Sr Examen

Otras calculadoras

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

Gráfico de la función y = sin(x^2+4)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          / 2    \
f(x) = sin\x  + 4/
f(x)=sin⁡(x2+4)f{\left(x \right)} = \sin{\left(x^{2} + 4 \right)} 
f = sin(x^2 + 4)
Gráfico de la función
02468-8-6-4-2-10102-2
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⁡(x2+4)=0\sin{\left(x^{2} + 4 \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−29.7501381671735x_{1} = -29.7501381671735
x2=−71.7291458141924x_{2} = -71.7291458141924
x3=66.1703993622357x_{3} = 66.1703993622357
x4=−23.562756467336x_{4} = -23.562756467336
x5=20.1878339011461x_{5} = 20.1878339011461
x6=−61.7497779276298x_{6} = -61.7497779276298
x7=60.1519436892029x_{7} = 60.1519436892029
x8=30.1695784225539x_{8} = 30.1695784225539
x9=−69.865417599354x_{9} = -69.865417599354
x10=4.24159740842157x_{10} = 4.24159740842157
x11=−26.4521597304287x_{11} = -26.4521597304287
x12=96.4600385267731x_{12} = 96.4600385267731
x13=58.0252437935581x_{13} = 58.0252437935581
x14=44.1240188319158x_{14} = 44.1240188319158
x15=28.1773762797711x_{15} = 28.1773762797711
x16=−7.67019250552396x_{16} = -7.67019250552396
x17=32.1848903732516x_{17} = 32.1848903732516
x18=−39.3439546928541x_{18} = -39.3439546928541
x19=−60.5683235036803x_{19} = -60.5683235036803
x20=−9.16321964707352x_{20} = -9.16321964707352
x21=25.1117844054368x_{21} = 25.1117844054368
x22=−68.0891500119669x_{22} = -68.0891500119669
x23=−46.5493342984682x_{23} = -46.5493342984682
x24=84.4984123771754x_{24} = 84.4984123771754
x25=−47.7815806751239x_{25} = -47.7815806751239
x26=−19.8741607874623x_{26} = -19.8741607874623
x27=24.024875460949x_{27} = 24.024875460949
x28=14.3696453606384x_{28} = 14.3696453606384
x29=−79.8731260944159x_{29} = -79.8731260944159
x30=−17.8769136967137x_{30} = -17.8769136967137
x31=−31.544040068475x_{31} = -31.544040068475
x32=−53.8116160253841x_{32} = -53.8116160253841
x33=−7.87232149530148x_{33} = -7.87232149530148
x34=8.26175713952941x_{34} = 8.26175713952941
x35=−3.85351215406657x_{35} = -3.85351215406657
x36=−100.386108727103x_{36} = -100.386108727103
x37=98.3308892485112x_{37} = 98.3308892485112
x38=−61.4181924971334x_{38} = -61.4181924971334
x39=−54.8810436074807x_{39} = -54.8810436074807
x40=49.1429175290208x_{40} = 49.1429175290208
x41=35.4369673376477x_{41} = 35.4369673376477
x42=−91.9240220552285x_{42} = -91.9240220552285
x43=18.138600312056x_{43} = 18.138600312056
x44=72.2528102077686x_{44} = 72.2528102077686
x45=16.023488704023x_{45} = 16.023488704023
x46=80.1871670191178x_{46} = 80.1871670191178
x47=−35.8336851017561x_{47} = -35.8336851017561
x48=−97.3676959837686x_{48} = -97.3676959837686
x49=−101.846357494056x_{49} = -101.846357494056
x50=76.1686498753518x_{50} = 76.1686498753518
x51=−1.51102127952573x_{51} = -1.51102127952573
x52=−91.306782307126x_{52} = -91.306782307126
x53=53.3719601425649x_{53} = 53.3719601425649
x54=−85.496360726975x_{54} = -85.496360726975
x55=94.3859012683263x_{55} = 94.3859012683263
x56=−41.5951817495917x_{56} = -41.5951817495917
x57=54.5652950899811x_{57} = 54.5652950899811
x58=−5.80509361191338x_{58} = -5.80509361191338
x59=88.3337859721332x_{59} = 88.3337859721332
x60=−31.2940649416723x_{60} = -31.2940649416723
x61=−43.8741141141805x_{61} = -43.8741141141805
x62=56.1262774839393x_{62} = 56.1262774839393
x63=−51.9400585491316x_{63} = -51.9400585491316
x64=2.32911527425531x_{64} = 2.32911527425531
x65=22.1894488274763x_{65} = 22.1894488274763
x66=40.3297233713932x_{66} = 40.3297233713932
x67=4.59703613524174x_{67} = 4.59703613524174
x68=−49.4932708805311x_{68} = -49.4932708805311
x69=−93.7178459637792x_{69} = -93.7178459637792
x70=74.1412755247421x_{70} = 74.1412755247421
x71=87.2603127713581x_{71} = 87.2603127713581
x72=10.1397312697158x_{72} = 10.1397312697158
x73=−66.5963243694603x_{73} = -66.5963243694603
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 sin(x^2 + 4).
sin⁡(02+4)\sin{\left(0^{2} + 4 \right)}
Resultado:
f(0)=sin⁡(4)f{\left(0 \right)} = \sin{\left(4 \right)}
Punto:
(0, sin(4))
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
2xcos⁡(x2+4)=02 x \cos{\left(x^{2} + 4 \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0
x2=−−ilog⁡(−ie−4i)x_{2} = - \sqrt{- i \log{\left(- i e^{- 4 i} \right)}}
x3=−ilog⁡(−ie−4i)x_{3} = \sqrt{- i \log{\left(- i e^{- 4 i} \right)}}
Signos de extremos en los puntos:
(0, sin(4))

