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Trazado el gráfico de la función/ 378/89+2*sin(2*x)-378*cos(x*sqrt(890)/10)/89-4*sqrt(890)*sin(x*sqrt(890)/10)/89

Gráfico de la función y = 378/89+2*sin(2*x)-378*cos(x*sqrt(890)/10)/89-4*sqrt(890)*sin(x*sqrt(890)/10)/89

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                                 /    _____\                /    _____\
                                 |x*\/ 890 |       _____    |x*\/ 890 |
                          378*cos|---------|   4*\/ 890 *sin|---------|
       378                       \    10   /                \    10   /
f(x) = --- + 2*sin(2*x) - ------------------ - ------------------------
        89                        89                      89
f(x)=−4890sin⁡(890x10)89+((2sin⁡(2x)+37889)−378cos⁡(890x10)89)f{\left(x \right)} = - \frac{4 \sqrt{890} \sin{\left(\frac{\sqrt{890} x}{10} \right)}}{89} + \left(\left(2 \sin{\left(2 x \right)} + \frac{378}{89}\right) - \frac{378 \cos{\left(\frac{\sqrt{890} x}{10} \right)}}{89}\right) 
f = -(4*sqrt(890))*sin((sqrt(890)*x)/10)/89 + 2*sin(2*x) + 378/89 - 378*cos((sqrt(890)*x)/10)/89
Gráfico de la función
0.00.51.01.52.02.53.03.54.04.55.05.56.020-10
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:
−4890sin⁡(890x10)89+((2sin⁡(2x)+37889)−378cos⁡(890x10)89)=0- \frac{4 \sqrt{890} \sin{\left(\frac{\sqrt{890} x}{10} \right)}}{89} + \left(\left(2 \sin{\left(2 x \right)} + \frac{378}{89}\right) - \frac{378 \cos{\left(\frac{\sqrt{890} x}{10} \right)}}{89}\right) = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=92.6733509530383x_{1} = 92.6733509530383
x2=−60.6347946763033x_{2} = -60.6347946763033
x3=−67.4604337164693x_{3} = -67.4604337164693
x4=−25.4402091140237x_{4} = -25.4402091140237
x5=−12.5162428610169x_{5} = -12.5162428610169
x6=90.3167765692715x_{6} = 90.3167765692715
x7=−23.3127273851637x_{7} = -23.3127273851637
x8=71.9720119363751x_{8} = 71.9720119363751
x9=−88.1391815992041x_{9} = -88.1391815992041
x10=−80.0687253292523x_{10} = -80.0687253292523
x11=67.7102247288082x_{11} = 67.7102247288082
x12=8.23488000519075x_{12} = 8.23488000519075
x13=−10.0843996872679x_{13} = -10.0843996872679
x14=−69.7400287186152x_{14} = -69.7400287186152
x15=80.3972540357323x_{15} = 80.3972540357323
x16=14.5456860825518x_{16} = 14.5456860825518
x17=83.9989075441596x_{17} = 83.9989075441596
x18=−16.4002909992973x_{18} = -16.4002909992973
x19=27.7981220700759x_{19} = 27.7981220700759
x20=−86.3715630324451x_{20} = -86.3715630324451
x21=−85.9371792775141x_{21} = -85.9371792775141
x22=−31.7752436778683x_{22} = -31.7752436778683
x23=109.275804253792x_{23} = 109.275804253792
x24=−73.7649695863916x_{24} = -73.7649695863916
x25=−54.3126154210667x_{25} = -54.3126154210667
x26=246918.723223988x_{26} = 246918.723223988
x27=−6.23287725373012x_{27} = -6.23287725373012
x28=5635.93915363489x_{28} = 5635.93915363489
x29=−66.9582017504141x_{29} = -66.9582017504141
x30=78.2808517870939x_{30} = 78.2808517870939
x31=−100.75432064893x_{31} = -100.75432064893
x32=−73.2829613430633x_{32} = -73.2829613430633
x33=−81.8323175671041x_{33} = -81.8323175671041
x34=61.2154984177943x_{34} = 61.2154984177943
x35=1.92451035802499x_{35} = 1.92451035802499
x36=74.0568655560213x_{36} = 74.0568655560213
x37=−47.9915584839098x_{37} = -47.9915584839098
