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Trazado el gráfico de la función/ abs(x-(1/x^2))-cos(x)/(log(abs(tan(x)-x)))*2*x^2+6*x+9

Gráfico de la función y = abs(x-(1/x^2))-cos(x)/(log(abs(tan(x)-x)))*2*x^2+6*x+9

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       |    1 |         cos(x)         2          
f(x) = |x - --| - -----------------*2*x  + 6*x + 9
       |     2|   log(|tan(x) - x|)               
       |    x |
f(x)=(6x+(−x2⋅2cos⁡(x)log⁡(∣−x+tan⁡(x)∣)+∣x−1x2∣))+9f{\left(x \right)} = \left(6 x + \left(- x^{2} \cdot 2 \frac{\cos{\left(x \right)}}{\log{\left(\left|{- x + \tan{\left(x \right)}}\right| \right)}} + \left|{x - \frac{1}{x^{2}}}\right|\right)\right) + 9 
f = 6*x - x^2*2*(cos(x)/log(Abs(-x + tan(x)))) + |x - 1/x^2| + 9
Gráfico de la función
02468-8-6-4-2-1010-500500
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:
(6x+(−x2⋅2cos⁡(x)log⁡(∣−x+tan⁡(x)∣)+∣x−1x2∣))+9=0\left(6 x + \left(- x^{2} \cdot 2 \frac{\cos{\left(x \right)}}{\log{\left(\left|{- x + \tan{\left(x \right)}}\right| \right)}} + \left|{x - \frac{1}{x^{2}}}\right|\right)\right) + 9 = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−29.5840925099944x_{1} = -29.5840925099944
x2=−1.19259845190913x_{2} = -1.19259845190913
x3=89.3585389006903x_{3} = 89.3585389006903
x4=−89.6608799353666x_{4} = -89.6608799353666
x5=−48.5066339739556x_{5} = -48.5066339739556
x6=83.0648105337411x_{6} = 83.0648105337411
x7=26.2412380070532x_{7} = 26.2412380070532
x8=−95.9376849504223x_{8} = -95.9376849504223
x9=57.8703512180766x_{9} = 57.8703512180766
x10=−35.8977749044837x_{10} = -35.8977749044837
x11=80.3087756891324x_{11} = 80.3087756891324
x12=−64.5636512938857x_{12} = -64.5636512938857
x13=−20.7745995057727x_{13} = -20.7745995057727
x14=32.5967018004127x_{14} = 32.5967018004127
x15=55.2452711405908x_{15} = 55.2452711405908
x16=−52.0251692380596x_{16} = -52.0251692380596
x17=−33.2464760709665x_{17} = -33.2464760709665
x18=92.8532948837997x_{18} = 92.8532948837997
x19=−92.5593332340683x_{19} = -92.5593332340683
x20=−14.5743503373268x_{20} = -14.5743503373268
x21=24.0829423022591x_{21} = 24.0829423022591
x22=−23.2600806167803x_{22} = -23.2600806167803
x23=67.771316219314x_{23} = 67.771316219314
x24=−83.3848774934392x_{24} = -83.3848774934392
x25=11.9611644819678x_{25} = 11.9611644819678
x26=−98.848214093588x_{26} = -98.848214093588
x27=48.9890258533454x_{27} = 48.9890258533454
x28=−45.7604285065276x_{28} = -45.7604285065276
x29=36.5006831893491x_{29} = 36.5006831893491
x30=17.9447090419093x_{30} = 17.9447090419093
x31=−79.979394577634x_{31} = -79.979394577634
x32=−54.805254115777x_{32} = -54.805254115777
x33=38.930352049396x_{33} = 38.930352049396
x34=13.3270262323497x_{34} = 13.3270262323497
x35=−39.5002051024911x_{35} = -39.5002051024911
x36=70.4724540846309x_{36} = 70.4724540846309
x37=64.1729512837875x_{37} = 64.1729512837875
x38=−16.920366948233x_{38} = -16.920366948233
x39=−67.3954387726728x_{39} = -67.3954387726728
x40=−86.2697718477369x_{40} = -86.2697718477369
x41=51.5636123768479x_{41} = 51.5636123768479
x42=−8.42091417831353x_{42} = -8.42091417831353
x43=−73.688026746585x_{43} = -73.688026746585
x44=95.6510338829714x_{44} = 95.6510338829714
x45=−77.1098460757445x_{45} = -77.1098460757445
x46=61.5064642047667x_{46} = 61.5064642047667
x47=−61.1013225001357x_{47} = -61.1013225001357
x48=42.7397293601088x_{48} = 42.7397293601088
x49=86.5803310311679x_{49} = 86.5803310311679
x50=−70.836005985521x_{50} = -70.836005985521
x51=−10.5649464554785x_{51} = -10.5649464554785
x52=−42.2045850897045x_{52} = -42.2045850897045
x53=19.8427215536707x_{53} = 19.8427215536707
x54=−27.0025826467422x_{54} = -27.0025826467422
x55=30.27779658204x_{55} = 30.27779658204
x56=−58.2931839968355x_{56} = -58.2931839968355
x57=45.2511412457967x_{57} = 45.2511412457967
x58=99.1274182279726x_{58} = 99.1274182279726
x59=74.0389568584821x_{59} = 74.0389568584821
x60=76.769572043515x_{60} = 76.769572043515
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 |x - 1/x^2| - (cos(x)/log(Abs(tan(x) - x)))*2*x^2 + 6*x + 9.
