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Trazado el gráfico de la función/ cosx/1-ctg^2x

Gráfico de la función y = cosx/1-ctg^2x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       cos(x)      2   
f(x) = ------ - cot (x)
         1
f(x)=cos⁡(x)1−cot⁡2(x)f{\left(x \right)} = \frac{\cos{\left(x \right)}}{1} - \cot^{2}{\left(x \right)} 
f = cos(x)/1 - cot(x)^2
Gráfico de la función
02468-8-6-4-2-1010-500000500000
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:
cos⁡(x)1−cot⁡2(x)=0\frac{\cos{\left(x \right)}}{1} - \cot^{2}{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
Solución numérica
x1=51.8362787842316x_{1} = 51.8362787842316
x2=93.3432227133914x_{2} = 93.3432227133914
x3=−36.1283155162826x_{3} = -36.1283155162826
x4=4.71238898038469x_{4} = 4.71238898038469
x5=89.5353906273091x_{5} = 89.5353906273091
x6=44.8868540445595x_{6} = 44.8868540445595
x7=−87.0600374062118x_{7} = -87.0600374062118
x8=−82.585965887637x_{8} = -82.585965887637
x9=68.2104814846731x_{9} = 68.2104814846731
x10=36.1283155162826x_{10} = 36.1283155162826
x11=32.3204834302003x_{11} = 32.3204834302003
x12=−4.71238898038469x_{12} = -4.71238898038469
x13=−42.4115008234622x_{13} = -42.4115008234622
x14=−19.7541128158411x_{14} = -19.7541128158411
x15=17.9449990272364x_{15} = 17.9449990272364
x16=82.585965887637x_{16} = 82.585965887637
x17=−80.1106126665397x_{17} = -80.1106126665397
x18=26.0372981230207x_{18} = 26.0372981230207
x19=7.85398163397448x_{19} = 7.85398163397448
x20=39.2699081698724x_{20} = 39.2699081698724
x21=−29.845130209103x_{21} = -29.845130209103
x22=76.9690200129499x_{22} = 76.9690200129499
x23=−23.5619449019235x_{23} = -23.5619449019235
x24=−17.2787595947439x_{24} = -17.2787595947439
x25=14.1371669411541x_{25} = 14.1371669411541
x26=49.3609255631343x_{26} = 49.3609255631343
x27=−61.9272961774935x_{27} = -61.9272961774935
x28=58.1194640914112x_{28} = 58.1194640914112
x29=61.261056745001x_{29} = 61.261056745001
x30=−51.8362787842316x_{30} = -51.8362787842316
x31=−32.3204834302003x_{31} = -32.3204834302003
x32=−39.2699081698724x_{32} = -39.2699081698724
x33=80.1106126665397x_{33} = 80.1106126665397
x34=−92.6769832808989x_{34} = -92.6769832808989
x35=99.626408020571x_{35} = 99.626408020571
x36=64.4026493985908x_{36} = 64.4026493985908
x37=67.5442420521806x_{37} = 67.5442420521806
x38=−20.4203522483337x_{38} = -20.4203522483337
x39=−70.0195952732778x_{39} = -70.0195952732778
x40=−5.37862841287721x_{40} = -5.37862841287721
x41=−98.9601685880785x_{41} = -98.9601685880785
x42=24.228184334416x_{42} = 24.228184334416
x43=−58.1194640914112x_{43} = -58.1194640914112
x44=86.3937979737193x_{44} = 86.3937979737193
x45=−54.9778714378214x_{45} = -54.9778714378214
x46=−64.4026493985908x_{46} = -64.4026493985908
x47=10.9955742875643x_{47} = 10.9955742875643
x48=−38.6036687373799x_{48} = -38.6036687373799
x49=−14.1371669411541x_{49} = -14.1371669411541
x50=−63.7364099660982x_{50} = -63.7364099660982
x51=11.6618137200568x_{51} = 11.6618137200568
x52=−48.6946861306418x_{52} = -48.6946861306418
x53=−26.7035375555132x_{53} = -26.7035375555132
x54=26.7035375555132x_{54} = 26.7035375555132
x55=23.5619449019235x_{55} = 23.5619449019235
x56=−26.0372981230207x_{56} = -26.0372981230207
