Sr Examen

Otras calculadoras

y=tgx-ctgx/4
Trazado el gráfico de la función/ y=tgx-ctgx/4

Gráfico de la función y = y=tgx-ctgx/4

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                cot(x)
f(x) = tan(x) - ------
                  4
f(x)=tan⁡(x)−cot⁡(x)4f{\left(x \right)} = \tan{\left(x \right)} - \frac{\cot{\left(x \right)}}{4} 
f = tan(x) - cot(x)/4
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:
tan⁡(x)−cot⁡(x)4=0\tan{\left(x \right)} - \frac{\cot{\left(x \right)}}{4} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−atan⁡(12)x_{1} = - \operatorname{atan}{\left(\frac{1}{2} \right)}
x2=atan⁡(12)x_{2} = \operatorname{atan}{\left(\frac{1}{2} \right)}
Solución numérica
x1=56.0850201556155x_{1} = 56.0850201556155
x2=16.1716108769498x_{2} = 16.1716108769498
x3=−3.6052402625906x_{3} = -3.6052402625906
x4=−2.67794504458899x_{4} = -2.67794504458899
x5=−18.385908312538x_{5} = -18.385908312538
x6=59.2266128092053x_{6} = 59.2266128092053
x7=−53.8707227200273x_{7} = -53.8707227200273
x8=−71.7929834235644x_{8} = -71.7929834235644
x9=−65.5097981163849x_{9} = -65.5097981163849
x10=−90.6425393451032x_{10} = -90.6425393451032
x11=−40.3770568876665x_{11} = -40.3770568876665
x12=85.2866492559252x_{12} = 85.2866492559252
x13=18.385908312538x_{13} = 18.385908312538
x14=5.81953769817878x_{14} = 5.81953769817878
x15=−57.0123153736171x_{15} = -57.0123153736171
x16=−63.2955006807967x_{16} = -63.2955006807967
x17=60.1539080272069x_{17} = 60.1539080272069
x18=−84.3593540379236x_{18} = -84.3593540379236
x19=−31.8795741448987x_{19} = -31.8795741448987
x20=−82.1450566023354x_{20} = -82.1450566023354
x21=−16.1716108769498x_{21} = -16.1716108769498
x22=37.2354642340767x_{22} = 37.2354642340767
x23=96.9257246522828x_{23} = 96.9257246522828
x24=2.67794504458899x_{24} = 2.67794504458899
x25=68.6513907699746x_{25} = 68.6513907699746
x26=71.7929834235644x_{26} = 71.7929834235644
x27=9.88842556977019x_{27} = 9.88842556977019
x28=27.8106862733073x_{28} = 27.8106862733073
x29=−79.0034639487456x_{29} = -79.0034639487456
x30=24.6690936197175x_{30} = 24.6690936197175
x31=−24.6690936197175x_{31} = -24.6690936197175
x32=53.8707227200273x_{32} = 53.8707227200273
x33=−9.88842556977019x_{33} = -9.88842556977019
x34=12.1027230053584x_{34} = 12.1027230053584
x35=88.428241909515x_{35} = 88.428241909515
x36=−49.8018348484359x_{36} = -49.8018348484359
x37=−87.5009466915134x_{37} = -87.5009466915134
x38=63.2955006807967x_{38} = 63.2955006807967
x39=47.5875374128477x_{39} = 47.5875374128477
x40=−25.5963888377192x_{40} = -25.5963888377192
x41=−74.9345760771542x_{41} = -74.9345760771542
x42=15.2443156589482x_{42} = 15.2443156589482
x43=−47.5875374128477x_{43} = -47.5875374128477
x44=97.8530198702844x_{44} = 97.8530198702844
x45=−85.2866492559252x_{45} = -85.2866492559252
x46=25.5963888377192x_{46} = 25.5963888377192
x47=74.9345760771542x_{47} = 74.9345760771542
x48=−78.076168730744x_{48} = -78.076168730744
x49=−21.5275009661277x_{49} = -21.5275009661277
x50=40.3770568876665x_{50} = 40.3770568876665
x51=−100.067317305873x_{51} = -100.067317305873
x52=−69.5786859879763x_{52} = -69.5786859879763
x53=−35.0211667984885x_{53} = -35.0211667984885
x54=69.5786859879763x_{54} = 69.5786859879763
x55=31.8795741448987x_{55} = 31.8795741448987
x56=−46.6602421948461x_{56} = -46.6602421948461
x57=82.1450566023354x_{57} = 82.1450566023354
x58=52.9434275020257x_{58} = 52.9434275020257
x59=66.4370933343865x_{59} = 66.4370933343865
x60=−52.9434275020257x_{60} = -52.9434275020257
x61=−60.1539080272069x_{61} = -60.1539080272069
x62=−91.5698345631048x_{62} = -91.5698345631048
x63=−30.9522789268971x_{63} = -30.9522789268971
x64=49.8018348484359x_{64} = 49.8018348484359
x65=41.3043521056681x_{65} = 41.3043521056681
x66=−43.5186495412563x_{66} = -43.5186495412563
x67=100.067317305873x_{67} = 100.067317305873
x68=−38.1627594520783x_{68} = -38.1627594520783
x69=34.0938715804869x_{69} = 34.0938715804869
x70=−96.9257246522828x_{70} = -96.9257246522828
x71=−19.3132035305396x_{71} = -19.3132035305396
x72=44.4459447592579x_{72} = 44.4459447592579
x73=−13.03001822336x_{73} = -13.03001822336
x74=38.1627594520783x_{74} = 38.1627594520783
x75=81.2177613843338x_{75} = 81.2177613843338
x76=90.6425393451032x_{76} = 90.6425393451032
x77=−41.3043521056681x_{77} = -41.3043521056681
x78=−6.74683291618039x_{78} = -6.74683291618039
x79=30.9522789268971x_{79} = 30.9522789268971
x80=46.6602421948461x_{80} = 46.6602421948461
x81=78.076168730744x_{81} = 78.076168730744
x82=−93.784131998693x_{82} = -93.784131998693
x83=−12.1027230053584x_{83} = -12.1027230053584
x84=91.5698345631048x_{84} = 91.5698345631048
x85=8.96113035176857x_{85} = 8.96113035176857
x86=−56.0850201556155x_{86} = -56.0850201556155
x87=75.8618712951559x_{87} = 75.8618712951559
x88=19.3132035305396x_{88} = 19.3132035305396
x89=−97.8530198702844x_{89} = -97.8530198702844
x90=93.784131998693x_{90} = 93.784131998693
x91=62.3682054627951x_{91} = 62.3682054627951
x92=−68.6513907699746x_{92} = -68.6513907699746
x93=−75.8618712951559x_{93} = -75.8618712951559
x94=22.4547961841294x_{94} = 22.4547961841294
x95=−34.0938715804869x_{95} = -34.0938715804869
x96=−5.81953769817878x_{96} = -5.81953769817878
x97=−27.8106862733073x_{97} = -27.8106862733073
x98=3.6052402625906x_{98} = 3.6052402625906
x99=−62.3682054627951x_{99} = -62.3682054627951
x100=84.3593540379236x_{100} = 84.3593540379236
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)+cot⁡2(x)4+54=0\tan^{2}{\left(x \right)} + \frac{\cot^{2}{\left(x \right)}}{4} + \frac{5}{4} = 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
2(tan⁡2(x)+1)tan⁡(x)−(cot⁡2(x)+1)cot⁡(x)2=02 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{\left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{2} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−atan⁡(22)x_{1} = - \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}
x2=atan⁡(22)x_{2} = \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}