x38=−41.6715329956941x_{38} = -41.6715329956941
x39=59.3530698707701x_{39} = 59.3530698707701
x40=40.4214514950331x_{40} = 40.4214514950331
x41=46.1067759425552x_{41} = 46.1067759425552
x42=−1103.1898126916x_{42} = -1103.1898126916
x43=−35.9287882218268x_{43} = -35.9287882218268
x44=−44.0039183043954x_{44} = -44.0039183043954
x45=65.6627512403372x_{45} = 65.6627512403372
x46=−94.4465236496853x_{46} = -94.4465236496853
x47=−92.2670711088846x_{47} = -92.2670711088846
x48=80.0762268512484x_{48} = 80.0762268512484
x49=−29.6209769179969x_{49} = -29.6209769179969
x50=−57.0917749203174x_{50} = -57.0917749203174
x51=96.6355030647524x_{51} = 96.6355030647524
x52=−63.4165802745949x_{52} = -63.4165802745949
x53=52.4205509027992x_{53} = 52.4205509027992
x54=124.702578024311x_{54} = 124.702578024311
x55=86.3737304792606x_{55} = 86.3737304792606
x56=0x_{56} = 0
x57=−76.0622456680179x_{57} = -76.0622456680179
x58=20.8569402585069x_{58} = 20.8569402585069
x59=−227.010443069804x_{59} = -227.010443069804
x60=−3.76917898152629x_{60} = -3.76917898152629
x61=−37.7022337466565x_{61} = -37.7022337466565
x62=34.1100213676882x_{62} = 34.1100213676882
x63=−50.3067236761556x_{63} = -50.3067236761556
x64=105.276936148122x_{64} = 105.276936148122
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en 378/89 + 2*sin(2*x) - 378*cos((x*sqrt(890))/10)/89 - (4*sqrt(890))*sin((x*sqrt(890))/10)/89.
(−378cos⁡(089010)89+(2sin⁡(0⋅2)+37889))−4890sin⁡(089010)89\left(- \frac{378 \cos{\left(\frac{0 \sqrt{890}}{10} \right)}}{89} + \left(2 \sin{\left(0 \cdot 2 \right)} + \frac{378}{89}\right)\right) - \frac{4 \sqrt{890} \sin{\left(\frac{0 \sqrt{890}}{10} \right)}}{89}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
189890sin⁡(890x10)445+4cos⁡(2x)−4cos⁡(890x10)=0\frac{189 \sqrt{890} \sin{\left(\frac{\sqrt{890} x}{10} \right)}}{445} + 4 \cos{\left(2 x \right)} - 4 \cos{\left(\frac{\sqrt{890} x}{10} \right)} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga extremos
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
−8sin⁡(2x)+2890sin⁡(890x10)5+189cos⁡(890x10)5=0- 8 \sin{\left(2 x \right)} + \frac{2 \sqrt{890} \sin{\left(\frac{\sqrt{890} x}{10} \right)}}{5} + \frac{189 \cos{\left(\frac{\sqrt{890} x}{10} \right)}}{5} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=23.8312047582098x_{1} = 23.8312047582098
x2=10.1745550186606x_{2} = 10.1745550186606
x3=18.4860232295894x_{3} = 18.4860232295894
x4=−23.5879302445965x_{4} = -23.5879302445965
x5=58.500688168163x_{5} = 58.500688168163
x6=−3.53514146629191x_{6} = -3.53514146629191
x7=−94.1587360720886x_{7} = -94.1587360720886
x8=−29.9021334661329x_{8} = -29.9021334661329
x9=81.7202275793511x_{9} = 81.7202275793511
x10=−74.1030322666738x_{10} = -74.1030322666738
x11=−83.6661141139838x_{11} = -83.6661141139838
x12=41.7640476926881x_{12} = 41.7640476926881
x13=85.9774681851565x_{13} = 85.9774681851565
x14=−96.3093904015378x_{14} = -96.3093904015378
x15=38.4712516322278x_{15} = 38.4712516322278
x16=99.6846800604598x_{16} = 99.6846800604598
x17=32.153162016904x_{17} = 32.153162016904
x18=−43.6423167091676x_{18} = -43.6423167091676
x19=−49.9572525416057x_{19} = -49.9572525416057
x20=78.5538844614693x_{20} = 78.5538844614693
x21=1.66890476639822x_{21} = 1.66890476639822
x22=48.0810141387391x_{22} = 48.0810141387391
x23=−45.7291434763743x_{23} = -45.7291434763743