x57=−61.261056745001x_{57} = -61.261056745001
x58=5.37862841287721x_{58} = 5.37862841287721
x59=−7.85398163397448x_{59} = -7.85398163397448
x60=−76.9690200129499x_{60} = -76.9690200129499
x61=45.553093477052x_{61} = 45.553093477052
x62=20.4203522483337x_{62} = 20.4203522483337
x63=−99.626408020571x_{63} = -99.626408020571
x64=−49.3609255631343x_{64} = -49.3609255631343
x65=−76.3027805804574x_{65} = -76.3027805804574
x66=−43.0777402559547x_{66} = -43.0777402559547
x67=−67.5442420521806x_{67} = -67.5442420521806
x68=76.3027805804574x_{68} = 76.3027805804574
x69=38.6036687373799x_{69} = 38.6036687373799
x70=73.8274273593601x_{70} = 73.8274273593601
x71=17.2787595947439x_{71} = 17.2787595947439
x72=−10.9955742875643x_{72} = -10.9955742875643
x73=−11.6618137200568x_{73} = -11.6618137200568
x74=−95.8185759344887x_{74} = -95.8185759344887
x75=42.4115008234622x_{75} = 42.4115008234622
x76=95.8185759344887x_{76} = 95.8185759344887
x77=−17.9449990272364x_{77} = -17.9449990272364
x78=29.845130209103x_{78} = 29.845130209103
x79=−73.8274273593601x_{79} = -73.8274273593601
x80=48.6946861306418x_{80} = 48.6946861306418
x81=−93.3432227133914x_{81} = -93.3432227133914
x82=70.6858347057703x_{82} = 70.6858347057703
x83=83.2522053201295x_{83} = 83.2522053201295
x84=−89.5353906273091x_{84} = -89.5353906273091
x85=−55.6441108703139x_{85} = -55.6441108703139
x86=55.6441108703139x_{86} = 55.6441108703139
x87=32.9867228626928x_{87} = 32.9867228626928
x88=−32.9867228626928x_{88} = -32.9867228626928
x89=61.9272961774935x_{89} = 61.9272961774935
x90=−45.553093477052x_{90} = -45.553093477052
x91=−1.5707963267949x_{91} = -1.5707963267949
x92=−70.6858347057703x_{92} = -70.6858347057703
x93=88.8691511948166x_{93} = 88.8691511948166
x94=−83.2522053201295x_{94} = -83.2522053201295
x95=−86.3937979737193x_{95} = -86.3937979737193
x96=98.9601685880785x_{96} = 98.9601685880785
x97=92.6769832808989x_{97} = 92.6769832808989
x98=54.9778714378214x_{98} = 54.9778714378214
x99=1.5707963267949x_{99} = 1.5707963267949
x100=70.0195952732778x_{100} = 70.0195952732778
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→−∞(cos⁡(x)1−cot⁡2(x))y = \lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{1} - \cot^{2}{\left(x \right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(cos⁡(x)1−cot⁡2(x))y = \lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{1} - \cot^{2}{\left(x \right)}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(x)/1 - cot(x)^2, 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→−∞(cos⁡(x)1−cot⁡2(x)x)y = x \lim_{x \to -\infty}\left(\frac{\frac{\cos{\left(x \right)}}{1} - \cot^{2}{\left(x \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(cos⁡(x)1−cot⁡2(x)x)y = x \lim_{x \to \infty}\left(\frac{\frac{\cos{\left(x \right)}}{1} - \cot^{2}{\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:
cos⁡(x)1−cot⁡2(x)=cos⁡(x)1−cot⁡2(x)\frac{\cos{\left(x \right)}}{1} - \cot^{2}{\left(x \right)} = \frac{\cos{\left(x \right)}}{1} - \cot^{2}{\left(x \right)}
- Sí
cos⁡(x)1−cot⁡2(x)=−cos⁡(x)1+cot⁡2(x)\frac{\cos{\left(x \right)}}{1} - \cot^{2}{\left(x \right)} = - \frac{\cos{\left(x \right)}}{1} + \cot^{2}{\left(x \right)}
- No
es decir, función
es
par
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