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y=|ctgx|-ctgx
Trazado el gráfico de la función/ y=|ctgx|-ctgx

Gráfico de la función y = y=|ctgx|-ctgx

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = |cot(x)| - cot(x)
f(x)=−cot⁡(x)+∣cot⁡(x)∣f{\left(x \right)} = - \cot{\left(x \right)} + \left|{\cot{\left(x \right)}}\right| 
f = -cot(x) + Abs(cot(x))
Gráfico de la función
02468-8-6-4-2-101002000
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:
−cot⁡(x)+∣cot⁡(x)∣=0- \cot{\left(x \right)} + \left|{\cot{\left(x \right)}}\right| = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=98.9601685880785x_{1} = 98.9601685880785
x2=−39.2699081698724x_{2} = -39.2699081698724
x3=26.7035375555132x_{3} = 26.7035375555132
x4=54x_{4} = 54
x5=−32.9867228626928x_{5} = -32.9867228626928
x6=−100x_{6} = -100
x7=−64.4026493985908x_{7} = -64.4026493985908
x8=66x_{8} = 66
x9=54.9778714378214x_{9} = 54.9778714378214
x10=−96x_{10} = -96
x11=−65.75x_{11} = -65.75
x12=1.5707963267949x_{12} = 1.5707963267949
x13=−28x_{13} = -28
x14=29.845130209103x_{14} = 29.845130209103
x15=60x_{15} = 60
x16=4x_{16} = 4
x17=−36.1283155162826x_{17} = -36.1283155162826
x18=−56x_{18} = -56
x19=98x_{19} = 98
x20=76.9690200129499x_{20} = 76.9690200129499
x21=−20.4203522483337x_{21} = -20.4203522483337
x22=−80.1106126665397x_{22} = -80.1106126665397
x23=−48.6946861306418x_{23} = -48.6946861306418
x24=−72x_{24} = -72
x25=32.9867228626928x_{25} = 32.9867228626928
x26=−8x_{26} = -8
x27=7.85398163397448x_{27} = 7.85398163397448
x28=−90x_{28} = -90
x29=10.9955742875643x_{29} = 10.9955742875643
x30=48x_{30} = 48
x31=94.25x_{31} = 94.25
x32=64x_{32} = 64
x33=23.5619449019235x_{33} = 23.5619449019235
x34=70.6858347057703x_{34} = 70.6858347057703
x35=82x_{35} = 82
x36=−4.71238898038469x_{36} = -4.71238898038469
x37=45.553093477052x_{37} = 45.553093477052
x38=−94x_{38} = -94
x39=48.6946861306418x_{39} = 48.6946861306418
x40=−2x_{40} = -2
x41=−78x_{41} = -78
x42=58x_{42} = 58
x43=95.8185759344887x_{43} = 95.8185759344887
x44=−34x_{44} = -34
x45=−61.261056745001x_{45} = -61.261056745001
x46=−83.2522053201295x_{46} = -83.2522053201295
x47=−26.7035375555132x_{47} = -26.7035375555132
x48=−52x_{48} = -52
x49=−92.6769832808989x_{49} = -92.6769832808989
x50=−62x_{50} = -62
x51=−17.2787595947439x_{51} = -17.2787595947439
x52=−30x_{52} = -30
x53=92x_{53} = 92
x54=−86.3937979737193x_{54} = -86.3937979737193
x55=−84x_{55} = -84
x56=70x_{56} = 70
x57=73.8274273593601x_{57} = 73.8274273593601
x58=−87.75x_{58} = -87.75
x59=−6x_{59} = -6
x60=26x_{60} = 26
x61=38x_{61} = 38
x62=−98.9601685880785x_{62} = -98.9601685880785
x63=44x_{63} = 44
x64=−50x_{64} = -50
x65=−42.4115008234622x_{65} = -42.4115008234622
x66=36x_{66} = 36
x67=−10.9955742875643x_{67} = -10.9955742875643
x68=17.2787595947439x_{68} = 17.2787595947439
x69=42x_{69} = 42
x70=22x_{70} = 22
x71=76x_{71} = 76
x72=−46x_{72} = -46
x73=16x_{73} = 16
x74=4.71238898038469x_{74} = 4.71238898038469
x75=−40x_{75} = -40
x76=−18x_{76} = -18
x77=−58.1194640914112x_{77} = -58.1194640914112
x78=89.5353906273091x_{78} = 89.5353906273091
x79=14x_{79} = 14
x80=−70.6858347057703x_{80} = -70.6858347057703
x81=86x_{81} = 86
x82=39.2699081698724x_{82} = 39.2699081698724
x83=−54.9778714378214x_{83} = -54.9778714378214
x84=−43.75x_{84} = -43.75
x85=67.5442420521806x_{85} = 67.5442420521806
x86=10x_{86} = 10
x87=−12x_{87} = -12
x88=−24x_{88} = -24
x89=88x_{89} = 88
x90=80x_{90} = 80
x91=−14.1371669411541x_{91} = -14.1371669411541
x92=−21.75x_{92} = -21.75
x93=83.2522053201295x_{93} = 83.2522053201295
x94=32x_{94} = 32
x95=61.261056745001x_{95} = 61.261056745001
x96=20x_{96} = 20
