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Trazado el gráfico de la función/ arctan(0.01*x+tan(x))

Gráfico de la función y = arctan(0.01*x+tan(x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
           / x          \
f(x) = atan|--- + tan(x)|
           \100         /
f(x)=atan⁡(x100+tan⁡(x))f{\left(x \right)} = \operatorname{atan}{\left(\frac{x}{100} + \tan{\left(x \right)} \right)} 
f = atan(x/100 + tan(x))
Gráfico de la función
02468-8-6-4-2-10105-5
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:
atan⁡(x100+tan⁡(x))=0\operatorname{atan}{\left(\frac{x}{100} + \tan{\left(x \right)} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=84.1236174891021x_{1} = 84.1236174891021
x2=−24.8888093394065x_{2} = -24.8888093394065
x3=−49.8034088143338x_{3} = -49.8034088143338
x4=81.000596562218x_{4} = 81.000596562218
x5=96.6211569411279x_{5} = 96.6211569411279
x6=−62.2748742762875x_{6} = -62.2748742762875
x7=65.3943035913842x_{7} = 65.3943035913842
x8=21.776728901746x_{8} = 21.776728901746
x9=15.5536629707846x_{9} = 15.5536629707846
x10=74.7562841841391x_{10} = 74.7562841841391
x11=−96.6211569411279x_{11} = -96.6211569411279
x12=−6.22105482782194x_{12} = -6.22105482782194
x13=−52.9203386534867x_{13} = -52.9203386534867
x14=9.3317301256938x_{14} = 9.3317301256938
x15=40.4562709483391x_{15} = 40.4562709483391
x16=−12.44258101586x_{16} = -12.44258101586
x17=−77.878148090763x_{17} = -77.878148090763
x18=−65.3943035913842x_{18} = -65.3943035913842
x19=−15.5536629707846x_{19} = -15.5536629707846
x20=−40.4562709483391x_{20} = -40.4562709483391
x21=93.4959819877802x_{21} = 93.4959819877802
x22=−56.0378910374261x_{22} = -56.0378910374261
x23=−9.3317301256938x_{23} = -9.3317301256938
x24=−28.0013130037828x_{24} = -28.0013130037828
x25=62.2748742762875x_{25} = 62.2748742762875
x26=28.0013130037828x_{26} = 28.0013130037828
x27=18.6650289501217x_{27} = 18.6650289501217
x28=24.8888093394065x_{28} = 24.8888093394065
x29=−68.514352943512x_{29} = -68.514352943512
x30=−59.1560693997803x_{30} = -59.1560693997803
x31=59.1560693997803x_{31} = 59.1560693997803
x32=34.2277406826553x_{32} = 34.2277406826553
x33=37.3417294715407x_{33} = 37.3417294715407
x34=−84.1236174891021x_{34} = -84.1236174891021
x35=56.0378910374261x_{35} = 56.0378910374261
x36=71.6350158838329x_{36} = 71.6350158838329
x37=−87.2471978954843x_{37} = -87.2471978954843
x38=−99.7468341842302x_{38} = -99.7468341842302
x39=−37.3417294715407x_{39} = -37.3417294715407
x40=52.9203386534867x_{40} = 52.9203386534867
x41=87.2471978954843x_{41} = 87.2471978954843
x42=−74.7562841841391x_{42} = -74.7562841841391
x43=6.22105482782194x_{43} = 6.22105482782194
x44=49.8034088143338x_{44} = 49.8034088143338
x45=−21.776728901746x_{45} = -21.776728901746
x46=−18.6650289501217x_{46} = -18.6650289501217
x47=−31.1142786101721x_{47} = -31.1142786101721
x48=46.6870949070171x_{48} = 46.6870949070171
x49=77.878148090763x_{49} = 77.878148090763
x50=12.44258101586x_{50} = 12.44258101586
x51=43.5713868705314x_{51} = 43.5713868705314
x52=99.7468341842302x_{52} = 99.7468341842302
x53=−34.2277406826553x_{53} = -34.2277406826553
x54=−46.6870949070171x_{54} = -46.6870949070171
x55=−81.000596562218x_{55} = -81.000596562218
x56=−3.11049770230558x_{56} = -3.11049770230558
x57=−43.5713868705314x_{57} = -43.5713868705314
x58=3.11049770230558x_{58} = 3.11049770230558
x59=−71.6350158838329x_{59} = -71.6350158838329
x60=−93.4959819877802x_{60} = -93.4959819877802
x61=−90.3713241215556x_{61} = -90.3713241215556
x62=0x_{62} = 0
x63=90.3713241215556x_{63} = 90.3713241215556
x64=31.1142786101721x_{64} = 31.1142786101721
x65=68.514352943512x_{65} = 68.514352943512
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 atan(x/100 + tan(x)).
atan⁡(0100+tan⁡(0))\operatorname{atan}{\left(\frac{0}{100} + \tan{\left(0 \right)} \right)}
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
tan⁡2(x)+101100(x100+tan⁡(x))2+1=0\frac{\tan^{2}{\left(x \right)} + \frac{101}{100}}{\left(\frac{x}{100} + \tan{\left(x \right)}\right)^{2} + 1} = 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
200(−(x+100tan⁡(x))(100tan⁡2(x)+101)2(x+100tan⁡(x))2+10000+100(tan⁡2(x)+1)tan⁡(x))(x+100tan⁡(x))2+10000=0\frac{200 \left(- \frac{\left(x + 100 \tan{\left(x \right)}\right) \left(100 \tan^{2}{\left(x \right)} + 101\right)^{2}}{\left(x + 100 \tan{\left(x \right)}\right)^{2} + 10000} + 100 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}\right)}{\left(x + 100 \tan{\left(x \right)}\right)^{2} + 10000} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=57.2089481403263x_{1} = 57.2089481403263
x2=76.0140857181823x_{2} = 76.0140857181823
x3=−55.6324199405873x_{3} = -55.6324199405873
x4=101.092768464544x_{4} = 101.092768464544
x5=5.56769883380512x_{5} = 5.56769883380512
x6=−5.56769883380512x_{6} = -5.56769883380512
x7=63.4769306935802x_{7} = 63.4769306935802
x8=−4.04266138735351x_{8} = -4.04266138735351
x9=−97.9576243794904x_{9} = -97.9576243794904
x10=97.9576243794904x_{10} = 97.9576243794904
x11=88.5527174231377x_{11} = 88.5527174231377
x12=−72.8796549646548x_{12} = -72.8796549646548
x13=−22.7470701515958x_{13} = -22.7470701515958
x14=−16.487615702182x_{14} = -16.487615702182
x15=85.4179249937426x_{15} = 85.4179249937426
x16=−44.6743835632362x_{16} = -44.6743835632362
x17=−11.7889267778908x_{17} = -11.7889267778908
x18=39.9647579863149x_{18} = 39.9647579863149
x19=29.010188902193x_{19} = 29.010188902193
x20=0x_{20} = 0
x21=54.0751198681782x_{21} = 54.0751198681782
x22=−69.7453175327784x_{22} = -69.7453175327784
x23=18.0420511899237x_{23} = 18.0420511899237
x24=47.8078262846906x_{24} = 47.8078262846906
x25=−13.3608030717749x_{25} = -13.3608030717749
x26=−57.2089481403263x_{26} = -57.2089481403263
x27=−60.342886915049x_{27} = -60.342886915049
x28=−32.142482539855x_{28} = -32.142482539855
x29=−79.1486083487238x_{29} = -79.1486083487238
x30=−91.6875984460841x_{30} = -91.6875984460841
x31=−63.4769306935802x_{31} = -63.4769306935802
x32=19.6166747731977x_{32} = 19.6166747731977
x33=44.6743835632362x_{33} = 44.6743835632362
x34=68.1687489515172x_{34} = 68.1687489515172
x35=38.4079956515257x_{35} = 38.4079956515257
x36=41.5410986955573x_{36} = 41.5410986955573
x37=79.1486083487238x_{37} = 79.1486083487238
x38=−10.2382408671037x_{38} = -10.2382408671037
x39=−38.4079956515257x_{39} = -38.4079956515257
x40=−76.0140857181823x_{40} = -76.0140857181823
x41=−82.2832217342947x_{41} = -82.2832217342947
x42=−39.9647579863149x_{42} = -39.9647579863149
x43=66.6110752957601x_{43} = 66.6110752957601
x44=24.3027423656289x_{44} = 24.3027423656289
x45=2.53297050706372x_{45} = 2.53297050706372
x46=46.2313816184006x_{46} = 46.2313816184006
x47=69.7453175327784x_{47} = 69.7453175327784
x48=16.487615702182x_{48} = 16.487615702182
x49=−33.6989434080338x_{49} = -33.6989434080338