x24=68.0851914891095x_{24} = 68.0851914891095
x25=−91.9734249677054x_{25} = -91.9734249677054
x26=34.353911224475x_{26} = 34.353911224475
x27=−101.577157891805x_{27} = -101.577157891805
x28=88.0440766034268x_{28} = 88.0440766034268
x29=−73.017429113259x_{29} = -73.017429113259
x30=−89.9879234543317x_{30} = -89.9879234543317
x31=−87.8444675035607x_{31} = -87.8444675035607
x32=30.154118066091x_{32} = 30.154118066091
x33=90.079260516863x_{33} = 90.079260516863
x34=−27.7443798036131x_{34} = -27.7443798036131
x35=−47.7473890638645x_{35} = -47.7473890638645
x36=45.8705727066345x_{36} = 45.8705727066345
x37=52.1855516918194x_{37} = 52.1855516918194
x38=−9.85052069673465x_{38} = -9.85052069673465
x39=12.1637941321656x_{39} = 12.1637941321656
x40=−25.7623703226005x_{40} = -25.7623703226005
x41=−69.9932053611041x_{41} = -69.9932053611041
x42=94.3678472966505x_{42} = 94.3678472966505
x43=−524.880672141935x_{43} = -524.880672141935
x44=25.8353996404x_{44} = 25.8353996404
x45=65.9254949536293x_{45} = 65.9254949536293
x46=56.4248495029579x_{46} = 56.4248495029579
x47=−14.1102223852298x_{47} = -14.1102223852298
x48=72.2396934313396x_{48} = 72.2396934313396
x49=63.7470832838042x_{49} = 63.7470832838042
x50=61.7641948680103x_{50} = 61.7641948680103
x51=−71.9990102438322x_{51} = -71.9990102438322
x52=−36.2163617718134x_{52} = -36.2163617718134
x53=−61.4738011852703x_{53} = -61.4738011852703
x54=54.3977013104231x_{54} = 54.3977013104231
x55=−54.0644396924683x_{55} = -54.0644396924683
x56=43.7777252561111x_{56} = 43.7777252561111
x57=−52.0525742061102x_{57} = -52.0525742061102
x58=−76.3105632475318x_{58} = -76.3105632475318
x59=−34.0654814082298x_{59} = -34.0654814082298
x60=−63.6755362069339x_{60} = -63.6755362069339
x61=36.4767608452946x_{61} = 36.4767608452946
x62=7.98318737515606x_{62} = 7.98318737515606
x63=−56.272050000712x_{63} = -56.272050000712
x64=74.4058242489835x_{64} = 74.4058242489835
x65=16.493115713592x_{65} = 16.493115713592
x66=−98.2928020524275x_{66} = -98.2928020524275
x67=96.3956592909252x_{67} = 96.3956592909252
x68=−15.1032678213626x_{68} = -15.1032678213626
x69=−12.0650398936258x_{69} = -12.0650398936258
x70=−81.5301459226731x_{70} = -81.5301459226731
x71=3.85564283510171x_{71} = 3.85564283510171
x72=50.1012086165312x_{72} = 50.1012086165312
x73=−41.4306218980963x_{73} = -41.4306218980963
x74=−32.0821250062869x_{74} = -32.0821250062869
x75=2815.53603060057x_{75} = 2815.53603060057
x76=83.7631100806301x_{76} = 83.7631100806301
x77=−1.46256036710973x_{77} = -1.46256036710973
x78=−85.6544087274576x_{78} = -85.6544087274576
x79=−16.1660933152203x_{79} = -16.1660933152203
x80=−65.6760230351375x_{80} = -65.6760230351375
x81=14.2975270349473x_{81} = 14.2975270349473
x82=70.0669453324981x_{82} = 70.0669453324981
x83=92.2927938454477x_{83} = 92.2927938454477
x84=21.7247424510756x_{84} = 21.7247424510756
x85=−67.7883623006727x_{85} = -67.7883623006727
x86=−7.78635797724167x_{86} = -7.78635797724167
x87=−5.7489590720429x_{87} = -5.7489590720429
x88=5.84188753760536x_{88} = 5.84188753760536
x89=28.0393791713571x_{89} = 28.0393791713571
x90=76.3871761656292x_{90} = 76.3871761656292

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
[2815.53603060057,∞)\left[2815.53603060057, \infty\right)
Convexa en los intervalos
