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- 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 =:
-
7-x-2*x^2
-
y=x^3
-
(3*x-4)^40/(x^2-2)^36
-
y=x^2-x
-
Expresiones idénticas
- - cuatro *x*tan(x^ dos + uno / dos)/(uno + dos *x^ dos)^ dos + dos *x*(uno +tan(x^ dos + uno / dos)^ dos)/(uno + dos *x^ dos)
- menos 4 multiplicar por x multiplicar por tangente de (x al cuadrado más 1 dividir por 2) dividir por (1 más 2 multiplicar por x al cuadrado ) al cuadrado más 2 multiplicar por x multiplicar por (1 más tangente de (x al cuadrado más 1 dividir por 2) al cuadrado ) dividir por (1 más 2 multiplicar por x al cuadrado )
- menos cuatro multiplicar por x multiplicar por tangente de (x en el grado dos más uno dividir por dos) dividir por (uno más dos multiplicar por x en el grado dos) en el grado dos más dos multiplicar por x multiplicar por (uno más tangente de (x en el grado dos más uno dividir por dos) en el grado dos) dividir por (uno más dos multiplicar por x en el grado dos)
- -4*x*tan(x2+1/2)/(1+2*x2)2+2*x*(1+tan(x2+1/2)2)/(1+2*x2)
- -4*x*tanx2+1/2/1+2*x22+2*x*1+tanx2+1/22/1+2*x2
- -4*x*tan(x²+1/2)/(1+2*x²)²+2*x*(1+tan(x²+1/2)²)/(1+2*x²)
- -4*x*tan(x en el grado 2+1/2)/(1+2*x en el grado 2) en el grado 2+2*x*(1+tan(x en el grado 2+1/2) en el grado 2)/(1+2*x en el grado 2)
- -4xtan(x^2+1/2)/(1+2x^2)^2+2x(1+tan(x^2+1/2)^2)/(1+2x^2)
- -4xtan(x2+1/2)/(1+2x2)2+2x(1+tan(x2+1/2)2)/(1+2x2)
- -4xtanx2+1/2/1+2x22+2x1+tanx2+1/22/1+2x2
- -4xtanx^2+1/2/1+2x^2^2+2x1+tanx^2+1/2^2/1+2x^2
- -4*x*tan(x^2+1 dividir por 2) dividir por (1+2*x^2)^2+2*x*(1+tan(x^2+1 dividir por 2)^2) dividir por (1+2*x^2)
-
Expresiones semejantes
- -4*x*tan(x^2+1/2)/(1+2*x^2)^2+2*x*(1+tan(x^2-1/2)^2)/(1+2*x^2)
- -4*x*tan(x^2+1/2)/(1+2*x^2)^2+2*x*(1-tan(x^2+1/2)^2)/(1+2*x^2)
- -4*x*tan(x^2+1/2)/(1+2*x^2)^2-2*x*(1+tan(x^2+1/2)^2)/(1+2*x^2)
- -4*x*tan(x^2-1/2)/(1+2*x^2)^2+2*x*(1+tan(x^2+1/2)^2)/(1+2*x^2)
- -4*x*tan(x^2+1/2)/(1-2*x^2)^2+2*x*(1+tan(x^2+1/2)^2)/(1+2*x^2)
- 4*x*tan(x^2+1/2)/(1+2*x^2)^2+2*x*(1+tan(x^2+1/2)^2)/(1+2*x^2)
- -4*x*tan(x^2+1/2)/(1+2*x^2)^2+2*x*(1+tan(x^2+1/2)^2)/(1-2*x^2)
-
Expresiones con funciones
- Tangente tan
- tan(-9*x/5)
- tan[x]/(x^4+x^2+x)
- tan(log(x^2+2))
- tan2x+sin2x
- tan(sqrt(exp(x+2)))/sin(((7*x)/4)-(1/1000))
- Tangente tan
- tan(-9*x/5)
- tan[x]/(x^4+x^2+x)
- tan(log(x^2+2))
- tan2x+sin2x
- tan(sqrt(exp(x+2)))/sin(((7*x)/4)-(1/1000))
Gráfico de la función y = -4*x*tan(x^2+1/2)/(1+2*x^2)^2+2*x*(1+tan(x^2+1/2)^2)/(1+2*x^2)
Solución
Ha introducido
[src]
/ 2 1\ / 2/ 2 1\\
-4*x*tan|x + -| 2*x*|1 + tan |x + -||
\ 2/ \ \ 2//
f(x) = ---------------- + ----------------------
2 2
/ 2\ 1 + 2*x
\1 + 2*x /
$$f{\left(x \right)} = \frac{2 x \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} + \frac{- 4 x \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}}$$
f = ((2*x)*(tan(x^2 + 1/2)^2 + 1))/(2*x^2 + 1) + ((-4*x)*tan(x^2 + 1/2))/(2*x^2 + 1)^2
Gráfico de la función
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:
$$\frac{2 x \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} + \frac{- 4 x \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 0$$
o sea hay que resolver la ecuación:
$$\frac{2 x \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} + \frac{- 4 x \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 0$$
