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Trazado el gráfico de la función/ 7^tg(x)-ctg(x)

Gráfico de la función y = 7^tg(x)-ctg(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        tan(x)         
f(x) = 7       - cot(x)
f(x)=7tan⁡(x)−cot⁡(x)f{\left(x \right)} = 7^{\tan{\left(x \right)}} - \cot{\left(x \right)} 
f = 7^tan(x) - cot(x)
Gráfico de la función
02468-8-6-4-2-1010-1e651e65
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:
7tan⁡(x)−cot⁡(x)=07^{\tan{\left(x \right)}} - \cot{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=88.3721212426512x_{1} = 88.3721212426512
x2=−32.9867228626928x_{2} = -32.9867228626928
x3=85.2305285890614x_{3} = 85.2305285890614
x4=91.513713896241x_{4} = 91.513713896241
x5=−98.9601685880785x_{5} = -98.9601685880785
x6=61.261056745001x_{6} = 61.261056745001
x7=−84.4154747047874x_{7} = -84.4154747047874
x8=47.5314167459839x_{8} = 47.5314167459839
x9=−10.9955742875642x_{9} = -10.9955742875642
x10=73.8274273593601x_{10} = 73.8274273593601
x11=76.9690200129499x_{11} = 76.9690200129499
x12=−58.1194640914112x_{12} = -58.1194640914112
x13=−64.4026493985908x_{13} = -64.4026493985908
x14=3.54911959572683x_{14} = 3.54911959572683
x15=−61.261056745001x_{15} = -61.261056745001
x16=−51.8362787842316x_{16} = -51.8362787842316
x17=25.5402681708554x_{17} = 25.5402681708554
x18=7.85398163397448x_{18} = 7.85398163397448
x19=−86.3937979737193x_{19} = -86.3937979737193
x20=58.1194640914112x_{20} = 58.1194640914112
x21=−100.123437972736x_{21} = -100.123437972736
x22=23.5619449019235x_{22} = 23.5619449019235
x23=45.5530934770521x_{23} = 45.5530934770521
x24=22.3986755172656x_{24} = 22.3986755172656
x25=75.8057506282921x_{25} = 75.8057506282921
x26=41.2482314388044x_{26} = 41.2482314388044
x27=38.1066387852146x_{27} = 38.1066387852146
x28=−43.5747702081201x_{28} = -43.5747702081201
x29=−70.6858347057703x_{29} = -70.6858347057703
x30=69.5225653211125x_{30} = 69.5225653211125
x31=82.0889359354717x_{31} = 82.0889359354717
x32=29.8451302091031x_{32} = 29.8451302091031
x33=−87.5570673583772x_{33} = -87.5570673583772
x34=−4.71238898038469x_{34} = -4.71238898038469
x35=−95.8185759344887x_{35} = -95.8185759344887
x36=−2.73406571145276x_{36} = -2.73406571145276
x37=−71.8491040904282x_{37} = -71.8491040904282
x38=9.83230490290642x_{38} = 9.83230490290642
x39=80.1106126665397x_{39} = 80.1106126665397
x40=54.9778714378214x_{40} = 54.9778714378214
x41=−76.9690200129499x_{41} = -76.9690200129499
x42=−80.1106126665397x_{42} = -80.1106126665397
x43=60.0977873603431x_{43} = 60.0977873603431
x44=51.8362787842316x_{44} = 51.8362787842316
x45=−49.8579555152997x_{45} = -49.8579555152997
x46=−29.845130209103x_{46} = -29.845130209103
x47=−24.7252142865813x_{47} = -24.7252142865813
x48=−20.4203522483337x_{48} = -20.4203522483337
x49=−48.6946861306418x_{49} = -48.6946861306418
x50=97.7968992034206x_{50} = 97.7968992034206
x51=−17.2787595947439x_{51} = -17.2787595947439
x52=67.5442420521806x_{52} = 67.5442420521806
x53=14.1371669411541x_{53} = 14.1371669411541
x54=−46.7163628617099x_{54} = -46.7163628617099
x55=−26.7035375555132x_{55} = -26.7035375555132
x56=19.2570828636758x_{56} = 19.2570828636758
x57=4.71238898038469x_{57} = 4.71238898038469
x58=70.6858347057703x_{58} = 70.6858347057703
x59=32.9867228626928x_{59} = 32.9867228626928
x60=−83.2522053201295x_{60} = -83.2522053201295
x61=−36.1283155162826x_{61} = -36.1283155162826
x62=44.3898240923941x_{62} = 44.3898240923941
x63=−27.8668069401711x_{63} = -27.8668069401711
x64=−78.1322893976078x_{64} = -78.1322893976078
x65=−92.6769832808989x_{65} = -92.6769832808989
x66=−34.1499922473507x_{66} = -34.1499922473507
x67=−56.1411408224792x_{67} = -56.1411408224792
x68=−68.7075114368384x_{68} = -68.7075114368384
x69=36.1283155162826x_{69} = 36.1283155162826
x70=1.57079632679491x_{70} = 1.57079632679491
x71=17.2787595947439x_{71} = 17.2787595947439
x72=10.9955742875643x_{72} = 10.9955742875643
x73=−12.1588436722221x_{73} = -12.1588436722221
