Sr Examen
Otras calculadoras
- Gráfico de la función implícita
- Derivadas paso a paso
- Integrales paso a paso
- Gráfico de la función paramétrica
- Gráfico en coordenadas polares
- Superficie especificada paramétricamente
- Superficie dada por la ecuación
- Curva en el espacio especificada paramétricamente
- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
3/2*x
-
3*((|x+7|))-x^2-13*x-42
-
(2x+3)e^(5x)
-
4/3*x+12
-
Expresiones idénticas
- sin(tres *t)*cos(t)+(cos(cinco *t))** dos
- seno de (3 multiplicar por t) multiplicar por coseno de (t) más ( coseno de (5 multiplicar por t)) multiplicar por multiplicar por 2
- seno de (tres multiplicar por t) multiplicar por coseno de (t) más ( coseno de (cinco multiplicar por t)) multiplicar por multiplicar por dos
- sin(3t)cos(t)+(cos(5t))2
- sin3tcost+cos5t2
-
Expresiones semejantes
- sin(3*t)*cos(t)-(cos(5*t))**2
-
Expresiones con funciones
- Seno sin
- sin(3*x^3)/x^3
- sin(x)^(2)+cos^(2)(x)
- sin(7*x)*cos(11*x)
- sin(x^2)+1
- sin(x)-16*x^3+3*cos(x)
Gráfico de la función y = sin(3*t)*cos(t)+(cos(5*t))**2
Solución
Ha introducido
[src]
2 f(t) = sin(3*t)*cos(t) + cos (5*t)
$$f{\left(t \right)} = \sin{\left(3 t \right)} \cos{\left(t \right)} + \cos^{2}{\left(5 t \right)}$$
f = sin(3*t)*cos(t) + cos(5*t)^2
Gráfico de la función
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje T con f = 0
o sea hay que resolver la ecuación:
$$\sin{\left(3 t \right)} \cos{\left(t \right)} + \cos^{2}{\left(5 t \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje T:
Solución numérica
$$t_{1} = 1.5707963267949$$
$$t_{2} = -7.06858347057703$$
$$t_{3} = -81.8453258326253$$
$$t_{4} = 18.3006069564618$$
$$t_{5} = -16.7070104388889$$
$$t_{6} = 32.9867228626928$$
$$t_{7} = -85.8220488178644$$
$$t_{8} = -59.8541772574967$$
$$t_{9} = 6.11926846788891$$
$$t_{10} = -29.0597320457056$$
$$t_{11} = 71.2575838616253$$
$$t_{12} = 49.4800842940392$$
$$t_{13} = 61.2208185498598$$
$$t_{14} = -69.9004365423729$$
$$t_{15} = 92.6769832808989$$
$$t_{16} = 72.0927141932746$$
$$t_{17} = -32.9867228626928$$
$$t_{18} = 23.5619449019235$$
$$t_{19} = -91.8915851175014$$
$$t_{20} = -9.97372692584633$$
$$t_{21} = 84.2740526818475$$
$$t_{22} = 90.5572379890271$$
$$t_{23} = -37.8630286823682$$
$$t_{24} = 77.7544181763474$$
$$t_{25} = 59.1413114531291$$
$$t_{26} = 8.63937979737193$$
$$t_{27} = 94.0838627684031$$
$$t_{28} = -51.2645296283766$$
$$t_{29} = 17.2385213996027$$
$$t_{30} = -73.2556782035052$$
$$t_{31} = 34.0085702244108$$
$$t_{32} = -4.14063982452974$$
$$t_{33} = -80.1106126665397$$
$$t_{34} = -19.8486030924787$$
$$t_{35} = -41.8397516676073$$
$$t_{36} = -125.827622982882$$
$$t_{37} = 90.3207887907066$$
$$t_{38} = -58.1597022865523$$
$$t_{39} = 55.5496205936763$$
$$t_{40} = -76.9690200129499$$
$$t_{41} = 45.5128552819108$$
$$t_{42} = 2.97767581429912$$
$$t_{43} = -53.9560240761034$$
$$t_{44} = 81.1324600282577$$
$$t_{45} = 86.3937979737193$$
$$t_{46} = -97.9383212263605$$
$$t_{47} = 2.14254548264985$$
$$t_{48} = -60.239209383283$$
$$t_{49} = 40.0553063332699$$
$$t_{50} = -25.9181393921158$$
$$t_{51} = -39.3101463650136$$
$$t_{52} = 50.101565618146$$
$$t_{53} = 491.659250286803$$
$$t_{54} = 65.1880475619882$$
$$t_{55} = 77.9908673746679$$
$$t_{56} = -36.1685537114238$$
$$t_{57} = 24.1336940577784$$
$$t_{58} = -14.1774051362952$$
$$t_{59} = -31.9648755009749$$
$$t_{60} = -29.8853684042442$$
$$t_{61} = -13.1153195794361$$
$$t_{62} = -75.947172651232$$
$$t_{63} = 12.0174216492822$$
$$t_{64} = -95.8185759344887$$
$$t_{65} = 58.9048622548086$$
$$t_{66} = 43.1968989868597$$
$$t_{67} = -4.75262717552585$$
$$t_{68} = -0.999047170939943$$
$$t_{69} = 68.1159912080355$$
$$t_{70} = 96.3903250903437$$
$$t_{71} = -95.0331777710912$$
$$t_{72} = 36.9137136796801$$
$$t_{73} = 28.1104170430175$$
$$t_{74} = -69.2789552182661$$
$$t_{75} = 7.81374343883332$$
$$t_{76} = 55.9997187995393$$
$$t_{77} = -63.6172512351933$$
$$t_{78} = 62.2829041067189$$
$$t_{79} = 99.9820159497964$$
$$t_{80} = -73.8274273593601$$
$$t_{81} = 14.9225651045515$$
$$t_{82} = 191.473235029687$$
$$t_{83} = -47.9092879672443$$
$$t_{84} = -15.8718801072396$$
$$t_{85} = -63.8309002427358$$
o sea hay que resolver la ecuación:
$$\sin{\left(3 t \right)} \cos{\left(t \right)} + \cos^{2}{\left(5 t \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje T:
Solución numérica
$$t_{1} = 1.5707963267949$$
$$t_{2} = -7.06858347057703$$
$$t_{3} = -81.8453258326253$$
$$t_{4} = 18.3006069564618$$
$$t_{5} = -16.7070104388889$$
$$t_{6} = 32.9867228626928$$
$$t_{7} = -85.8220488178644$$
$$t_{8} = -59.8541772574967$$
$$t_{9} = 6.11926846788891$$
$$t_{10} = -29.0597320457056$$
$$t_{11} = 71.2575838616253$$
$$t_{12} = 49.4800842940392$$
$$t_{13} = 61.2208185498598$$
$$t_{14} = -69.9004365423729$$
$$t_{15} = 92.6769832808989$$
$$t_{16} = 72.0927141932746$$
$$t_{17} = -32.9867228626928$$
$$t_{18} = 23.5619449019235$$
$$t_{19} = -91.8915851175014$$
$$t_{20} = -9.97372692584633$$
$$t_{21} = 84.2740526818475$$
$$t_{22} = 90.5572379890271$$
$$t_{23} = -37.8630286823682$$
$$t_{24} = 77.7544181763474$$
$$t_{25} = 59.1413114531291$$
$$t_{26} = 8.63937979737193$$
$$t_{27} = 94.0838627684031$$
$$t_{28} = -51.2645296283766$$
$$t_{29} = 17.2385213996027$$
$$t_{30} = -73.2556782035052$$
$$t_{31} = 34.0085702244108$$
$$t_{32} = -4.14063982452974$$
$$t_{33} = -80.1106126665397$$
$$t_{34} = -19.8486030924787$$
$$t_{35} = -41.8397516676073$$
$$t_{36} = -125.827622982882$$
$$t_{37} = 90.3207887907066$$
$$t_{38} = -58.1597022865523$$
$$t_{39} = 55.5496205936763$$
$$t_{40} = -76.9690200129499$$
$$t_{41} = 45.5128552819108$$
$$t_{42} = 2.97767581429912$$
$$t_{43} = -53.9560240761034$$
$$t_{44} = 81.1324600282577$$
$$t_{45} = 86.3937979737193$$
$$t_{46} = -97.9383212263605$$
$$t_{47} = 2.14254548264985$$
$$t_{48} = -60.239209383283$$
$$t_{49} = 40.0553063332699$$
$$t_{50} = -25.9181393921158$$
$$t_{51} = -39.3101463650136$$
$$t_{52} = 50.101565618146$$
$$t_{53} = 491.659250286803$$
$$t_{54} = 65.1880475619882$$
$$t_{55} = 77.9908673746679$$
$$t_{56} = -36.1685537114238$$
$$t_{57} = 24.1336940577784$$
$$t_{58} = -14.1774051362952$$
$$t_{59} = -31.9648755009749$$
$$t_{60} = -29.8853684042442$$
$$t_{61} = -13.1153195794361$$
$$t_{62} = -75.947172651232$$
$$t_{63} = 12.0174216492822$$
$$t_{64} = -95.8185759344887$$
$$t_{65} = 58.9048622548086$$
$$t_{66} = 43.1968989868597$$
$$t_{67} = -4.75262717552585$$
$$t_{68} = -0.999047170939943$$
$$t_{69} = 68.1159912080355$$
$$t_{70} = 96.3903250903437$$
$$t_{71} = -95.0331777710912$$
$$t_{72} = 36.9137136796801$$
$$t_{73} = 28.1104170430175$$
$$t_{74} = -69.2789552182661$$
$$t_{75} = 7.81374343883332$$
$$t_{76} = 55.9997187995393$$
$$t_{77} = -63.6172512351933$$
$$t_{78} = 62.2829041067189$$
$$t_{79} = 99.9820159497964$$
$$t_{80} = -73.8274273593601$$
$$t_{81} = 14.9225651045515$$
$$t_{82} = 191.473235029687$$
$$t_{83} = -47.9092879672443$$
$$t_{84} = -15.8718801072396$$
$$t_{85} = -63.8309002427358$$
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d t} f{\left(t \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d t} f{\left(t \right)} = $$
primera derivada
$$- \sin{\left(t \right)} \sin{\left(3 t \right)} - 10 \sin{\left(5 t \right)} \cos{\left(5 t \right)} + 3 \cos{\left(t \right)} \cos{\left(3 t \right)} = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga extremos
$$\frac{d}{d t} f{\left(t \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d t} f{\left(t \right)} = $$
primera derivada
$$- \sin{\left(t \right)} \sin{\left(3 t \right)} - 10 \sin{\left(5 t \right)} \cos{\left(5 t \right)} + 3 \cos{\left(t \right)} \cos{\left(3 t \right)} = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga extremos
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con t->+oo y t->-oo
$$\lim_{t \to -\infty}\left(\sin{\left(3 t \right)} \cos{\left(t \right)} + \cos^{2}{\left(5 t \right)}\right) = \left\langle -1, 2\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -1, 2\right\rangle$$
$$\lim_{t \to \infty}\left(\sin{\left(3 t \right)} \cos{\left(t \right)} + \cos^{2}{\left(5 t \right)}\right) = \left\langle -1, 2\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -1, 2\right\rangle$$
$$\lim_{t \to -\infty}\left(\sin{\left(3 t \right)} \cos{\left(t \right)} + \cos^{2}{\left(5 t \right)}\right) = \left\langle -1, 2\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -1, 2\right\rangle$$
$$\lim_{t \to \infty}\left(\sin{\left(3 t \right)} \cos{\left(t \right)} + \cos^{2}{\left(5 t \right)}\right) = \left\langle -1, 2\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -1, 2\right\rangle$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(3*t)*cos(t) + cos(5*t)^2, dividida por t con t->+oo y t ->-oo
$$\lim_{t \to -\infty}\left(\frac{\sin{\left(3 t \right)} \cos{\left(t \right)} + \cos^{2}{\left(5 t \right)}}{t}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{t \to \infty}\left(\frac{\sin{\left(3 t \right)} \cos{\left(t \right)} + \cos^{2}{\left(5 t \right)}}{t}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{t \to -\infty}\left(\frac{\sin{\left(3 t \right)} \cos{\left(t \right)} + \cos^{2}{\left(5 t \right)}}{t}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{t \to \infty}\left(\frac{\sin{\left(3 t \right)} \cos{\left(t \right)} + \cos^{2}{\left(5 t \right)}}{t}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-t) и f = -f(-t).
Pues, comprobamos:
$$\sin{\left(3 t \right)} \cos{\left(t \right)} + \cos^{2}{\left(5 t \right)} = - \sin{\left(3 t \right)} \cos{\left(t \right)} + \cos^{2}{\left(5 t \right)}$$
- No
$$\sin{\left(3 t \right)} \cos{\left(t \right)} + \cos^{2}{\left(5 t \right)} = \sin{\left(3 t \right)} \cos{\left(t \right)} - \cos^{2}{\left(5 t \right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\sin{\left(3 t \right)} \cos{\left(t \right)} + \cos^{2}{\left(5 t \right)} = - \sin{\left(3 t \right)} \cos{\left(t \right)} + \cos^{2}{\left(5 t \right)}$$
- No
$$\sin{\left(3 t \right)} \cos{\left(t \right)} + \cos^{2}{\left(5 t \right)} = \sin{\left(3 t \right)} \cos{\left(t \right)} - \cos^{2}{\left(5 t \right)}$$
- No
es decir, función
no es
par ni impar