     __________________                           
    /       /    -4*I\      /         /    -4*I\\ 
(-\/  -I*log\-I*e    /, sin\4 - I*log\-I*e    //)

    __________________                           
   /       /    -4*I\      /         /    -4*I\\ 
(\/  -I*log\-I*e    /, sin\4 - I*log\-I*e    //)


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=−−ilog⁡(−ie−4i)x_{1} = - \sqrt{- i \log{\left(- i e^{- 4 i} \right)}}
x2=−ilog⁡(−ie−4i)x_{2} = \sqrt{- i \log{\left(- i e^{- 4 i} \right)}}
Puntos máximos de la función:
x2=0x_{2} = 0
Decrece en los intervalos
[−−ilog⁡(−ie−4i),0]∪[−ilog⁡(−ie−4i),∞)\left[- \sqrt{- i \log{\left(- i e^{- 4 i} \right)}}, 0\right] \cup \left[\sqrt{- i \log{\left(- i e^{- 4 i} \right)}}, \infty\right)
Crece en los intervalos
(−∞,−−ilog⁡(−ie−4i)]∪[0,−ilog⁡(−ie−4i)]\left(-\infty, - \sqrt{- i \log{\left(- i e^{- 4 i} \right)}}\right] \cup \left[0, \sqrt{- i \log{\left(- i e^{- 4 i} \right)}}\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
2(−2x2sin⁡(x2+4)+cos⁡(x2+4))=02 \left(- 2 x^{2} \sin{\left(x^{2} + 4 \right)} + \cos{\left(x^{2} + 4 \right)}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−9.50017470962807x_{1} = -9.50017470962807
x2=−63.7277099609022x_{2} = -63.7277099609022
x3=−20.4967362860769x_{3} = -20.4967362860769
x4=30.1695875265468x_{4} = 30.1695875265468
x5=10.4451863730613x_{5} = 10.4451863730613
x6=46.3803060878027x_{6} = 46.3803060878027
x7=21.3230773368568x_{7} = 21.3230773368568
x8=60.1519448378615x_{8} = 60.1519448378615
x9=−17.8769574549053x_{9} = -17.8769574549053
x10=−1.57542912672467x_{10} = -1.57542912672467
x11=−40.7944358890199x_{11} = -40.7944358890199
x12=−53.8116176297812x_{12} = -53.8116176297812
x13=80.1871675039879x_{13} = 80.1871675039879
x14=−81.8736855772618x_{14} = -81.8736855772618
x15=50.1553337835779x_{15} = 50.1553337835779
x16=−61.749778989408x_{16} = -61.749778989408
x17=−83.9201548675793x_{17} = -83.9201548675793
x18=−41.5951852234398x_{18} = -41.5951852234398
x19=78.2037247894758x_{19} = 78.2037247894758
x20=−8.63403286188247x_{20} = -8.63403286188247
x21=−5.80637076114012x_{21} = -5.80637076114012
x22=−66.241577907437x_{22} = -66.241577907437
x23=8.26220039766855x_{23} = 8.26220039766855
x24=61.6479426668884x_{24} = 61.6479426668884
x25=−35.8336905350776x_{25} = -35.8336905350776
x26=−69.8878978890777x_{26} = -69.8878978890777
x27=−55.8457093964186x_{27} = -55.8457093964186
x28=−49.5250003428779x_{28} = -49.5250003428779
x29=20.4199562853161x_{29} = 20.4199562853161
x30=0.499515792912275x_{30} = 0.499515792912275
x31=66.0991458964268x_{31} = 66.0991458964268
x32=−89.7976057388256x_{32} = -89.7976057388256
x33=16.0235494705067x_{33} = 16.0235494705067
x34=44.1240217420616x_{34} = 44.1240217420616
x35=−68.7549172210702x_{35} = -68.7549172210702
x36=−60.0473987583507x_{36} = -60.0473987583507
x37=39.4635511565776x_{37} = 39.4635511565776
x38=−76.0861152878029x_{38} = -76.0861152878029
x39=−71.7510421457999x_{39} = -71.7510421457999
x40=11.985755511645x_{40} = 11.985755511645
x41=−0.499515792912275x_{41} = -0.499515792912275
x42=−34.3566711716287x_{42} = -34.3566711716287
x43=−28.8930067222169x_{43} = -28.8930067222169