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
[atan⁡(22),∞)\left[\operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}, \infty\right)
Convexa en los intervalos
(−∞,−atan⁡(22)]\left(-\infty, - \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}\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→−∞(tan⁡(x)−cot⁡(x)4)y = \lim_{x \to -\infty}\left(\tan{\left(x \right)} - \frac{\cot{\left(x \right)}}{4}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(tan⁡(x)−cot⁡(x)4)y = \lim_{x \to \infty}\left(\tan{\left(x \right)} - \frac{\cot{\left(x \right)}}{4}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(x) - cot(x)/4, 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→−∞(tan⁡(x)−cot⁡(x)4x)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} - \frac{\cot{\left(x \right)}}{4}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(tan⁡(x)−cot⁡(x)4x)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} - \frac{\cot{\left(x \right)}}{4}}{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:
tan⁡(x)−cot⁡(x)4=−tan⁡(x)+cot⁡(x)4\tan{\left(x \right)} - \frac{\cot{\left(x \right)}}{4} = - \tan{\left(x \right)} + \frac{\cot{\left(x \right)}}{4}
- No
tan⁡(x)−cot⁡(x)4=tan⁡(x)−cot⁡(x)4\tan{\left(x \right)} - \frac{\cot{\left(x \right)}}{4} = \tan{\left(x \right)} - \frac{\cot{\left(x \right)}}{4}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = y=tgx-ctgx/4
    © Sr Examen – Калькулятор