x97=−74x_{97} = -74
x98=51.8362787842316x_{98} = 51.8362787842316
x99=−76.9690200129499x_{99} = -76.9690200129499
x100=−68x_{100} = -68
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
(−cot⁡2(x)−1)sign⁡(cot⁡(x))+cot⁡2(x)+1=0\left(- \cot^{2}{\left(x \right)} - 1\right) \operatorname{sign}{\left(\cot{\left(x \right)} \right)} + \cot^{2}{\left(x \right)} + 1 = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−33.38561299476x_{1} = -33.38561299476
x2=54x_{2} = 54
x3=−100x_{3} = -100
x4=66x_{4} = 66
x5=−96x_{5} = -96
x6=−92.7703341398873x_{6} = -92.7703341398873
x7=−65.75x_{7} = -65.75
x8=−28x_{8} = -28
x9=10.7442050079644x_{9} = 10.7442050079644
x10=−77.5408431813856x_{10} = -77.5408431813856
x11=60x_{11} = 60
x12=76.9061633594664x_{12} = 76.9061633594664
x13=−42.7160706375309x_{13} = -42.7160706375309
x14=−56x_{14} = -56
x15=98x_{15} = 98
x16=−58.5444355187604x_{16} = -58.5444355187604
x17=−72x_{17} = -72
x18=−8x_{18} = -8
x19=−90x_{19} = -90
x20=70.0756770314154x_{20} = 70.0756770314154
x21=48x_{21} = 48
x22=94.25x_{22} = 94.25
x23=64x_{23} = 64
x24=72.5007206006979x_{24} = 72.5007206006979
x25=−14.9682117128838x_{25} = -14.9682117128838
x26=54.8557441540359x_{26} = 54.8557441540359
x27=82x_{27} = 82
x28=−64.7330839489965x_{28} = -64.7330839489965
x29=50.4480595376821x_{29} = 50.4480595376821
x30=−11.318049831998x_{30} = -11.318049831998
x31=−26.7067729887365x_{31} = -26.7067729887365
x32=−2x_{32} = -2
x33=−78x_{33} = -78
x34=58x_{34} = 58
x35=−99.6313800125244x_{35} = -99.6313800125244
x36=−34x_{36} = -34
x37=−20.6995601091398x_{37} = -20.6995601091398
x38=−83.75x_{38} = -83.75
x39=23.332047952558x_{39} = 23.332047952558
x40=−55.4594614389181x_{40} = -55.4594614389181
x41=−52x_{41} = -52
x42=−23.75x_{42} = -23.75
x43=−62x_{43} = -62
x44=−30x_{44} = -30
x45=−48.7272430134738x_{45} = -48.7272430134738
x46=98.953605307344x_{46} = 98.953605307344
x47=89.3762475917371x_{47} = 89.3762475917371
x48=70x_{48} = 70
x49=−87.75x_{49} = -87.75
x50=−93.75x_{50} = -93.75
x51=45.3471894767407x_{51} = 45.3471894767407
x52=−17.3986217195782x_{52} = -17.3986217195782
x53=−6x_{53} = -6
x54=−70.7484138174238x_{54} = -70.7484138174238
x55=−61.652487121769x_{55} = -61.652487121769
x56=26x_{56} = 26
x57=94.5504223650703x_{57} = 94.5504223650703
x58=59.7277281186083x_{58} = 59.7277281186083
x59=38x_{59} = 38
x60=16.9999788791615x_{60} = 16.9999788791615
x61=28.3921582488827x_{61} = 28.3921582488827
x62=44x_{62} = 44
x63=67.3619164241845x_{63} = 67.3619164241845
x64=−39.5148469139329x_{64} = -39.5148469139329
x65=32.8019279088066x_{65} = 32.8019279088066
x66=−50x_{66} = -50
x67=25.5269252341341x_{67} = 25.5269252341341
x68=36x_{68} = 36
x69=42x_{69} = 42
x70=22x_{70} = 22
x71=4.70113063979927x_{71} = 4.70113063979927
x72=76x_{72} = 76
x73=81.8782751902011x_{73} = 81.8782751902011
x74=−46x_{74} = -46
x75=16x_{75} = 16
x76=47.8512676053609x_{76} = 47.8512676053609
x77=−8.52322650975524x_{77} = -8.52322650975524
x78=6.33269911147606x_{78} = 6.33269911147606
x79=−40x_{79} = -40
x80=92.25x_{80} = 92.25
x81=−18x_{81} = -18
x82=14x_{82} = 14
x83=86x_{83} = 86
x84=−83.8196156882187x_{84} = -83.8196156882187
x85=−43.75x_{85} = -43.75
x86=−86.7506259137328x_{86} = -86.7506259137328
x87=10x_{87} = 10
x88=−12x_{88} = -12
x89=88x_{89} = 88
x90=80x_{90} = 80
x91=35.91266153345x_{91} = 35.91266153345
x92=−21.75x_{92} = -21.75
x93=32x_{93} = 32
x94=−80.7543623138673x_{94} = -80.7543623138673
x95=1.31647185509841x_{95} = 1.31647185509841
x96=20x_{96} = 20
x97=−74x_{97} = -74
x98=−36.8520619269931x_{98} = -36.8520619269931
Signos de extremos en los puntos:
(-33.38561299476004, 0)