x44=28.1773874545224x_{44} = 28.1773874545224
x45=−29.3247090552322x_{45} = -29.3247090552322
x46=76.4157210350253x_{46} = 76.4157210350253
x47=−43.8741170743382x_{47} = -43.8741170743382
x48=4.24486632455045x_{48} = 4.24486632455045
x49=−7.87283382728091x_{49} = -7.87283382728091
x50=−7.24970211419189x_{50} = -7.24970211419189
x51=19.0674517366313x_{51} = 19.0674517366313
x52=−77.355514873214x_{52} = -77.355514873214
x53=−3.85786707343277x_{53} = -3.85786707343277
x54=−19.8741926346905x_{54} = -19.8741926346905
x55=−93.717846267499x_{55} = -93.717846267499
x56=6.80266761696267x_{56} = 6.80266761696267
x57=32.1848978719108x_{57} = 32.1848978719108
x58=19.7949981150871x_{58} = 19.7949981150871
x59=5.52936448112481x_{59} = 5.52936448112481
x60=20.1878642868281x_{60} = 20.1878642868281
x61=96.5088797169989x_{61} = 96.5088797169989
x62=6.07077183009749x_{62} = 6.07077183009749
x63=25.607327462205x_{63} = 25.607327462205
x64=−59.4691101495173x_{64} = -59.4691101495173
x65=97.7380422586888x_{65} = 97.7380422586888
x66=−47.8144461765317x_{66} = -47.8144461765317
x67=57.9439754267967x_{67} = 57.9439754267967
x68=72.2745478802369x_{68} = 72.2745478802369
x69=−51.940060333287x_{69} = -51.940060333287
x70=−23.4290677985505x_{70} = -23.4290677985505
x71=2.34844406355589x_{71} = 2.34844406355589
x72=−10.2937098065693x_{72} = -10.2937098065693
x73=3.4278986325124x_{73} = 3.4278986325124
x74=−29.7501476616856x_{74} = -29.7501476616856
x75=−97.4644439256926x_{75} = -97.4644439256926
x76=−4.59960545847001x_{76} = -4.59960545847001
x77=68.3654267021995x_{77} = 68.3654267021995
x78=−79.8731265850276x_{78} = -79.8731265850276
x79=63.9000169387531x_{79} = 63.9000169387531
x80=18.1386422035504x_{80} = 18.1386422035504
x81=34.7205067127919x_{81} = 34.7205067127919
x82=12.9919420484305x_{82} = 12.9919420484305
x83=22.189471709751x_{83} = 22.189471709751
x84=77.1725427066804x_{84} = 77.1725427066804
x85=56.1262788979128x_{85} = 56.1262788979128
x86=9.98386566916112x_{86} = 9.98386566916112
x87=10.1399710599954x_{87} = 10.1399710599954
x88=83.751525713878x_{88} = 83.751525713878

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
[77.1725427066804,∞)\left[77.1725427066804, \infty\right)
Convexa en los intervalos
(−∞,−89.7976057388256]\left(-\infty, -89.7976057388256\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞sin⁡(x2+4)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(x^{2} + 4 \right)} = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞sin⁡(x2+4)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(x^{2} + 4 \right)} = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x^2 + 4), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(x2+4)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x^{2} + 4 \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡(x2+4)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x^{2} + 4 \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:
sin⁡(x2+4)=sin⁡(x2+4)\sin{\left(x^{2} + 4 \right)} = \sin{\left(x^{2} + 4 \right)}
- Sí
sin⁡(x2+4)=−sin⁡(x2+4)\sin{\left(x^{2} + 4 \right)} = - \sin{\left(x^{2} + 4 \right)}
- No
es decir, función
es
par
    © Sr Examen – Калькулятор