x74=95.8185759344887x_{74} = 95.8185759344887
x75=−14.1371669411541x_{75} = -14.1371669411541
x76=−93.8402526655568x_{76} = -93.8402526655568
x77=−18.4420289794017x_{77} = -18.4420289794017
x78=16.115490210086x_{78} = 16.115490210086
x79=−39.2699081698724x_{79} = -39.2699081698724
x80=53.8146020531635x_{80} = 53.8146020531635
x81=−21.5836216329915x_{81} = -21.5836216329915
x82=−65.5659187832486x_{82} = -65.5659187832486
x83=83.2522053201295x_{83} = 83.2522053201295
x84=26.7035375555132x_{84} = 26.7035375555132
x85=−40.4331775545303x_{85} = -40.4331775545303
x86=98.9601685880785x_{86} = 98.9601685880785
x87=48.6946861306418x_{87} = 48.6946861306418
x88=−5.87565836504255x_{88} = -5.87565836504255
x89=39.2699081698724x_{89} = 39.2699081698724
x90=−7.85398163397448x_{90} = -7.85398163397448
x91=89.5353906273093x_{91} = 89.5353906273093
x92=31.823453478035x_{92} = 31.823453478035
x93=66.3809726675227x_{93} = 66.3809726675227
x94=−62.4243261296588x_{94} = -62.4243261296588
x95=−73.8274273593601x_{95} = -73.8274273593601
x96=−54.9778714378214x_{96} = -54.9778714378214
x97=63.2393800139329x_{97} = 63.2393800139329
x98=92.6769832808989x_{98} = 92.6769832808989
x99=−90.698660011967x_{99} = -90.698660011967
x100=−42.4115008234622x_{100} = -42.4115008234622
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
7tan⁡(x)(tan⁡2(x)+1)log⁡(7)+cot⁡2(x)+1=07^{\tan{\left(x \right)}} \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(7 \right)} + \cot^{2}{\left(x \right)} + 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
7tan⁡(x)(tan⁡2(x)+1)2log⁡(7)2+2⋅7tan⁡(x)(tan⁡2(x)+1)log⁡(7)tan⁡(x)−2(cot⁡2(x)+1)cot⁡(x)=07^{\tan{\left(x \right)}} \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(7 \right)}^{2} + 2 \cdot 7^{\tan{\left(x \right)}} \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(7 \right)} \tan{\left(x \right)} - 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−62.3769176845804x_{1} = -62.3769176845804
x2=−98.9601685880785x_{2} = -98.9601685880785
x3=53.8620104982419x_{3} = 53.8620104982419
x4=92.6769832808989x_{4} = 92.6769832808989
x5=−48.6946861306418x_{5} = -48.6946861306418
x6=−100.076029527658x_{6} = -100.076029527658
x7=−64.4026493985908x_{7} = -64.4026493985908
x8=−12.1114352271437x_{8} = -12.1114352271437
x9=91.5611223413195x_{9} = 91.5611223413195
x10=73.8274273593601x_{10} = 73.8274273593601
x11=45.553093477052x_{11} = 45.553093477052
x12=89.5353906273091x_{12} = 89.5353906273091
x13=31.8708619231134x_{13} = 31.8708619231134
x14=76.9690200129499x_{14} = 76.9690200129499
x15=−93.7928442204783x_{15} = -93.7928442204783
x16=60.1451958054215x_{16} = 60.1451958054215
x17=63.2867884590113x_{17} = 63.2867884590113
x18=−58.1194640914112x_{18} = -58.1194640914112
x19=88.4195296877297x_{19} = 88.4195296877297
x20=−61.261056745001x_{20} = -61.261056745001
x21=−51.8362787842316x_{21} = -51.8362787842316
x22=3.59652804080525x_{22} = 3.59652804080525
x23=7.85398163397448x_{23} = 7.85398163397448
x24=58.1194640914112x_{24} = 58.1194640914112
x25=23.5619449019235x_{25} = 23.5619449019235
x26=−46.6689544166314x_{26} = -46.6689544166314
x27=75.8531590733705x_{27} = 75.8531590733705
x28=−78.0848809525294x_{28} = -78.0848809525294
x29=−34.1025838022723x_{29} = -34.1025838022723
x30=−32.9867228626928x_{30} = -32.9867228626928
x31=82.1363443805501x_{31} = 82.1363443805501
x32=−70.6858347057703x_{32} = -70.6858347057703
x33=25.5876766159338x_{33} = 25.5876766159338
x34=−49.8105470702212x_{34} = -49.8105470702212
x35=−95.8185759344887x_{35} = -95.8185759344887
x36=−26.7035375555132x_{36} = -26.7035375555132
x37=−68.66010299176x_{37} = -68.66010299176
x38=85.2779370341399x_{38} = 85.2779370341399
x39=29.845130209103x_{39} = 29.845130209103
x40=−24.6778058415029x_{40} = -24.6778058415029
x41=80.1106126665397x_{41} = 80.1106126665397
x42=16.1628986551644x_{42} = 16.1628986551644
x43=−76.9690200129499x_{43} = -76.9690200129499
x44=61.261056745001x_{44} = 61.261056745001