(54, 0)

(-100, 0)

(66, 0)

(-96, 0)

(-92.7703341398873, 0)

(-65.75, 0)

(-28, 0)

(10.744205007964386, 0)

(-77.54084318138558, 0)

(60, 0)

(76.90616335946636, 0)

(-42.71607063753089, 0)

(-56, 0)

(98, 0)

(-58.54443551876036, 0)

(-72, 0)

(-8, 0)

(-90, 0)

(70.07567703141541, 0)

(48, 0)

(94.25, 0)

(64, 0)

(72.50072060069789, 0)

(-14.968211712883823, 0)

(54.85574415403593, 0)

(82, 0)

(-64.73308394899654, 0)

(50.4480595376821, 0)

(-11.31804983199799, 0)

(-26.706772988736542, 0)

(-2, 0)

(-78, 0)

(58, 0)

(-99.63138001252443, 0)

(-34, 0)

(-20.69956010913982, 0)

(-83.75, 0)

(23.332047952558007, 0)

(-55.45946143891815, 0)

(-52, 0)

(-23.75, 0)

(-62, 0)

(-30, 0)

(-48.72724301347383, 0)

(98.953605307344, 0)

(89.37624759173707, 0)

(70, 0)

(-87.75, 0)

(-93.75, 0)

(45.347189476740716, 0)

(-17.398621719578234, 0)

(-6, 0)

(-70.74841381742375, 0)

(-61.65248712176901, 0)

(26, 0)

(94.55042236507028, 0)

(59.727728118608255, 0)

(38, 0)

(16.999978879161517, 0)

(28.392158248882744, 0)

(44, 0)

(67.36191642418451, 0)

(-39.51484691393293, 0)

(32.80192790880663, 0)

(-50, 0)

(25.526925234134108, 0)

(36, 0)

(42, 0)

(22, 0)

(4.701130639799267, 0)

(76, 0)

(81.87827519020108, 0)

(-46, 0)

(16, 0)

(47.851267605360945, 0)

(-8.523226509755236, 0)

(6.332699111476061, 0)

(-40, 0)

(92.25, 0)

(-18, 0)

(14, 0)

(86, 0)

(-83.81961568821875, 0)

(-43.75, 0)

(-86.75062591373276, 0)

(10, 0)

(-12, 0)

(88, 0)

(80, 0)

(35.91266153345002, 0)

(-21.75, 0)

(32, 0)

(-80.75436231386733, 0)

(1.3164718550984087, 0)

(20, 0)

(-74, 0)

(-36.852061926993095, 0)


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:
La función no tiene puntos mínimos
La función no tiene puntos máximos
No cambia el valor en todo el eje numérico
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(cot⁡2(x)+1)((cot⁡2(x)+1)δ(cot⁡(x))+cot⁡(x)sign⁡(cot⁡(x))−cot⁡(x))=02 \left(\cot^{2}{\left(x \right)} + 1\right) \left(\left(\cot^{2}{\left(x \right)} + 1\right) \delta\left(\cot{\left(x \right)}\right) + \cot{\left(x \right)} \operatorname{sign}{\left(\cot{\left(x \right)} \right)} - \cot{\left(x \right)}\right) = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
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→−∞(−cot⁡(x)+∣cot⁡(x)∣)y = \lim_{x \to -\infty}\left(- \cot{\left(x \right)} + \left|{\cot{\left(x \right)}}\right|\right)
True

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

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(−cot⁡(x)+∣cot⁡(x)∣x)y = x \lim_{x \to \infty}\left(\frac{- \cot{\left(x \right)} + \left|{\cot{\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:
−cot⁡(x)+∣cot⁡(x)∣=cot⁡(x)+∣cot⁡(x)∣- \cot{\left(x \right)} + \left|{\cot{\left(x \right)}}\right| = \cot{\left(x \right)} + \left|{\cot{\left(x \right)}}\right|
- No
−cot⁡(x)+∣cot⁡(x)∣=−cot⁡(x)−∣cot⁡(x)∣- \cot{\left(x \right)} + \left|{\cot{\left(x \right)}}\right| = - \cot{\left(x \right)} - \left|{\cot{\left(x \right)}}\right|
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = y=|ctgx|-ctgx
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