x50=−61.9003751119859x_{50} = -61.9003751119859
x51=−19.6166747731977x_{51} = -19.6166747731977
x52=−29.010188902193x_{52} = -29.010188902193
x53=−41.5410986955573x_{53} = -41.5410986955573
x54=13.3608030717749x_{54} = 13.3608030717749
x55=4.04266138735351x_{55} = 4.04266138735351
x56=25.8783306571506x_{56} = 25.8783306571506
x57=−66.6110752957601x_{57} = -66.6110752957601
x58=22.7470701515958x_{58} = 22.7470701515958
x59=50.9414093641438x_{59} = 50.9414093641438
x60=−2.53297050706372x_{60} = -2.53297050706372
x61=−94.8225675747847x_{61} = -94.8225675747847
x62=−85.4179249937426x_{62} = -85.4179249937426
x63=−83.841364376541x_{63} = -83.841364376541
x64=60.342886915049x_{64} = 60.342886915049
x65=11.7889267778908x_{65} = 11.7889267778908
x66=−54.0751198681782x_{66} = -54.0751198681782
x67=90.1110521775279x_{67} = 90.1110521775279
x68=−25.8783306571506x_{68} = -25.8783306571506
x69=72.8796549646548x_{69} = 72.8796549646548
x70=91.6875984460841x_{70} = 91.6875984460841
x71=−35.2751078006808x_{71} = -35.2751078006808
x72=−18.0420511899237x_{72} = -18.0420511899237
x73=−101.092768464544x_{73} = -101.092768464544
x74=−88.5527174231377x_{74} = -88.5527174231377
x75=−47.8078262846906x_{75} = -47.8078262846906
x76=−50.9414093641438x_{76} = -50.9414093641438
x77=94.8225675747847x_{77} = 94.8225675747847
x78=32.142482539855x_{78} = 32.142482539855
x79=10.2382408671037x_{79} = 10.2382408671037
x80=82.2832217342947x_{80} = 82.2832217342947
x81=61.9003751119859x_{81} = 61.9003751119859
x82=35.2751078006808x_{82} = 35.2751078006808
x83=−24.3027423656289x_{83} = -24.3027423656289

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
[101.092768464544,∞)\left[101.092768464544, \infty\right)
Convexa en los intervalos
(−∞,−101.092768464544]\left(-\infty, -101.092768464544\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→−∞atan⁡(x100+tan⁡(x))y = \lim_{x \to -\infty} \operatorname{atan}{\left(\frac{x}{100} + \tan{\left(x \right)} \right)}
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞atan⁡(x100+tan⁡(x))y = \lim_{x \to \infty} \operatorname{atan}{\left(\frac{x}{100} + \tan{\left(x \right)} \right)}
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función atan(x/100 + tan(x)), 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→−∞(atan⁡(x100+tan⁡(x))x)y = x \lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{x}{100} + \tan{\left(x \right)} \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(atan⁡(x100+tan⁡(x))x)y = x \lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{x}{100} + \tan{\left(x \right)} \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:
atan⁡(x100+tan⁡(x))=−atan⁡(x100+tan⁡(x))\operatorname{atan}{\left(\frac{x}{100} + \tan{\left(x \right)} \right)} = - \operatorname{atan}{\left(\frac{x}{100} + \tan{\left(x \right)} \right)}
- No
atan⁡(x100+tan⁡(x))=atan⁡(x100+tan⁡(x))\operatorname{atan}{\left(\frac{x}{100} + \tan{\left(x \right)} \right)} = \operatorname{atan}{\left(\frac{x}{100} + \tan{\left(x \right)} \right)}
- Sí
es decir, función
es
impar
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