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Trazado el gráfico de la función/ cosx×ctgx

Gráfico de la función y = cosx×ctgx

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = cos(x)*cot(x)
f(x)=cos⁡(x)cot⁡(x)f{\left(x \right)} = \cos{\left(x \right)} \cot{\left(x \right)} 
f = cos(x)*cot(x)
Gráfico de la función
02468-8-6-4-2-1010-10001000
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)cot⁡(x)=0\cos{\left(x \right)} \cot{\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=42.4115007257482x_{1} = 42.4115007257482
x2=45.553093765886x_{2} = 45.553093765886
x3=20.4203521458051x_{3} = 20.4203521458051
x4=−80.1106125767207x_{4} = -80.1106125767207
x5=−95.8185758679732x_{5} = -95.8185758679732
x6=−45.5530935939383x_{6} = -45.5530935939383
x7=95.8185760670326x_{7} = 95.8185760670326
x8=−89.5353907537234x_{8} = -89.5353907537234
x9=67.5442423466955x_{9} = 67.5442423466955
x10=23.5619451851288x_{10} = 23.5619451851288
x11=−54.9778705470218x_{11} = -54.9778705470218
x12=76.9690195630622x_{12} = 76.9690195630622
x13=14.1371671153205x_{13} = 14.1371671153205
x14=76.9690210413954x_{14} = 76.9690210413954
x15=10.9955738135239x_{15} = 10.9955738135239
x16=−10.9955733545245x_{16} = -10.9955733545245
x17=−14.1371668348422x_{17} = -14.1371668348422
x18=54.9778709800313x_{18} = 54.9778709800313
x19=83.2522058525039x_{19} = 83.2522058525039
x20=−36.1283154153854x_{20} = -36.1283154153854
x21=73.8274274867432x_{21} = 73.8274274867432
x22=−29.8451300946099x_{22} = -29.8451300946099
x23=−4.71238987596373x_{23} = -4.71238987596373
x24=−86.3937977179549x_{24} = -86.3937977179549
x25=−98.9601677364429x_{25} = -98.9601677364429
x26=61.2610559072772x_{26} = 61.2610559072772
x27=7.85398174563321x_{27} = 7.85398174563321
x28=−58.1194639960073x_{28} = -58.1194639960073
x29=89.5353909275596x_{29} = 89.5353909275596
x30=−48.6946870685899x_{30} = -48.6946870685899
x31=−76.969019142094x_{31} = -76.969019142094
x32=32.9867223968539x_{32} = 32.9867223968539
x33=−26.7035384718653x_{33} = -26.7035384718653
x34=64.4026493057109x_{34} = 64.4026493057109
x35=−42.4115005568527x_{35} = -42.4115005568527
x36=−26.7035370683564x_{36} = -26.7035370683564
x37=−92.676982818755x_{37} = -92.676982818755
x38=54.9778724408964x_{38} = 54.9778724408964
x39=−92.6769842647274x_{39} = -92.6769842647274
x40=−98.9601691056411x_{40} = -98.9601691056411
x41=58.1194643607763x_{41} = 58.1194643607763
x42=−17.2787598652139x_{42} = -17.2787598652139
x43=83.2522045003433x_{43} = 83.2522045003433
x44=−76.9690205214496x_{44} = -76.9690205214496
x45=−70.6858342354489x_{45} = -70.6858342354489
x46=−51.8362786889097x_{46} = -51.8362786889097
x47=−4.71238848455677x_{47} = -4.71238848455677
x48=−83.2522056070609x_{48} = -83.2522056070609
x49=61.2610572679436x_{49} = 61.2610572679436
x50=92.6769830369374x_{50} = 92.6769830369374
x51=17.2787587191641x_{51} = 17.2787587191641
x52=−20.4203519762382x_{52} = -20.4203519762382
x53=98.9601696430262x_{53} = 98.9601696430262
x54=−1.57079643412171x_{54} = -1.57079643412171
x55=−73.8274272796526x_{55} = -73.8274272796526
x56=51.836278906418x_{56} = 51.836278906418
x57=29.8451303260505x_{57} = 29.8451303260505
x58=−48.6946856519848x_{58} = -48.6946856519848
x59=−54.9778719374446x_{59} = -54.9778719374446
x60=−7.85398149471446x_{60} = -7.85398149471446
x61=−70.6858356661912x_{61} = -70.6858356661912
x62=−39.2699084457792x_{62} = -39.2699084457792
x63=−61.2610570263942x_{63} = -61.2610570263942
x64=17.2787600994214x_{64} = 17.2787600994214
x65=36.1283159260006x_{65} = 36.1283159260006
x66=−32.9867233536188x_{66} = -32.9867233536188
x67=86.3937978856968x_{67} = 86.3937978856968
x68=98.960168145952x_{68} = 98.960168145952
x69=39.2699086835855x_{69} = 39.2699086835855
x70=70.6858344565908x_{70} = 70.6858344565908
x71=−67.5442421738335x_{71} = -67.5442421738335
x72=10.9955752430018x_{72} = 10.9955752430018
x73=−64.4026491374242x_{73} = -64.4026491374242
x74=26.7035372957759x_{74} = 26.7035372957759
x75=−23.5619450140352x_{75} = -23.5619450140352
x76=1.57079660442153x_{76} = 1.57079660442153
x77=−10.9955747699649x_{77} = -10.9955747699649
x78=39.2699073135637x_{78} = 39.2699073135637
x79=−32.9867219511814x_{79} = -32.9867219511814
x80=48.6946858762043x_{80} = 48.6946858762043
x81=80.1106131287292x_{81} = 80.1106131287292
x82=4.71238871530383x_{82} = 4.71238871530383
x83=32.9867238414546x_{83} = 32.9867238414546
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 cos(x)*cot(x).
cos⁡(0)cot⁡(0)\cos{\left(0 \right)} \cot{\left(0 \right)}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
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)cot⁡(x))y = \lim_{x \to -\infty}\left(\cos{\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→∞(cos⁡(x)cot⁡(x))y = \lim_{x \to \infty}\left(\cos{\left(x \right)} \cot{\left(x \right)}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(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→−∞(cos⁡(x)cot⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{\cos{\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→∞(cos⁡(x)cot⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{\cos{\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:
cos⁡(x)cot⁡(x)=−cos⁡(x)cot⁡(x)\cos{\left(x \right)} \cot{\left(x \right)} = - \cos{\left(x \right)} \cot{\left(x \right)}
- No
cos⁡(x)cot⁡(x)=cos⁡(x)cot⁡(x)\cos{\left(x \right)} \cot{\left(x \right)} = \cos{\left(x \right)} \cot{\left(x \right)}
- Sí
es decir, función
es
impar
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