((−02⋅2cos⁡(0)log⁡(∣tan⁡(0)−0∣)+∣−102∣)+0⋅6)+9\left(\left(- 0^{2} \cdot 2 \frac{\cos{\left(0 \right)}}{\log{\left(\left|{\tan{\left(0 \right)} - 0}\right| \right)}} + \left|{- \frac{1}{0^{2}}}\right|\right) + 0 \cdot 6\right) + 9
Resultado:
f(0)=∞f{\left(0 \right)} = \infty
signof no cruza Y
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
x2(2sin⁡(x)log⁡(∣−x+tan⁡(x)∣)+2cos⁡(x)tan⁡2(x)sign⁡(−x+tan⁡(x))log⁡(∣−x+tan⁡(x)∣)2∣−x+tan⁡(x)∣)−4xcos⁡(x)log⁡(∣−x+tan⁡(x)∣)+(1+2x3)sign⁡(x−1x2)+6=0x^{2} \left(\frac{2 \sin{\left(x \right)}}{\log{\left(\left|{- x + \tan{\left(x \right)}}\right| \right)}} + \frac{2 \cos{\left(x \right)} \tan^{2}{\left(x \right)} \operatorname{sign}{\left(- x + \tan{\left(x \right)} \right)}}{\log{\left(\left|{- x + \tan{\left(x \right)}}\right| \right)}^{2} \left|{- x + \tan{\left(x \right)}}\right|}\right) - \frac{4 x \cos{\left(x \right)}}{\log{\left(\left|{- x + \tan{\left(x \right)}}\right| \right)}} + \left(1 + \frac{2}{x^{3}}\right) \operatorname{sign}{\left(x - \frac{1}{x^{2}} \right)} + 6 = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=18.9264058354201x_{1} = 18.9264058354201
x2=50.2998142615969x_{2} = 50.2998142615969
x3=−53.4410077699145x_{3} = -53.4410077699145
x4=−12.7614185116853x_{4} = -12.7614185116853
x5=44.0208807826813x_{5} = 44.0208807826813
x6=72.2871602270437x_{6} = 72.2871602270437
x7=100.549257995514x_{7} = 100.549257995514
x8=−81.7075317813637x_{8} = -81.7075317813637
x9=−75.4266346827908x_{9} = -75.4266346827908
x10=−59.7208761622848x_{10} = -59.7208761622848
x11=−37.758412241665x_{11} = -37.758412241665
x12=−66.001338580364x_{12} = -66.001338580364
x13=40.8973432278724x_{13} = 40.8973432278724
x14=−72.2822481243589x_{14} = -72.2822481243589
x15=−106.833892440804x_{15} = -106.833892440804
x16=69.1408591269651x_{16} = 69.1408591269651
x17=−87.9887673331621x_{17} = -87.9887673331621
x18=78.5677422051829x_{18} = 78.5677422051829
x19=−44.0325759301551x_{19} = -44.0325759301551
x20=−78.563502726215x_{20} = -78.563502726215
x21=−6.68642743061798x_{21} = -6.68642743061798
x22=−100.551993225384x_{22} = -100.551993225384
x23=97.4115904152189x_{23} = 97.4115904152189
x24=−40.884054909174x_{24} = -40.884054909174
x25=−15.8067010908366x_{25} = -15.8067010908366
x26=−91.1267738597789x_{26} = -91.1267738597789
x27=25.1942813811105x_{27} = 25.1942813811105
x28=−25.2245963745014x_{28} = -25.2245963745014
x29=15.871906339153x_{29} = 15.871906339153
x30=−3.4855577220419x_{30} = -3.4855577220419
x31=59.7277486157697x_{31} = 59.7277486157697
x32=56.5795967393325x_{32} = 56.5795967393325
x33=−69.1461717480462x_{33} = -69.1461717480462
x34=81.7035764585142x_{34} = 81.7035764585142
x35=−9.57178791799043x_{35} = -9.57178791799043
x36=−47.1619528823701x_{36} = -47.1619528823701
x37=75.4220778711379x_{37} = 75.4220778711379
x38=53.4493547996909x_{38} = 53.4493547996909
x39=−84.8450292391343x_{39} = -84.8450292391343
x40=9.71336796192775x_{40} = 9.71336796192775
x41=94.2672035142578x_{41} = 94.2672035142578
x42=12.6687201304769x_{42} = 12.6687201304769
x43=−1.77810988385759x_{43} = -1.77810988385759
x44=22.1036243063686x_{44} = 22.1036243063686
x45=62.8599964822152x_{45} = 62.8599964822152
x46=−94.2702717458136x_{46} = -94.2702717458136
x47=34.6255785634057x_{47} = 34.6255785634057
x48=−22.0658159146584x_{48} = -22.0658159146584
x49=47.172329623982x_{49} = 47.172329623982
x50=6.43748560224016x_{50} = 6.43748560224016
x51=−34.6078835525902x_{51} = -34.6078835525902
x52=−31.4880803517962x_{52} = -31.4880803517962
x53=91.1300326089527x_{53} = 91.1300326089527
x54=87.9852989815678x_{54} = 87.9852989815678
x55=−56.5871516552196x_{55} = -56.5871516552196
x56=−18.9751325961568x_{56} = -18.9751325961568
x57=−62.8662782477935x_{57} = -62.8662782477935
x58=37.7431608214804x_{58} = 37.7431608214804
x59=−50.3090917332535x_{59} = -50.3090917332535
x60=28.3593255982933x_{60} = 28.3593255982933
x61=84.8487286205416x_{61} = 84.8487286205416
x62=−28.3344594475311x_{62} = -28.3344594475311
x63=−97.408696123488x_{63} = -97.408696123488
x64=0.672717116300184x_{64} = 0.672717116300184
x65=66.0071040861331x_{65} = 66.0071040861331
x66=31.4672625504045x_{66} = 31.4672625504045
Signos de extremos en los puntos:
(18.92640583542006, -101.769343354913)