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$- \frac{8 x^{2} \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{\left(2 x^{2} + 1\right)^{2}} + \frac{32 x^{2} \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{3}} + \frac{8 x^{2} \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right) \tan{\left(x^{2} + \frac{1}{2} \right)} + 2 \tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 2}{2 x^{2} + 1} + \frac{- 8 x^{2} \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right) - 4 \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 24.2276516091942$$
$$x_{2} = -79.9736379062586$$
$$x_{3} = 38.2554922699397$$
$$x_{4} = -45.9759960258234$$
$$x_{5} = -11.8691603685014$$
$$x_{6} = -1.68473749010703$$
$$x_{7} = -83.9035710993377$$
$$x_{8} = -89.7471094627101$$
$$x_{9} = 28.127962881328$$
$$x_{10} = -57.7025860600124$$
$$x_{11} = -76.0471709199656$$
$$x_{12} = -91.7549366557361$$
$$x_{13} = -70.1148533151679$$
$$x_{14} = -77.9646195916378$$
$$x_{15} = -44.3057083278542$$
$$x_{16} = 87.3343406755818$$
$$x_{17} = -85.7919092295442$$
$$x_{18} = 16.2294985469974$$
$$x_{19} = -67.9532292328637$$
$$x_{20} = 96.2499704747411$$
$$x_{21} = -52.0039570801788$$
$$x_{22} = -97.9806472089682$$
$$x_{23} = 20.2743793800958$$
$$x_{24} = -7.6943345390805$$
$$x_{25} = 93.1481561580688$$
$$x_{26} = -31.6491517223897$$
$$x_{27} = 43.8065463376575$$
$$x_{28} = 31.8470589866078$$
$$x_{29} = 100.466100104885$$
$$x_{30} = 92.2500655105387$$
$$x_{31} = -25.7976504428415$$
$$x_{32} = 33.9942163486092$$
$$x_{33} = -17.9746031910921$$
$$x_{34} = 66.196842242554$$
$$x_{35} = 4.28821644642643$$
$$x_{36} = 48.2433464683252$$
$$x_{37} = -63.9273980841703$$
$$x_{38} = -29.7561754047794$$
$$x_{39} = 90.2532539182077$$
$$x_{40} = 30.0189593964374$$
$$x_{41} = 98.4285112204771$$
$$x_{42} = -13.7112922211213$$
$$x_{43} = 68.0456297887513$$
$$x_{44} = -3.90574871969606$$
$$x_{45} = 78.2060165150451$$
$$x_{46} = 18.4063634635748$$
$$x_{47} = 72.2117992350915$$
$$x_{48} = 58.0283340044788$$
$$x_{49} = -21.9126729494334$$
$$x_{50} = -5.56234042020343$$
$$x_{51} = 8.28407265099272$$
$$x_{52} = -65.8399425496541$$
$$x_{53} = -23.7694762992186$$
$$x_{54} = -53.7857517670918$$
$$x_{55} = 86.2484289973703$$
$$x_{56} = -33.9479771749304$$
$$x_{57} = -81.7222496864894$$
$$x_{58} = 56.2413018983851$$
$$x_{59} = -73.9952439834427$$
$$x_{60} = -89.5017394259607$$
$$x_{61} = 22.2682093962144$$
$$x_{62} = 42.0501734594177$$
$$x_{63} = 64.2460343405985$$
$$x_{64} = -15.7381332897777$$
$$x_{65} = -55.9893713795814$$
$$x_{66} = 60.0242199598979$$
$$x_{67} = 82.2204810339713$$
$$x_{68} = 10.0019203972396$$
$$x_{69} = -59.9980448791643$$
$$x_{70} = 61.9558424851141$$
$$x_{71} = 70.2491438435124$$
$$x_{72} = 25.9796753372099$$
$$x_{73} = 80.2481467468931$$
$$x_{74} = -35.7509272117533$$
$$x_{75} = 14.27260285079$$
$$x_{76} = 12.3872073720398$$
$$x_{77} = 76.4591659424083$$
$$x_{78} = -93.5855779692324$$
$$x_{79} = -39.1885097414717$$
$$x_{80} = 83.9971261785332$$
$$x_{81} = 40.2562078674106$$
$$x_{82} = -27.7342959949159$$
$$x_{83} = 46.2485129234309$$
$$x_{84} = 5.83772950085635$$
$$x_{85} = -96.0375765638132$$
$$x_{86} = -61.7017867903039$$
$$x_{87} = 74.2495470900384$$
$$x_{88} = 36.0135855463228$$
$$x_{89} = -10.0019203972396$$
$$x_{90} = -41.7502617099517$$
$$x_{91} = -47.7524511969237$$
$$x_{92} = 2.42790391927647$$
$$x_{93} = 52.1246383328403$$
$$x_{94} = -39.7457114582492$$
$$x_{95} = -49.7501287834332$$