(−∞,−524.880672141935]\left(-\infty, -524.880672141935\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(−4890sin⁡(890x10)89+((2sin⁡(2x)+37889)−378cos⁡(890x10)89))=⟨−2,93489⟩+890⟨−489,489⟩\lim_{x \to -\infty}\left(- \frac{4 \sqrt{890} \sin{\left(\frac{\sqrt{890} x}{10} \right)}}{89} + \left(\left(2 \sin{\left(2 x \right)} + \frac{378}{89}\right) - \frac{378 \cos{\left(\frac{\sqrt{890} x}{10} \right)}}{89}\right)\right) = \left\langle -2, \frac{934}{89}\right\rangle + \sqrt{890} \left\langle - \frac{4}{89}, \frac{4}{89}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−2,93489⟩+890⟨−489,489⟩y = \left\langle -2, \frac{934}{89}\right\rangle + \sqrt{890} \left\langle - \frac{4}{89}, \frac{4}{89}\right\rangle
lim⁡x→∞(−4890sin⁡(890x10)89+((2sin⁡(2x)+37889)−378cos⁡(890x10)89))=⟨−2,93489⟩+890⟨−489,489⟩\lim_{x \to \infty}\left(- \frac{4 \sqrt{890} \sin{\left(\frac{\sqrt{890} x}{10} \right)}}{89} + \left(\left(2 \sin{\left(2 x \right)} + \frac{378}{89}\right) - \frac{378 \cos{\left(\frac{\sqrt{890} x}{10} \right)}}{89}\right)\right) = \left\langle -2, \frac{934}{89}\right\rangle + \sqrt{890} \left\langle - \frac{4}{89}, \frac{4}{89}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−2,93489⟩+890⟨−489,489⟩y = \left\langle -2, \frac{934}{89}\right\rangle + \sqrt{890} \left\langle - \frac{4}{89}, \frac{4}{89}\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función 378/89 + 2*sin(2*x) - 378*cos((x*sqrt(890))/10)/89 - (4*sqrt(890))*sin((x*sqrt(890))/10)/89, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−4890sin⁡(890x10)89+((2sin⁡(2x)+37889)−378cos⁡(890x10)89)x)=0\lim_{x \to -\infty}\left(\frac{- \frac{4 \sqrt{890} \sin{\left(\frac{\sqrt{890} x}{10} \right)}}{89} + \left(\left(2 \sin{\left(2 x \right)} + \frac{378}{89}\right) - \frac{378 \cos{\left(\frac{\sqrt{890} x}{10} \right)}}{89}\right)}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(−4890sin⁡(890x10)89+((2sin⁡(2x)+37889)−378cos⁡(890x10)89)x)=0\lim_{x \to \infty}\left(\frac{- \frac{4 \sqrt{890} \sin{\left(\frac{\sqrt{890} x}{10} \right)}}{89} + \left(\left(2 \sin{\left(2 x \right)} + \frac{378}{89}\right) - \frac{378 \cos{\left(\frac{\sqrt{890} x}{10} \right)}}{89}\right)}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
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:
−4890sin⁡(890x10)89+((2sin⁡(2x)+37889)−378cos⁡(890x10)89)=−2sin⁡(2x)+4890sin⁡(890x10)89−378cos⁡(890x10)89+37889- \frac{4 \sqrt{890} \sin{\left(\frac{\sqrt{890} x}{10} \right)}}{89} + \left(\left(2 \sin{\left(2 x \right)} + \frac{378}{89}\right) - \frac{378 \cos{\left(\frac{\sqrt{890} x}{10} \right)}}{89}\right) = - 2 \sin{\left(2 x \right)} + \frac{4 \sqrt{890} \sin{\left(\frac{\sqrt{890} x}{10} \right)}}{89} - \frac{378 \cos{\left(\frac{\sqrt{890} x}{10} \right)}}{89} + \frac{378}{89}
- No
−4890sin⁡(890x10)89+((2sin⁡(2x)+37889)−378cos⁡(890x10)89)=2sin⁡(2x)−4890sin⁡(890x10)89+378cos⁡(890x10)89−37889- \frac{4 \sqrt{890} \sin{\left(\frac{\sqrt{890} x}{10} \right)}}{89} + \left(\left(2 \sin{\left(2 x \right)} + \frac{378}{89}\right) - \frac{378 \cos{\left(\frac{\sqrt{890} x}{10} \right)}}{89}\right) = 2 \sin{\left(2 x \right)} - \frac{4 \sqrt{890} \sin{\left(\frac{\sqrt{890} x}{10} \right)}}{89} + \frac{378 \cos{\left(\frac{\sqrt{890} x}{10} \right)}}{89} - \frac{378}{89}
- No
es decir, función
no es
par ni impar
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