(50.299814261596886, -929.876887570917)

(-53.441007769914535, 1176.85724039848)

(-12.761418511685255, -181.056914073388)

(44.020880782681346, -706.376980136213)

(72.28716022704367, 2955.53576436538)

(100.54925799551361, -3672.16202495061)

(-81.70753178136374, -3431.1551358651)

(-75.42663468279082, -2999.25302228719)

(-59.720876162284796, 1453.98336979912)

(-37.75841224166503, -963.997921524853)

(-66.00133858036405, 1757.87635605271)

(40.89734322787239, 1195.58035392363)

(-72.28224812435886, 2088.11969585778)

(-106.83389244080402, -5411.07629160685)

(69.14085912696508, -1763.4456595883)

(-87.98876733316212, -3888.56033769412)

(78.5677422051829, 3387.13213710201)

(-44.032575930155076, -1234.69669950805)

(-78.56350272621502, 2444.34499072505)

(-6.686427430617981, -69.250631480034)

(-100.55199322538377, -4878.7693347274)

(97.41159041521888, 4834.69503681404)

(-40.88405490917401, 704.893146521838)

(-15.806701090836608, 110.523561063908)

(-91.12677385977888, 3233.45578763719)

(25.19428138111053, -207.642321248861)

(-25.224596374501434, -510.152565583937)

(15.871906339152966, 300.588075280485)

(-3.4855577220419027, 11.7169804220918)

(59.727748615769734, 2170.67456051776)

(56.57959673933248, -1180.88577034959)

(-69.1461717480462, -2593.1674279521)

(81.70357645851425, -2450.68878533177)

(-9.571787917990433, 41.9559234841178)

(-47.16195288237008, 926.975186202914)

(75.42207787113792, -2094.1610975484)

(53.4493547996909, 1818.19872013581)

(-84.84502923913432, 2826.22279639294)

(9.713367961927753, 157.651261315906)

(94.26720351425783, -3239.96011571782)

(12.668720130476869, -28.486265328065)

(-1.7781098838575915, 1.1192391859289)

(22.103624306368623, 477.888974648464)

