xx=0]
f2:diff(f1,x); (%o211) −

x−1

*xx*log

x

*

log

x

+1

+xx*

log

1−x

−log

1−x

*x

*

log

x

+1

+xx*log

x

+

x−1

*xx−1+xx*

x1−x−11−x−log

1−x

1−x

x
−

x1−x−log

1−x

*

x−1

*xx*log

x

+xx*

log

1−x

−log

1−x

*x

1−x

x
f2:ratsimp(%); (%o213) −xx*

x2−x

*log

x

2
+xx*

2*log

1−x

*x−2*log

1−x

*x2

*log

x

+xx*

log

1−x

2*x2−log

1−x

2
*x−1

1−x

x
*x
solve(f2,x); (%o214) [x=−log

x

2
−2*log

1−x

*log

x

+log

1−x

2
+4
−log

x

+log

1−x

2*log

x

−2*log

1−x

,x=log

x

2
−2*log

1−x

*log

x

+log

1−x

2
+4
+log

x

−log

1−x

2*log

x

−2*log

1−x

,xx−1=0]
plot2d([f,f1,f2],[x,0,1],[y,-4,4])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped. (8/2) f:atan(x)-log(1+x^2)/2; (%o215) atan

x

−log

x2+1

2
solve(f,x); (%o216) [atan

x

=log

x2+1

2
]
f1:diff(f,x); (%o217) 1x2+1−xx2+1 solve(f1,x); (%o218) [x=1] f2:diff(f1,x); (%o219) −1x2+1+2*x2

x2+1

2
−2*x

x2+1

2
solve(f2,x); (%o220) [x=1−2,x=2+1] plot2d([f,f1,f2],[x,-4,4])$ (8/3) f:atan(x)-x/(1+x); (%o234) atan

x

−xx+1
solve(f,x); (%o235) [x=−atan

x

atan

x

−1
]
f1:diff(f,x); (%o236) 1x2+1−1x+1+x

x+1

2
solve(f1,x); (%o237) [x=0] f2:diff(f1,x); (%o238) −2*x

x2+1

2
+2

x+1

2
−2*x

x+1

3
solve(f2,x); (%o239) [x=−8*

3*%i2+−12

81*

2*19372+134729

13
+

2*19372+134729

13
*

−12−3*%i2

−19,x=

2*19372+134729

13
*

3*%i2+−12

−8*

−12−3*%i2

81*

2*19372+134729

13
−19,x=

2*19372+134729

13
−881*

2*19372+134729

13
−19]
plot2d([f,f1,f2],[x,-4,4],[y,-8,8])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped. (9/1) f:x^4/(1+x)^3; (%o243) x4

x+1

3
solve(f,x); (%o244) [x=0] f1:diff(f,x); (%o245) 4*x3

x+1

3
−3*x4

x+1

4
solve(f1,x); (%o246) [x=−4,x=0] f2:diff(f1,x); (%o247) 12*x2

x+1

3
−24*x3

x+1

4
+12*x4

x+1

5
solve(f2,x); (%o248) [x=0] plot2d([f,f1,f2],[x,-12,12],[y,-12,12])$ plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped. (9/2) f:exp(x)/(1+x); (%o1) %exx+1 solve(f,x); (%o2) [] f1:diff(f,x); (%o3) %exx+1−%ex

x+1

2
solve(f1,x); (%o4) [x=0] f2:diff(f1,x); (%o5) %exx+1−2*%ex

x+1

2
+2*%ex

x+1

3
solve(f2,x); (%o6) [x=−%i,x=%i] plot2d([f,f1,f2],[x,-4,4],[y,-4,4])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped. (9/3) f:exp(x)/sinh(x); (%o8) %exsinh

x

solve(f,x); (%o9) [] f1:diff(f,x); (%o10) %exsinh

x

−%ex*cosh

x

sinh

x

2
solve(f1,x); (%o11) [sinh

x

=cosh

x

]
f2:diff(f1,x); (%o12) 2*%ex*cosh

x

2
sinh

x

3
−2*%ex*cosh

x

sinh

x

2
solve(f2,x); solve: using arc−trig functions to get a solution.Some solutions will be lost.(%o13) [x=acosh

0

,sinh

x

=cosh

x

]
plot2d([f,f1,f2],[x,-4,4],[y,-4,4])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped. (10/1) f:x^(1/log(x)); (%o15) x1log

x

solve(f,x); (%o16) [x1log

x

=0]
f1:diff(f,x); (%o17) 0 solve(f1,x); (%o18) all f2:diff(f1,x); (%o19) 0 solve(f2,x); (%o20) all plot2d([f,f1,f2],[x,-4,4],[y,-4,4])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range. (10/2) f:exp(-x)*((1-x^2)*sin(x)-(1+x^2)*cos(x))/2; (%o1) %e−x*

1−x2

*sin

x

−

x2+1

*cos

x

2
solve(f,x); (%o2) [x2=sin

x

−cos

x

sin

x

+cos

x

]
f1:diff(f,x); (%o3) %e−x*

x2+1

*sin

x

−2*x*sin

x

+

1−x2

*cos

x

−2*x*cos

x

2
−%e−x*

1−x2

*sin

x

−

x2+1

*cos

x

2
f1:ratsimp(%); (%o4) %e−x*

x2−x

*sin

x

+

1−x

*cos

x

solve(f1,x); (%o5) [x=1,x=cos

x

sin

x

]
f2:diff(f1,x); (%o6) %e−x*

2*x−1

*sin

x

−

1−x

*sin

x

+

x2−x

*cos

x

−cos

x

−%e−x*

x2−x

*sin

x

+

1−x

*cos

x

f2:ratsimp(%); (%o7) −%e−x*

x2−4*x+2

*sin

x

+

2−x2

*cos

x

solve(f2,x); (%o8) [x=−2*sin

x

2
+cos

x

2
−2*sin

x

sin

x

−cos

x

,x=2*sin

x

2
+cos

x

2
+2*sin

x

sin

x

−cos

x

]
trigsimp(%); (%o9) [x=2*sin

x

−2
sin

x

−cos

x

,x=2*sin

x

+2
sin

x

−cos

x

]
plot2d([f,f1,f2],[x,-5,5],[y,-12,12])$ plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped. (10/3) assume(n>0); (%o14) [n>0] f:x^(2*n+1)*exp(-x^2); (%o13) x2*n+1*%e−x2 solve(f,x); (%o15) [x=0] f1:diff(f,x); (%o16)

2*n+1

*x2*n*%e−x2−2*x2*n+2*%e−x2
solve(f1,x); (%o17) [xn=−2*x2*n+22*n+1,xn=2*x2*n+22*n+1] f2:diff(f1,x); (%o18) 4*x2*n+3*%e−x2−2*

2*n+2

*x2*n+1*%e−x2−2*

2*n+1

*x2*n+1*%e−x2+2*n*

2*n+1

*x2*n−1*%e−x2
solve(f2,x); (%o19) [x2*n+3=

4*n+3

*x2*n+1+

−2*n2−n

*x2*n−1
2
]
n:1; (%o20) 1 plot2d([f,f1,f2],[x,-4,4])$ (11-13) etc. Feladat -exp(-1/abs(x)); sgn(x)*exp(-1/abs(x)); (%o72) −%e−1x(%o73) %e−1x*sgn

x

Feladat f:sin(x)/x; (%o292) sin

x

x
limit(f,x,0,plus); (%o297) 1 limit(f,x,%pi/2,minus); (%o298) 2%pi f1:diff(f,x); (%o293) cos

x

x
−sin

x

x2
solve(f1,x); (%o294) [x=sin

x

cos

x

]
tan(x)-x; (%o295) tan

x

−x
diff(%,x); (%o296) sec

x

2
−1
Feladat Ha a a másik befogó, v a sebesség, akkor az idő (a-x)/v+3/2/v*sqrt(b^2+x^2); (%o1) 3*x2+b22*v+a−xv azaz f:a-x+3/2*sqrt(b^2+x^2); (%o11) 3*x2+b22−x+a minimuma kell: 0=diff(f,x); (%o12) 0=3*x2*x2+b2−1 %+1; (%o13) 1=3*x2*x2+b2 %*2*sqrt(b^2+x^2); (%o14) 2*x2+b2=3*x eq:%^2; (%o15) 4*

x2+b2

=9*x2
solve(eq,x); (%o16) [x=−2*b5,x=2*b5] Feladat Legyen a törő egyenes az x tengely, a fénysugár induljon az A1=(x1,y1), y1 0 pontból és érkezzen az A2=(x2,y2), y2 0 pontba. A felső közegben sebessége v1, az alsóban v2. A töréspont P=(x,0). Legyen B1=(x1,0) és B2=(x2,0). Feltehetjük, hogy x1 x2. Legyen a B1-A1-P szög α1, a P-A2-B2 szög α2. Az idő t:((x-x1)^2+y1^2)^(1/2)/v1+ ((x2-x)^2+y2^2)^(1/2)/v2; (%o28) y22+

x2−x

2
v2
+y12+

x−x1

2
v1
Ha abs(x) nagy, t nagy, de van minimum, itt a derivált 0. t1:diff(t,x); (%o29) x−x1v1*y12+

x−x1

2
−x2−xv2*y22+

x2−x

2
A t1=0 egyenlet ekvivalens a következővel: sin(α1)/v1=sin(α2)/v2; (%o30) sin

α1

v1
=sin

α2

v2
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