$$x_{96} = 54.2510144714636$$
$$x_{97} = -100.058762044599$$
$$x_{98} = -19.7245890568195$$
$$x_{99} = 51.5184052042485$$
$$x_{100} = -71.7754282289635$$
Signos de extremos en los puntos:
Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
$$x_{1} = 24.2276516091942$$
$$x_{2} = 38.2554922699397$$
$$x_{3} = 28.127962881328$$
$$x_{4} = 87.3343406755818$$
$$x_{5} = 16.2294985469974$$
$$x_{6} = 96.2499704747411$$
$$x_{7} = 20.2743793800958$$
$$x_{8} = 93.1481561580688$$
$$x_{9} = 43.8065463376575$$
$$x_{10} = 31.8470589866078$$
$$x_{11} = 100.466100104885$$
$$x_{12} = 92.2500655105387$$
$$x_{13} = 33.9942163486092$$
$$x_{14} = 66.196842242554$$
$$x_{15} = 4.28821644642643$$
$$x_{16} = 48.2433464683252$$
$$x_{17} = 90.2532539182077$$
$$x_{18} = 30.0189593964374$$
$$x_{19} = 98.4285112204771$$
$$x_{20} = 68.0456297887513$$
$$x_{21} = 78.2060165150451$$
$$x_{22} = 18.4063634635748$$
$$x_{23} = 72.2117992350915$$
$$x_{24} = 58.0283340044788$$
$$x_{25} = 8.28407265099272$$
$$x_{26} = 86.2484289973703$$
$$x_{27} = 56.2413018983851$$
$$x_{28} = 22.2682093962144$$
$$x_{29} = 42.0501734594177$$
$$x_{30} = 64.2460343405985$$
$$x_{31} = 60.0242199598979$$
$$x_{32} = 82.2204810339713$$
$$x_{33} = 10.0019203972396$$
$$x_{34} = 61.9558424851141$$
$$x_{35} = 70.2491438435124$$
$$x_{36} = 25.9796753372099$$
$$x_{37} = 80.2481467468931$$
$$x_{38} = 14.27260285079$$
$$x_{39} = 12.3872073720398$$
$$x_{40} = 76.4591659424083$$
$$x_{41} = 83.9971261785332$$
$$x_{42} = 40.2562078674106$$
$$x_{43} = 46.2485129234309$$
$$x_{44} = 5.83772950085635$$
$$x_{45} = 74.2495470900384$$
$$x_{46} = 36.0135855463228$$
$$x_{47} = 2.42790391927647$$
$$x_{48} = 52.1246383328403$$
$$x_{49} = 54.2510144714636$$
$$x_{50} = 51.5184052042485$$
Puntos máximos de la función:
$$x_{50} = -79.9736379062586$$
$$x_{50} = -45.9759960258234$$
$$x_{50} = -11.8691603685014$$
$$x_{50} = -1.68473749010703$$
$$x_{50} = -83.9035710993377$$
$$x_{50} = -89.7471094627101$$
$$x_{50} = -57.7025860600124$$
$$x_{50} = -76.0471709199656$$
$$x_{50} = -91.7549366557361$$
$$x_{50} = -70.1148533151679$$
$$x_{50} = -77.9646195916378$$
$$x_{50} = -44.3057083278542$$
$$x_{50} = -85.7919092295442$$
$$x_{50} = -67.9532292328637$$
$$x_{50} = -52.0039570801788$$
$$x_{50} = -97.9806472089682$$
$$x_{50} = -7.6943345390805$$
$$x_{50} = -31.6491517223897$$
$$x_{50} = -25.7976504428415$$
$$x_{50} = -17.9746031910921$$
$$x_{50} = -63.9273980841703$$
$$x_{50} = -29.7561754047794$$
$$x_{50} = -13.7112922211213$$
$$x_{50} = -3.90574871969606$$
$$x_{50} = -21.9126729494334$$
$$x_{50} = -5.56234042020343$$
$$x_{50} = -65.8399425496541$$
$$x_{50} = -23.7694762992186$$
$$x_{50} = -53.7857517670918$$
$$x_{50} = -33.9479771749304$$
$$x_{50} = -81.7222496864894$$
$$x_{50} = -73.9952439834427$$
$$x_{50} = -89.5017394259607$$
$$x_{50} = -15.7381332897777$$
$$x_{50} = -55.9893713795814$$
$$x_{50} = -59.9980448791643$$
$$x_{50} = -35.7509272117533$$
$$x_{50} = -93.5855779692324$$
$$x_{50} = -39.1885097414717$$
$$x_{50} = -27.7342959949159$$
$$x_{50} = -96.0375765638132$$
$$x_{50} = -61.7017867903039$$
$$x_{50} = -10.0019203972396$$
$$x_{50} = -41.7502617099517$$
$$x_{50} = -47.7524511969237$$
$$x_{50} = -39.7457114582492$$
$$x_{50} = -49.7501287834332$$
$$x_{50} = -100.058762044599$$
$$x_{50} = -19.7245890568195$$
$$x_{50} = -71.7754282289635$$
Decrece en los intervalos
$$\left[100.466100104885, \infty\right)$$