(62.8599964822152, -1458.89025191094)

(-94.27027174581356, -4371.18474257796)

(34.62557856340566, 926.671800233348)

(-22.06581591465841, 212.876223129989)

(47.17232962398199, 1492.97999009949)

(6.437485602240162, 9.47244179660039)

(-34.607883552590195, 511.272666028188)

(-31.48808035179616, -722.173164164373)

(91.13003260895273, 4326.99638602441)

(87.9852989815678, -2832.71619765727)

(-56.58715165521956, -1859.88563961156)

(-18.97513259615679, -329.17342635265)

(-62.86627824779348, -2213.2473964156)

(37.743160821480416, -510.989865039299)

(-50.309091733253545, -1533.52953922317)

(28.3593255982933, 687.0837332611)

(84.8487286205416, 3844.38506630239)

(-28.334459447531096, 346.923439388858)

(-97.40869612348799, 3665.77352805373)

(0.6727171163001838, 14.912374220297)

(66.00710408613314, 2549.92655473021)

(31.46726255040447, -344.44307756174)


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=18.9264058354201x_{1} = 18.9264058354201
x2=50.2998142615969x_{2} = 50.2998142615969
x3=−12.7614185116853x_{3} = -12.7614185116853
x4=44.0208807826813x_{4} = 44.0208807826813
x5=100.549257995514x_{5} = 100.549257995514
x6=−81.7075317813637x_{6} = -81.7075317813637
x7=−75.4266346827908x_{7} = -75.4266346827908
x8=−37.758412241665x_{8} = -37.758412241665
x9=−106.833892440804x_{9} = -106.833892440804
x10=69.1408591269651x_{10} = 69.1408591269651
x11=−87.9887673331621x_{11} = -87.9887673331621
x12=−44.0325759301551x_{12} = -44.0325759301551
x13=−6.68642743061798x_{13} = -6.68642743061798
x14=−100.551993225384x_{14} = -100.551993225384
x15=25.1942813811105x_{15} = 25.1942813811105
x16=−25.2245963745014x_{16} = -25.2245963745014
x17=56.5795967393325x_{17} = 56.5795967393325
x18=−69.1461717480462x_{18} = -69.1461717480462
x19=81.7035764585142x_{19} = 81.7035764585142
x20=75.4220778711379x_{20} = 75.4220778711379
x21=94.2672035142578x_{21} = 94.2672035142578
x22=12.6687201304769x_{22} = 12.6687201304769
x23=−1.77810988385759x_{23} = -1.77810988385759
x24=62.8599964822152x_{24} = 62.8599964822152
x25=−94.2702717458136x_{25} = -94.2702717458136
x26=6.43748560224016x_{26} = 6.43748560224016
x27=−31.4880803517962x_{27} = -31.4880803517962
x28=87.9852989815678x_{28} = 87.9852989815678
x29=−56.5871516552196x_{29} = -56.5871516552196
x30=−18.9751325961568x_{30} = -18.9751325961568
x31=−62.8662782477935x_{31} = -62.8662782477935
x32=37.7431608214804x_{32} = 37.7431608214804
x33=−50.3090917332535x_{33} = -50.3090917332535
x34=0.672717116300184x_{34} = 0.672717116300184
x35=31.4672625504045x_{35} = 31.4672625504045
Puntos máximos de la función:
x35=−53.4410077699145x_{35} = -53.4410077699145
x35=72.2871602270437x_{35} = 72.2871602270437
x35=−59.7208761622848x_{35} = -59.7208761622848
x35=−66.001338580364x_{35} = -66.001338580364
x35=40.8973432278724x_{35} = 40.8973432278724
x35=−72.2822481243589x_{35} = -72.2822481243589
x35=78.5677422051829x_{35} = 78.5677422051829
x35=−78.563502726215x_{35} = -78.563502726215
x35=97.4115904152189x_{35} = 97.4115904152189
x35=−40.884054909174x_{35} = -40.884054909174
x35=−15.8067010908366x_{35} = -15.8067010908366
x35=−91.1267738597789x_{35} = -91.1267738597789
x35=15.871906339153x_{35} = 15.871906339153
x35=−3.4855577220419x_{35} = -3.4855577220419
x35=59.7277486157697x_{35} = 59.7277486157697