x45=66.4283811126011x_{45} = 66.4283811126011
x46=−80.1106126665397x_{46} = -80.1106126665397
x47=51.8362787842316x_{47} = 51.8362787842316
x48=−29.845130209103x_{48} = -29.845130209103
x49=−5.82824991996413x_{49} = -5.82824991996413
x50=−20.4203522483337x_{50} = -20.4203522483337
x51=−65.5185103381702x_{51} = -65.5185103381702
x52=−17.2787595947439x_{52} = -17.2787595947439
x53=−43.5273617630417x_{53} = -43.5273617630417
x54=67.5442420521806x_{54} = 67.5442420521806
x55=−90.6512515668885x_{55} = -90.6512515668885
x56=4.71238898038469x_{56} = 4.71238898038469
x57=70.6858347057703x_{57} = 70.6858347057703
x58=−71.8016956453498x_{58} = -71.8016956453498
x59=32.9867228626928x_{59} = 32.9867228626928
x60=−83.2522053201295x_{60} = -83.2522053201295
x61=−36.1283155162826x_{61} = -36.1283155162826
x62=−21.5362131879131x_{62} = -21.5362131879131
x63=9.87971334798483x_{63} = 9.87971334798483
x64=1.5707963267949x_{64} = 1.5707963267949
x65=−92.6769832808989x_{65} = -92.6769832808989
x66=−27.8193984950927x_{66} = -27.8193984950927
x67=−87.5096589132988x_{67} = -87.5096589132988
x68=22.446083962344x_{68} = 22.446083962344
x69=17.2787595947439x_{69} = 17.2787595947439
x70=10.9955742875643x_{70} = 10.9955742875643
x71=19.3044913087542x_{71} = 19.3044913087542
x72=95.8185759344887x_{72} = 95.8185759344887
x73=−14.1371669411541x_{73} = -14.1371669411541
x74=−40.3857691094519x_{74} = -40.3857691094519
x75=54.9778714378214x_{75} = 54.9778714378214
x76=−4.71238898038469x_{76} = -4.71238898038469
x77=83.2522053201295x_{77} = 83.2522053201295
x78=14.1371669411541x_{78} = 14.1371669411541
x79=98.9601685880785x_{79} = 98.9601685880785
x80=−84.368066259709x_{80} = -84.368066259709
x81=69.5699737661909x_{81} = 69.5699737661909
x82=26.7035375555132x_{82} = 26.7035375555132
x83=−18.3946205343233x_{83} = -18.3946205343233
x84=48.6946861306418x_{84} = 48.6946861306418
x85=41.2956398838828x_{85} = 41.2956398838828
x86=−86.3937979737193x_{86} = -86.3937979737193
x87=97.844307648499x_{87} = 97.844307648499
x88=44.4372325374726x_{88} = 44.4372325374726
x89=38.154047230293x_{89} = 38.154047230293
x90=39.2699081698724x_{90} = 39.2699081698724
x91=−7.85398163397448x_{91} = -7.85398163397448
x92=47.5788251910624x_{92} = 47.5788251910624
x93=−39.2699081698724x_{93} = -39.2699081698724
x94=−56.0937323774008x_{94} = -56.0937323774008
x95=−10.9955742875643x_{95} = -10.9955742875643
x96=−73.8274273593601x_{96} = -73.8274273593601
x97=−2.68665726637434x_{97} = -2.68665726637434
x98=−54.9778714378214x_{98} = -54.9778714378214
x99=−42.4115008234622x_{99} = -42.4115008234622
x100=36.1283155162826x_{100} = 36.1283155162826

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
[98.9601685880785,∞)\left[98.9601685880785, \infty\right)
Convexa en los intervalos
(−∞,−100.076029527658]\left(-\infty, -100.076029527658\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→−∞(7tan⁡(x)−cot⁡(x))y = \lim_{x \to -\infty}\left(7^{\tan{\left(x \right)}} - \cot{\left(x \right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(7tan⁡(x)−cot⁡(x))y = \lim_{x \to \infty}\left(7^{\tan{\left(x \right)}} - \cot{\left(x \right)}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función 7^tan(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→−∞(7tan⁡(x)−cot⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{7^{\tan{\left(x \right)}} - \cot{\left(x \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(7tan⁡(x)−cot⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{7^{\tan{\left(x \right)}} - \cot{\left(x \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:
7tan⁡(x)−cot⁡(x)=cot⁡(x)+7−tan⁡(x)7^{\tan{\left(x \right)}} - \cot{\left(x \right)} = \cot{\left(x \right)} + 7^{- \tan{\left(x \right)}}
- No
7tan⁡(x)−cot⁡(x)=−cot⁡(x)−7−tan⁡(x)7^{\tan{\left(x \right)}} - \cot{\left(x \right)} = - \cot{\left(x \right)} - 7^{- \tan{\left(x \right)}}
- No
es decir, función
no es
par ni impar
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