Crece en los intervalos
$$\left[-1.68473749010703, 2.42790391927647\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$- \frac{8 x^{2} \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{\left(2 x^{2} + 1\right)^{2}} + \frac{32 x^{2} \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{3}} + \frac{8 x^{2} \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right) \tan{\left(x^{2} + \frac{1}{2} \right)} + 2 \tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 2}{2 x^{2} + 1} + \frac{- 8 x^{2} \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right) - 4 \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 24.2276516091942$$
$$x_{2} = -79.9736379062586$$
$$x_{3} = 38.2554922699397$$
$$x_{4} = -45.9759960258234$$
$$x_{5} = -11.8691603685014$$
$$x_{6} = -1.68473749010703$$
$$x_{7} = -83.9035710993377$$
$$x_{8} = -89.7471094627101$$
$$x_{9} = 28.127962881328$$
$$x_{10} = -57.7025860600124$$
$$x_{11} = -76.0471709199656$$
$$x_{12} = -91.7549366557361$$
$$x_{13} = -70.1148533151679$$
$$x_{14} = -77.9646195916378$$
$$x_{15} = -44.3057083278542$$
$$x_{16} = 87.3343406755818$$
$$x_{17} = -85.7919092295442$$
$$x_{18} = 16.2294985469974$$
$$x_{19} = -67.9532292328637$$
$$x_{20} = 96.2499704747411$$
$$x_{21} = -52.0039570801788$$
$$x_{22} = -97.9806472089682$$
$$x_{23} = 20.2743793800958$$
$$x_{24} = -7.6943345390805$$
$$x_{25} = 93.1481561580688$$
$$x_{26} = -31.6491517223897$$
$$x_{27} = 43.8065463376575$$
$$x_{28} = 31.8470589866078$$
$$x_{29} = 100.466100104885$$
$$x_{30} = 92.2500655105387$$
$$x_{31} = -25.7976504428415$$
$$x_{32} = 33.9942163486092$$
$$x_{33} = -17.9746031910921$$
$$x_{34} = 66.196842242554$$
$$x_{35} = 4.28821644642643$$
$$x_{36} = 48.2433464683252$$
$$x_{37} = -63.9273980841703$$
$$x_{38} = -29.7561754047794$$
$$x_{39} = 90.2532539182077$$
$$x_{40} = 30.0189593964374$$
$$x_{41} = 98.4285112204771$$
$$x_{42} = -13.7112922211213$$
$$x_{43} = 68.0456297887513$$
$$x_{44} = -3.90574871969606$$
$$x_{45} = 78.2060165150451$$
$$x_{46} = 18.4063634635748$$
$$x_{47} = 72.2117992350915$$
$$x_{48} = 58.0283340044788$$
$$x_{49} = -21.9126729494334$$
$$x_{50} = -5.56234042020343$$
$$x_{51} = 8.28407265099272$$
$$x_{52} = -65.8399425496541$$
$$x_{53} = -23.7694762992186$$
$$x_{54} = -53.7857517670918$$
$$x_{55} = 86.2484289973703$$
$$x_{56} = -33.9479771749304$$
$$x_{57} = -81.7222496864894$$
$$x_{58} = 56.2413018983851$$
$$x_{59} = -73.9952439834427$$
$$x_{60} = -89.5017394259607$$
$$x_{61} = 22.2682093962144$$
$$x_{62} = 42.0501734594177$$
$$x_{63} = 64.2460343405985$$
$$x_{64} = -15.7381332897777$$
$$x_{65} = -55.9893713795814$$
$$x_{66} = 60.0242199598979$$
$$x_{67} = 82.2204810339713$$
$$x_{68} = 10.0019203972396$$
$$x_{69} = -59.9980448791643$$
$$x_{70} = 61.9558424851141$$
$$x_{71} = 70.2491438435124$$
$$x_{72} = 25.9796753372099$$
$$x_{73} = 80.2481467468931$$
$$x_{74} = -35.7509272117533$$
$$x_{75} = 14.27260285079$$
$$x_{76} = 12.3872073720398$$
$$x_{77} = 76.4591659424083$$
$$x_{78} = -93.5855779692324$$
$$x_{79} = -39.1885097414717$$
$$x_{80} = 83.9971261785332$$
$$x_{81} = 40.2562078674106$$
$$x_{82} = -27.7342959949159$$
$$x_{83} = 46.2485129234309$$
$$x_{84} = 5.83772950085635$$
$$x_{85} = -96.0375765638132$$
$$x_{86} = -61.7017867903039$$
$$x_{87} = 74.2495470900384$$
$$x_{88} = 36.0135855463228$$
$$x_{89} = -10.0019203972396$$
$$x_{90} = -41.7502617099517$$
$$x_{91} = -47.7524511969237$$
$$x_{92} = 2.42790391927647$$
$$x_{93} = 52.1246383328403$$
$$x_{94} = -39.7457114582492$$
$$x_{95} = -49.7501287834332$$
$$x_{96} = 54.2510144714636$$
$$x_{97} = -100.058762044599$$
$$x_{98} = -19.7245890568195$$
$$x_{99} = 51.5184052042485$$
$$x_{100} = -71.7754282289635$$
Signos de extremos en los puntos:
(24.22765160919423, 0.0412400004298584)
(-79.97363790625862, -0.0125031429255024)
(38.25549226993971, 0.0261311074336407)
(-45.97599602582338, -0.0217453357995279)
(-11.869160368501433, -0.083953199267024)
(-1.6847374901070293, -0.494581413436108)
(-83.90357109933768, -0.0119175973502458)
(-89.74710946271009, -0.0111417284655362)
(28.127962881328, 0.0355293463178467)
(-57.70258606001235, -0.01732764344202)
(-76.04717091996564, -0.0131485962203823)
(-91.75493665573607, -0.010897948902026)
(-70.11485331516795, -0.0142608626920029)
(-77.96461959163776, -0.0128252757410824)
(-44.30570832785416, -0.0225647064733198)
(87.33434067558181, 0.0114494990084249)
(-85.79190922954416, -0.0116553189884025)
(16.22949854699737, 0.0614992894007723)
(-67.95322923286372, -0.0147144106885241)
(96.24997047474106, 0.0103890528356644)
(-52.003957080178814, -0.0192257509155601)
(-97.98064720896824, -0.0102055655576201)
(20.274379380095777, 0.0492633563199694)
(-7.694334539080495, -0.128870495478659)
(93.14815615806877, 0.0107349669137288)
(-31.649151722389664, -0.0315806535405489)
(43.806546337657494, 0.0228216910491224)
(31.84705898660784, 0.0313845959513192)
(100.46610010488516, 0.00995311316385467)
(92.2500655105387, 0.0108394638143351)
(-25.7976504428415, -0.0387341029380789)
(33.994216348609214, 0.0294040422485603)
(-17.97460319109211, -0.0555479871812111)
(66.19684224255403, 0.0151047371264909)
(4.288216446426431, 0.226902372610224)
(48.24334646832518, 0.0207237941524377)
(-63.927398084170335, -0.0156408314114131)
(-29.75617540477937, -0.0335874946583318)
(90.25325391820775, 0.0110792528255155)
(30.01895939643742, 0.0332937997574656)
(98.42851122047708, 0.0101591335573691)
(-13.711292221121342, -0.0727387459186462)
(68.04562978875133, 0.0146944340070731)
(-3.905748719696056, -0.247714990854652)
(78.20601651504505, 0.0127856947057117)
(18.406363463574827, 0.0542488866279504)
(72.21179923509148, 0.0138468245803103)
(58.0283340044788, 0.0172304019269274)
(-21.9126729494334, -0.0455881838797124)
(-5.562340420203429, -0.176887332858119)
(8.284072650992723, 0.119835701923925)
(-65.8399425496541, -0.0151865967927645)
(-23.769476299218624, -0.0420335395357972)
(-53.7857517670918, -0.0185890712335803)
(86.24842899737028, 0.011593634777292)
(-33.947977174930436, -0.0294440575758166)
(-81.72224968648939, -0.0122356535511347)
(56.24130189838509, 0.0177777167122112)
(-73.99524398344275, -0.0135131479943154)
(-89.50173942596075, -0.0111722698328472)
(22.268209396214353, 0.0448617989338514)
(42.05017345941771, 0.023774390616066)
(64.24603434059848, 0.0155632775903121)
(-15.738133289777657, -0.0634117366166786)
(-55.98937137958142, -0.0178576840998968)
(60.024219959897906, 0.0166576296811097)
(82.2204810339713, 0.0121615201780202)
(10.001920397239605, 0.0994817210465885)
(-59.99804487916435, -0.0166648948127904)
(61.955842485114054, 0.016138425462689)
(70.24914384351239, 0.0142336066968371)
(25.97967533720991, 0.0384631186423616)
(80.24814674689313, 0.0124603794537106)
(-35.75092721175329, -0.0279603612935909)
(14.272602850789992, 0.0698924382623759)
(12.38720737203982, 0.0804656052696414)
(76.45916594240833, 0.0130777580555251)
(-93.58557796923236, -0.0106847970974881)
(-39.18850974147173, -0.0255093765245862)
(83.99712617853316, 0.0119043255432757)
(40.256207867410616, 0.02483322552945)
(-27.734295994915918, -0.0360330066631058)
(46.24851292343087, 0.0216172626573556)
(5.837729500856349, 0.168795789612955)
(-96.03757656381325, -0.010412026474297)
(-61.70178679030391, -0.0162048576407179)
(74.24954709003842, 0.0134668741621358)
(36.01358554632282, 0.0277565954787635)
(-10.001920397239605, -0.0994817210465885)
(-41.75026170995174, -0.023945075594561)
(-47.75245119692366, -0.0209367418267805)
(2.4279039192764706, 0.377786140487174)
(52.12463833284034, 0.0191812550159427)
(-39.745711458249154, -0.025151984375679)
(-49.75012878343321, -0.0200963901119944)
(54.25101447146359, 0.0184297037070368)
(-100.05876204459865, -0.00999362813311856)
(-19.724589056819497, -0.0506330074022565)
(51.51840520424845, 0.0194068822667605)
(-71.77542822896348, -0.0139309924289532)
Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
$$x_{1} = 24.2276516091942$$
$$x_{2} = 38.2554922699397$$
$$x_{3} = 28.127962881328$$
$$x_{4} = 87.3343406755818$$
$$x_{5} = 16.2294985469974$$
$$x_{6} = 96.2499704747411$$
$$x_{7} = 20.2743793800958$$
$$x_{8} = 93.1481561580688$$
$$x_{9} = 43.8065463376575$$
$$x_{10} = 31.8470589866078$$
$$x_{11} = 100.466100104885$$
$$x_{12} = 92.2500655105387$$
$$x_{13} = 33.9942163486092$$
$$x_{14} = 66.196842242554$$
$$x_{15} = 4.28821644642643$$
$$x_{16} = 48.2433464683252$$
$$x_{17} = 90.2532539182077$$
$$x_{18} = 30.0189593964374$$
$$x_{19} = 98.4285112204771$$
$$x_{20} = 68.0456297887513$$
$$x_{21} = 78.2060165150451$$
$$x_{22} = 18.4063634635748$$
$$x_{23} = 72.2117992350915$$
$$x_{24} = 58.0283340044788$$
$$x_{25} = 8.28407265099272$$
$$x_{26} = 86.2484289973703$$
$$x_{27} = 56.2413018983851$$
$$x_{28} = 22.2682093962144$$
$$x_{29} = 42.0501734594177$$
$$x_{30} = 64.2460343405985$$
$$x_{31} = 60.0242199598979$$
$$x_{32} = 82.2204810339713$$
$$x_{33} = 10.0019203972396$$
$$x_{34} = 61.9558424851141$$
$$x_{35} = 70.2491438435124$$
$$x_{36} = 25.9796753372099$$
$$x_{37} = 80.2481467468931$$
$$x_{38} = 14.27260285079$$
$$x_{39} = 12.3872073720398$$
$$x_{40} = 76.4591659424083$$
$$x_{41} = 83.9971261785332$$
$$x_{42} = 40.2562078674106$$
$$x_{43} = 46.2485129234309$$
$$x_{44} = 5.83772950085635$$
$$x_{45} = 74.2495470900384$$
$$x_{46} = 36.0135855463228$$
$$x_{47} = 2.42790391927647$$
$$x_{48} = 52.1246383328403$$
$$x_{49} = 54.2510144714636$$
$$x_{50} = 51.5184052042485$$
Puntos máximos de la función:
$$x_{50} = -79.9736379062586$$
$$x_{50} = -45.9759960258234$$
$$x_{50} = -11.8691603685014$$
$$x_{50} = -1.68473749010703$$
$$x_{50} = -83.9035710993377$$
$$x_{50} = -89.7471094627101$$
$$x_{50} = -57.7025860600124$$
$$x_{50} = -76.0471709199656$$
$$x_{50} = -91.7549366557361$$
$$x_{50} = -70.1148533151679$$
$$x_{50} = -77.9646195916378$$
$$x_{50} = -44.3057083278542$$
$$x_{50} = -85.7919092295442$$
$$x_{50} = -67.9532292328637$$
$$x_{50} = -52.0039570801788$$
$$x_{50} = -97.9806472089682$$
$$x_{50} = -7.6943345390805$$
$$x_{50} = -31.6491517223897$$
$$x_{50} = -25.7976504428415$$
$$x_{50} = -17.9746031910921$$
$$x_{50} = -63.9273980841703$$
$$x_{50} = -29.7561754047794$$
$$x_{50} = -13.7112922211213$$
$$x_{50} = -3.90574871969606$$
$$x_{50} = -21.9126729494334$$
$$x_{50} = -5.56234042020343$$
$$x_{50} = -65.8399425496541$$
$$x_{50} = -23.7694762992186$$
$$x_{50} = -53.7857517670918$$
$$x_{50} = -33.9479771749304$$
$$x_{50} = -81.7222496864894$$
$$x_{50} = -73.9952439834427$$
$$x_{50} = -89.5017394259607$$
$$x_{50} = -15.7381332897777$$
$$x_{50} = -55.9893713795814$$
$$x_{50} = -59.9980448791643$$
$$x_{50} = -35.7509272117533$$
$$x_{50} = -93.5855779692324$$
$$x_{50} = -39.1885097414717$$
$$x_{50} = -27.7342959949159$$
$$x_{50} = -96.0375765638132$$
$$x_{50} = -61.7017867903039$$
$$x_{50} = -10.0019203972396$$
$$x_{50} = -41.7502617099517$$
$$x_{50} = -47.7524511969237$$
$$x_{50} = -39.7457114582492$$
$$x_{50} = -49.7501287834332$$
$$x_{50} = -100.058762044599$$
$$x_{50} = -19.7245890568195$$
$$x_{50} = -71.7754282289635$$
Decrece en los intervalos
$$\left[100.466100104885, \infty\right)$$
Crece en los intervalos
$$\left[-1.68473749010703, 2.42790391927647\right]$$
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$\frac{8 x \left(- \frac{4 x^{2} \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right) \tan{\left(x^{2} + \frac{1}{2} \right)}}{2 x^{2} + 1} + \frac{16 x^{2} \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{\left(2 x^{2} + 1\right)^{2}} - \frac{48 x^{2} \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{3}} + \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right) \left(2 x^{2} \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right) + 4 x^{2} \tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 3 \tan{\left(x^{2} + \frac{1}{2} \right)}\right) - \frac{\left(4 x^{2} \tan{\left(x^{2} + \frac{1}{2} \right)} + 3\right) \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} - \frac{2 \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} - \frac{4 x^{2} \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right) \tan{\left(x^{2} + \frac{1}{2} \right)} + \tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1}{2 x^{2} + 1} + \frac{4 \left(2 x^{2} \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right) + \tan{\left(x^{2} + \frac{1}{2} \right)}\right)}{\left(2 x^{2} + 1\right)^{2}} + \frac{8 \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}}\right)}{2 x^{2} + 1} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 0$$
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
$$\left[0, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, 0\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$\frac{8 x \left(- \frac{4 x^{2} \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right) \tan{\left(x^{2} + \frac{1}{2} \right)}}{2 x^{2} + 1} + \frac{16 x^{2} \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{\left(2 x^{2} + 1\right)^{2}} - \frac{48 x^{2} \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{3}} + \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right) \left(2 x^{2} \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right) + 4 x^{2} \tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 3 \tan{\left(x^{2} + \frac{1}{2} \right)}\right) - \frac{\left(4 x^{2} \tan{\left(x^{2} + \frac{1}{2} \right)} + 3\right) \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} - \frac{2 \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} - \frac{4 x^{2} \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right) \tan{\left(x^{2} + \frac{1}{2} \right)} + \tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1}{2 x^{2} + 1} + \frac{4 \left(2 x^{2} \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right) + \tan{\left(x^{2} + \frac{1}{2} \right)}\right)}{\left(2 x^{2} + 1\right)^{2}} + \frac{8 \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}}\right)}{2 x^{2} + 1} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 0$$
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
$$\left[0, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, 0\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{x \to -\infty}\left(\frac{2 x \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} + \frac{- 4 x \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(\frac{2 x \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} + \frac{- 4 x \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{x \to -\infty}\left(\frac{2 x \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} + \frac{- 4 x \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(\frac{2 x \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} + \frac{- 4 x \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}}\right)$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función ((-4*x)*tan(x^2 + 1/2))/(1 + 2*x^2)^2 + ((2*x)*(1 + tan(x^2 + 1/2)^2))/(1 + 2*x^2), dividida por x con x->+oo y x ->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{\frac{2 x \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} + \frac{- 4 x \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}}}{x}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\frac{2 x \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} + \frac{- 4 x \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}}}{x}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{\frac{2 x \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} + \frac{- 4 x \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}}}{x}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\frac{2 x \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} + \frac{- 4 x \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}}}{x}\right)$$
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:
$$\frac{2 x \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} + \frac{- 4 x \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}} = - \frac{2 x \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} + \frac{4 x \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}}$$
- No
$$\frac{2 x \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} + \frac{- 4 x \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}} = \frac{2 x \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} - \frac{4 x \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{2 x \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} + \frac{- 4 x \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}} = - \frac{2 x \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} + \frac{4 x \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}}$$
- No
$$\frac{2 x \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} + \frac{- 4 x \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}} = \frac{2 x \left(\tan^{2}{\left(x^{2} + \frac{1}{2} \right)} + 1\right)}{2 x^{2} + 1} - \frac{4 x \tan{\left(x^{2} + \frac{1}{2} \right)}}{\left(2 x^{2} + 1\right)^{2}}$$
- No
es decir, función
no es
par ni impar