x35=−9.57178791799043x_{35} = -9.57178791799043
x35=−47.1619528823701x_{35} = -47.1619528823701
x35=53.4493547996909x_{35} = 53.4493547996909
x35=−84.8450292391343x_{35} = -84.8450292391343
x35=9.71336796192775x_{35} = 9.71336796192775
x35=22.1036243063686x_{35} = 22.1036243063686
x35=34.6255785634057x_{35} = 34.6255785634057
x35=−22.0658159146584x_{35} = -22.0658159146584
x35=47.172329623982x_{35} = 47.172329623982
x35=−34.6078835525902x_{35} = -34.6078835525902
x35=91.1300326089527x_{35} = 91.1300326089527
x35=28.3593255982933x_{35} = 28.3593255982933
x35=84.8487286205416x_{35} = 84.8487286205416
x35=−28.3344594475311x_{35} = -28.3344594475311
x35=−97.408696123488x_{35} = -97.408696123488
x35=66.0071040861331x_{35} = 66.0071040861331
Decrece en los intervalos
[100.549257995514,∞)\left[100.549257995514, \infty\right)
Crece en los intervalos
(−∞,−106.833892440804]\left(-\infty, -106.833892440804\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→−∞((6x+(−x2⋅2cos⁡(x)log⁡(∣−x+tan⁡(x)∣)+∣x−1x2∣))+9)y = \lim_{x \to -\infty}\left(\left(6 x + \left(- x^{2} \cdot 2 \frac{\cos{\left(x \right)}}{\log{\left(\left|{- x + \tan{\left(x \right)}}\right| \right)}} + \left|{x - \frac{1}{x^{2}}}\right|\right)\right) + 9\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞((6x+(−x2⋅2cos⁡(x)log⁡(∣−x+tan⁡(x)∣)+∣x−1x2∣))+9)y = \lim_{x \to \infty}\left(\left(6 x + \left(- x^{2} \cdot 2 \frac{\cos{\left(x \right)}}{\log{\left(\left|{- x + \tan{\left(x \right)}}\right| \right)}} + \left|{x - \frac{1}{x^{2}}}\right|\right)\right) + 9\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función |x - 1/x^2| - (cos(x)/log(Abs(tan(x) - x)))*2*x^2 + 6*x + 9, 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→−∞((6x+(−x2⋅2cos⁡(x)log⁡(∣−x+tan⁡(x)∣)+∣x−1x2∣))+9x)y = x \lim_{x \to -\infty}\left(\frac{\left(6 x + \left(- x^{2} \cdot 2 \frac{\cos{\left(x \right)}}{\log{\left(\left|{- x + \tan{\left(x \right)}}\right| \right)}} + \left|{x - \frac{1}{x^{2}}}\right|\right)\right) + 9}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞((6x+(−x2⋅2cos⁡(x)log⁡(∣−x+tan⁡(x)∣)+∣x−1x2∣))+9x)y = x \lim_{x \to \infty}\left(\frac{\left(6 x + \left(- x^{2} \cdot 2 \frac{\cos{\left(x \right)}}{\log{\left(\left|{- x + \tan{\left(x \right)}}\right| \right)}} + \left|{x - \frac{1}{x^{2}}}\right|\right)\right) + 9}{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:
(6x+(−x2⋅2cos⁡(x)log⁡(∣−x+tan⁡(x)∣)+∣x−1x2∣))+9=−2x2cos⁡(x)log⁡(∣x−tan⁡(x)∣)−6x+∣x+1x2∣+9\left(6 x + \left(- x^{2} \cdot 2 \frac{\cos{\left(x \right)}}{\log{\left(\left|{- x + \tan{\left(x \right)}}\right| \right)}} + \left|{x - \frac{1}{x^{2}}}\right|\right)\right) + 9 = - \frac{2 x^{2} \cos{\left(x \right)}}{\log{\left(\left|{x - \tan{\left(x \right)}}\right| \right)}} - 6 x + \left|{x + \frac{1}{x^{2}}}\right| + 9
- No
(6x+(−x2⋅2cos⁡(x)log⁡(∣−x+tan⁡(x)∣)+∣x−1x2∣))+9=2x2cos⁡(x)log⁡(∣x−tan⁡(x)∣)+6x−∣x+1x2∣−9\left(6 x + \left(- x^{2} \cdot 2 \frac{\cos{\left(x \right)}}{\log{\left(\left|{- x + \tan{\left(x \right)}}\right| \right)}} + \left|{x - \frac{1}{x^{2}}}\right|\right)\right) + 9 = \frac{2 x^{2} \cos{\left(x \right)}}{\log{\left(\left|{x - \tan{\left(x \right)}}\right| \right)}} + 6 x - \left|{x + \frac{1}{x^{2}}}\right| - 9
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор