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Trazado el gráfico de la función/ sin(3x/2)+ctg(4x/3)

Gráfico de la función y = sin(3x/2)+ctg(4x/3)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          /3*x\      /4*x\
f(x) = sin|---| + cot|---|
          \ 2 /      \ 3 /
f(x)=sin⁡(3x2)+cot⁡(4x3)f{\left(x \right)} = \sin{\left(\frac{3 x}{2} \right)} + \cot{\left(\frac{4 x}{3} \right)} 
f = sin((3*x)/2) + cot((4*x)/3)
Gráfico de la función
02468-8-6-4-2-1010-250250
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
sin⁡(3x2)+cot⁡(4x3)=0\sin{\left(\frac{3 x}{2} \right)} + \cot{\left(\frac{4 x}{3} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=64.8645090516301x_{1} = 64.8645090516301
x2=−36.0841441552738x_{2} = -36.0841441552738
x3=52.6854009807195x_{3} = 52.6854009807195
x4=18.1925937148376x_{4} = 18.1925937148376
x5=−6.09461541278416x_{5} = -6.09461541278416
x6=7.72845440271755x_{6} = 7.72845440271755
x7=2.95941847872391x_{7} = 2.95941847872391
x8=−67.6697692834375x_{8} = -67.6697692834375
x9=94.9047418143949x_{9} = 94.9047418143949
x10=19.5065181282399x_{10} = 19.5065181282399
x11=−13.5462004162692x_{11} = -13.5462004162692
x12=72.4388052074311x_{12} = 72.4388052074311
x13=−43.7937272558617x_{13} = -43.7937272558617
x14=78.357642164879x_{14} = 78.357642164879
x15=36.0841441552738x_{15} = 36.0841441552738
x16=98.1110463915906x_{16} = 98.1110463915906
x17=−73.7832559983513x_{17} = -73.7832559983513
x18=55.8917055579152x_{18} = 55.8917055579152
x19=73.7832559983513x_{19} = 73.7832559983513
x20=83.1266780888726x_{20} = 83.1266780888726
x21=−78.357642164879x_{21} = -78.357642164879
x22=34.7396933643536x_{22} = 34.7396933643536
x23=−27.1653972085526x_{23} = -27.1653972085526
x24=69.3036082733709x_{24} = 69.3036082733709
x25=−77.0131913739588x_{25} = -77.0131913739588
x26=39.3140795308812x_{26} = 39.3140795308812
x27=−52.6854009807195x_{27} = -52.6854009807195
x28=85.9319383206799x_{28} = 85.9319383206799
x29=61.8520232698858x_{29} = 61.8520232698858
x30=−55.8917055579152x_{30} = -55.8917055579152
x31=−29.97065744036x_{31} = -29.97065744036
x32=40.6585303218014x_{32} = 40.6585303218014
x33=−69.3036082733709x_{33} = -69.3036082733709
x34=−22.7128227054356x_{34} = -22.7128227054356
x35=31.6044964302934x_{35} = 31.6044964302934
x36=60.4119345485131x_{36} = 60.4119345485131
x37=22.7128227054356x_{37} = 22.7128227054356
x38=−7.72845440271755x_{38} = -7.72845440271755
x39=−39.3140795308812x_{39} = -39.3140795308812
x40=−1.61496768780373x_{40} = -1.61496768780373
x41=6.09461541278416x_{41} = 6.09461541278416
x42=−51.2453122593467x_{42} = -51.2453122593467
x43=77.0131913739588x_{43} = 77.0131913739588
x44=93.5908174009927x_{44} = 93.5908174009927
x45=48.2328264776024x_{45} = 48.2328264776024
x46=−85.9319383206799x_{46} = -85.9319383206799
x47=−45.4275662457951x_{47} = -45.4275662457951
x48=1.61496768780373x_{48} = 1.61496768780373
x49=90.384512823797x_{49} = 90.384512823797
x50=−19.5065181282399x_{50} = -19.5065181282399
x51=81.4928390989392x_{51} = 81.4928390989392
x52=24.1529114268083x_{52} = 24.1529114268083
x53=−24.1529114268083x_{53} = -24.1529114268083
x54=−10.5337146345249x_{54} = -10.5337146345249
x55=−98.1110463915906x_{55} = -98.1110463915906
x56=−93.5908174009927x_{56} = -93.5908174009927
x57=−61.8520232698858x_{57} = -61.8520232698858
x58=14.9862891376419x_{58} = 14.9862891376419
x59=−81.4928390989392x_{59} = -81.4928390989392
x60=−72.4388052074311x_{60} = -72.4388052074311
x61=67.6697692834375x_{61} = 67.6697692834375
x62=10.5337146345249x_{62} = 10.5337146345249
x63=−90.384512823797x_{63} = -90.384512823797
x64=−99.5511351129633x_{64} = -99.5511351129633
x65=−14.9862891376419x_{65} = -14.9862891376419
x66=−88.9444241024243x_{66} = -88.9444241024243
x67=−60.4119345485131x_{67} = -60.4119345485131
x68=−31.6044964302934x_{68} = -31.6044964302934
x69=43.7937272558617x_{69} = 43.7937272558617
x70=−48.2328264776024x_{70} = -48.2328264776024
x71=−64.8645090516301x_{71} = -64.8645090516301
x72=51.2453122593467x_{72} = 51.2453122593467
x73=−18.1925937148376x_{73} = -18.1925937148376
x74=29.97065744036x_{74} = 29.97065744036
x75=27.1653972085526x_{75} = 27.1653972085526
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
3cos⁡(3x2)2−4cot⁡2(4x3)3−43=0\frac{3 \cos{\left(\frac{3 x}{2} \right)}}{2} - \frac{4 \cot^{2}{\left(\frac{4 x}{3} \right)}}{3} - \frac{4}{3} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=293.484286571338x_{1} = 293.484286571338
x2=100.237393538035x_{2} = 100.237393538035
x3=83.5068318594376x_{3} = 83.5068318594376
x4=−83.5068318594376x_{4} = -83.5068318594376
x5=12.7481910272038x_{5} = 12.7481910272038
x6=8.10860817328255x_{6} = 8.10860817328255
x7=62.6500326589512x_{7} = 62.6500326589512
x8=−1063.68373977945x_{8} = -1063.68373977945
x9=−50.5590538342749x_{9} = -50.5590538342749
x10=62.5382816949577x_{10} = 62.5382816949577
x11=−352.151948578895x_{11} = -352.151948578895
x12=12.8599419911974x_{12} = 12.8599419911974
x13=50.4473028702814x_{13} = 50.4473028702814
x14=46.1859120311904x_{14} = 46.1859120311904
x15=−83.8850238742679x_{15} = -83.8850238742679
x16=−88.1464147133589x_{16} = -88.1464147133589
x17=158.905055545593x_{17} = 158.905055545593
x18=88.1464147133589x_{18} = 88.1464147133589
x19=83.8850238742679x_{19} = 83.8850238742679
x20=−24.9509208158737x_{20} = -24.9509208158737
x21=66.9114234980422x_{21} = 66.9114234980422
x22=67.2896155128725x_{22} = 67.2896155128725
x23=252533.602045749x_{23} = 252533.602045749
x24=24.8391698518802x_{24} = 24.8391698518802
x25=−104.98872735595x_{25} = -104.98872735595
x26=−88.2581656773524x_{26} = -88.2581656773524
x27=−24.8391698518802x_{27} = -24.8391698518802
x28=−4645.47755688665x_{28} = -4645.47755688665
x29=−67.2896155128725x_{29} = -67.2896155128725
x30=−137.936505381113x_{30} = -137.936505381113
x31=−29.590503669795x_{31} = -29.590503669795
x32=−100.237393538035x_{32} = -100.237393538035
x33=−12.8599419911974x_{33} = -12.8599419911974
x34=24.9509208158737x_{34} = 24.9509208158737
x35=50.5590538342749x_{35} = 50.5590538342749
x36=29.2123116549653x_{36} = 29.2123116549653
x37=−8.10860817328255x_{37} = -8.10860817328255
x38=−50.4473028702814x_{38} = -50.4473028702814
x39=29.590503669795x_{39} = 29.590503669795
x40=−12.7481910272038x_{40} = -12.7481910272038
x41=−62.5382816949577x_{41} = -62.5382816949577
x42=−4272.74782929496x_{42} = -4272.74782929496
x43=−45.8077200163601x_{43} = -45.8077200163601
x44=88.2581656773524x_{44} = 88.2581656773524
Signos de extremos en los puntos:
(293.4842865713376, 0.206396274705096)

(100.23739353803519, -0.559207809241328)

(83.5068318594376, -0.206396274705078)

(-83.5068318594376, 0.206396274705078)

(12.748191027203838, 0.558185315871553)

(8.108608173282551, -0.206396274705081)

(62.650032658951204, -0.558185315871546)

(-1063.6837397794532, 0.206396274705187)

(-50.55905383427488, -0.559207809241328)

(62.538281694957675, -0.559207809241335)

(-352.15194857889503, -0.559207809241405)

(12.85994199119736, 0.559207809241335)

(50.447302870281355, 0.558185315871553)

(46.185912031190405, -0.168468423313238)

(-83.88502387426792, 0.168468423313237)

(-88.14641471335888, -0.558185315871558)

(158.90505554559303, -0.206396274705073)

(88.14641471335888, 0.558185315871558)

(83.88502387426792, -0.168468423313237)

(-24.950920815873683, 0.558185315871553)

(66.91142349804215, 0.168468423313237)

(67.28961551287249, 0.206396274705076)

(252533.60204574908, -0.55818531585053)

(24.839169851880158, -0.559207809241335)

(-104.98872735595, -0.206396274705083)

(-88.2581656773524, -0.559207809241335)

(-24.839169851880158, 0.559207809241335)

(-4645.477556886648, 0.168468423312564)

(-67.28961551287249, -0.206396274705076)

(-137.9365053811127, 0.559207809241347)

(-29.590503669794966, -0.206396274705084)

(-100.23739353803519, 0.559207809241328)

(-12.85994199119736, -0.559207809241335)

(24.950920815873683, -0.558185315871553)

(50.55905383427488, 0.559207809241328)

(29.2123116549653, 0.16846842331324)

(-8.108608173282551, 0.206396274705081)

(-50.447302870281355, -0.558185315871553)

(29.590503669794966, 0.206396274705084)

(-12.748191027203838, -0.558185315871553)

(-62.538281694957675, 0.559207809241335)

(-4272.747829294964, -0.558185315872208)

(-45.80772001636007, 0.206396274705077)

(88.2581656773524, 0.559207809241335)


Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
x1=100.237393538035x_{1} = 100.237393538035
x2=83.5068318594376x_{2} = 83.5068318594376
x3=12.7481910272038x_{3} = 12.7481910272038
x4=8.10860817328255x_{4} = 8.10860817328255
x5=−50.5590538342749x_{5} = -50.5590538342749
x6=62.5382816949577x_{6} = 62.5382816949577
x7=−352.151948578895x_{7} = -352.151948578895
x8=50.4473028702814x_{8} = 50.4473028702814
x9=−83.8850238742679x_{9} = -83.8850238742679
x10=158.905055545593x_{10} = 158.905055545593
x11=88.1464147133589x_{11} = 88.1464147133589
x12=−24.9509208158737x_{12} = -24.9509208158737
x13=66.9114234980422x_{13} = 66.9114234980422
x14=24.8391698518802x_{14} = 24.8391698518802
x15=−104.98872735595x_{15} = -104.98872735595
x16=−88.2581656773524x_{16} = -88.2581656773524
x17=−4645.47755688665x_{17} = -4645.47755688665
x18=−67.2896155128725x_{18} = -67.2896155128725
x19=−29.590503669795x_{19} = -29.590503669795
x20=−12.8599419911974x_{20} = -12.8599419911974
x21=29.2123116549653x_{21} = 29.2123116549653
Puntos máximos de la función:
x21=293.484286571338x_{21} = 293.484286571338
x21=−83.5068318594376x_{21} = -83.5068318594376
x21=62.6500326589512x_{21} = 62.6500326589512
x21=−1063.68373977945x_{21} = -1063.68373977945
x21=12.8599419911974x_{21} = 12.8599419911974
x21=46.1859120311904x_{21} = 46.1859120311904
x21=−88.1464147133589x_{21} = -88.1464147133589
x21=83.8850238742679x_{21} = 83.8850238742679
x21=67.2896155128725x_{21} = 67.2896155128725
x21=−24.8391698518802x_{21} = -24.8391698518802
x21=−137.936505381113x_{21} = -137.936505381113
x21=−100.237393538035x_{21} = -100.237393538035
x21=24.9509208158737x_{21} = 24.9509208158737
x21=50.5590538342749x_{21} = 50.5590538342749
x21=−8.10860817328255x_{21} = -8.10860817328255
x21=−50.4473028702814x_{21} = -50.4473028702814
x21=29.590503669795x_{21} = 29.590503669795
x21=−12.7481910272038x_{21} = -12.7481910272038
x21=−62.5382816949577x_{21} = -62.5382816949577
x21=−4272.74782929496x_{21} = -4272.74782929496
x21=−45.8077200163601x_{21} = -45.8077200163601
x21=88.2581656773524x_{21} = 88.2581656773524
Decrece en los intervalos
[158.905055545593,∞)\left[158.905055545593, \infty\right)
Crece en los intervalos
(−∞,−4645.47755688665]\left(-\infty, -4645.47755688665\right]
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
128(cot⁡2(4x3)+1)cot⁡(4x3)−81sin⁡(3x2)36=0\frac{128 \left(\cot^{2}{\left(\frac{4 x}{3} \right)} + 1\right) \cot{\left(\frac{4 x}{3} \right)} - 81 \sin{\left(\frac{3 x}{2} \right)}}{36} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=55.0683030659931x_{1} = 55.0683030659931
x2=95.7281443063169x_{2} = 95.7281443063169
x3=38.5374842623253x_{3} = 38.5374842623253
x4=−91.0634460133268x_{4} = -91.0634460133268
x5=−48.5102721900561x_{5} = -48.5102721900561
x6=12.8032580851988x_{6} = 12.8032580851988
x7=64.5870633391765x_{7} = 64.5870633391765
x8=−5.56758855939267x_{8} = -5.56758855939267
x9=−100.294077444034x_{9} = -100.294077444034
x10=−43.2667004024702x_{10} = -43.2667004024702
x11=−0.83837241924776x_{11} = -0.83837241924776
x12=−15.6652223271718x_{12} = -15.6652223271718
x13=−58.0290324632394x_{13} = -58.0290324632394
x14=−10.8111603469786x_{14} = -10.8111603469786
x15=−53.3643341702493x_{15} = -53.3643341702493
x16=53.3643341702493x_{16} = 53.3643341702493
x17=−3.77458504698695x_{17} = -3.77458504698695
x18=43.2667004024702x_{18} = 43.2667004024702
x19=50.5023699282764x_{19} = 50.5023699282764
x20=80.9658122455477x_{20} = 80.9658122455477
x21=−26.8879514960989x_{21} = -26.8879514960989
x22=−95.7281443063169x_{22} = -95.7281443063169
x23=32.1315232836848x_{23} = 32.1315232836848
x24=22.0338895159058x_{24} = 22.0338895159058
x25=−12.8032580851988x_{25} = -12.8032580851988
x26=−71.6236386391681x_{26} = -71.6236386391681
x27=−64.5870633391765x_{27} = -64.5870633391765
x28=17.3691912229156x_{28} = 17.3691912229156
x29=−86.2093840331336x_{29} = -86.2093840331336
x30=−45.9999370794909x_{30} = -45.9999370794909
x31=76.2365961054028x_{31} = 76.2365961054028
x32=−50.5023699282764x_{32} = -50.5023699282764
x33=41.4736968900645x_{33} = 41.4736968900645
x34=−36.8607394238298x_{34} = -36.8607394238298
x35=8.30082523641336x_{35} = 8.30082523641336
x36=−33.9245267960906x_{36} = -33.9245267960906
x37=100.294077444034x_{37} = 100.294077444034
x38=−69.8306351267624x_{38} = -69.8306351267624
x39=−20.3299206201619x_{39} = -20.3299206201619
x40=62.5949656009562x_{40} = 62.5949656009562
x41=−8.30082523641336x_{41} = -8.30082523641336
x42=−41.4736968900645x_{42} = -41.4736968900645
x43=−97.4321132020608x_{43} = -97.4321132020608
x44=−55.0683030659931x_{44} = -55.0683030659931
x45=10.8111603469786x_{45} = 10.8111603469786
x46=−83.6990489225684x_{46} = -83.6990489225684
x47=71.6236386391681x_{47} = 71.6236386391681
x48=24.8958537578787x_{48} = 24.8958537578787
x49=59.7330013589833x_{49} = 59.7330013589833
x50=−80.9658122455477x_{50} = -80.9658122455477
x51=97.4321132020608x_{51} = 97.4321132020608
x52=15.6652223271718x_{52} = 15.6652223271718
x53=−76.2365961054028x_{53} = -76.2365961054028
x54=26.8879514960989x_{54} = 26.8879514960989
x55=−62.5949656009562x_{55} = -62.5949656009562
x56=−32.1315232836848x_{56} = -32.1315232836848
x57=0.83837241924776x_{57} = 0.83837241924776
x58=−67.0973984497417x_{58} = -67.0973984497417
x59=83.6990489225684x_{59} = 83.6990489225684
x60=−29.3982866066642x_{60} = -29.3982866066642
x61=−79.172808733142x_{61} = -79.172808733142
x62=58.0290324632394x_{62} = 58.0290324632394
x63=29.3982866066642x_{63} = 29.3982866066642
x64=−17.3691912229156x_{64} = -17.3691912229156
x65=67.0973984497417x_{65} = 67.0973984497417
x66=92.7674149090707x_{66} = 92.7674149090707
x67=20.3299206201619x_{67} = 20.3299206201619
x68=−88.2014817713539x_{68} = -88.2014817713539
x69=−22.0338895159058x_{69} = -22.0338895159058
x70=5.56758855939267x_{70} = 5.56758855939267
x71=3.77458504698695x_{71} = 3.77458504698695
x72=36.8607394238298x_{72} = 36.8607394238298
x73=33.9245267960906x_{73} = 33.9245267960906
x74=45.9999370794909x_{74} = 45.9999370794909
x75=−74.5598512669073x_{75} = -74.5598512669073
x76=−59.7330013589833x_{76} = -59.7330013589833
x77=−92.7674149090707x_{77} = -92.7674149090707
x78=74.5598512669073x_{78} = 74.5598512669073
x79=86.2093840331336x_{79} = 86.2093840331336
x80=48.5102721900561x_{80} = 48.5102721900561
x81=−24.8958537578787x_{81} = -24.8958537578787
x82=79.172808733142x_{82} = 79.172808733142
x83=88.2014817713539x_{83} = 88.2014817713539
x84=−38.5374842623253x_{84} = -38.5374842623253
x85=69.8306351267624x_{85} = 69.8306351267624
x86=91.0634460133268x_{86} = 91.0634460133268

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
(−∞,−100.294077444034]\left(-\infty, -100.294077444034\right]
Convexa en los intervalos
[100.294077444034,∞)\left[100.294077444034, \infty\right)
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞(sin⁡(3x2)+cot⁡(4x3))y = \lim_{x \to -\infty}\left(\sin{\left(\frac{3 x}{2} \right)} + \cot{\left(\frac{4 x}{3} \right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(sin⁡(3x2)+cot⁡(4x3))y = \lim_{x \to \infty}\left(\sin{\left(\frac{3 x}{2} \right)} + \cot{\left(\frac{4 x}{3} \right)}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin((3*x)/2) + cot((4*x)/3), dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞(sin⁡(3x2)+cot⁡(4x3)x)y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(\frac{3 x}{2} \right)} + \cot{\left(\frac{4 x}{3} \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(sin⁡(3x2)+cot⁡(4x3)x)y = x \lim_{x \to \infty}\left(\frac{\sin{\left(\frac{3 x}{2} \right)} + \cot{\left(\frac{4 x}{3} \right)}}{x}\right)
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
sin⁡(3x2)+cot⁡(4x3)=−sin⁡(3x2)−cot⁡(4x3)\sin{\left(\frac{3 x}{2} \right)} + \cot{\left(\frac{4 x}{3} \right)} = - \sin{\left(\frac{3 x}{2} \right)} - \cot{\left(\frac{4 x}{3} \right)}
- No
sin⁡(3x2)+cot⁡(4x3)=sin⁡(3x2)+cot⁡(4x3)\sin{\left(\frac{3 x}{2} \right)} + \cot{\left(\frac{4 x}{3} \right)} = \sin{\left(\frac{3 x}{2} \right)} + \cot{\left(\frac{4 x}{3} \right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор