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- Gráfico de la función implícita
- Derivadas paso a paso
- Integrales paso a paso
- Gráfico de la función paramétrica
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- Superficie especificada paramétricamente
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- 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 =:
-
-x^4+8*x^2-16
-
x^3/
-
x^4-4x
-
(x^4+1)/x^3
-
Expresiones idénticas
- sin(x)+sin(diez / tres *x)+log(x)-(cero , ochenta y cuatro *x+ tres)
- seno de (x) más seno de (10 dividir por 3 multiplicar por x) más logaritmo de (x) menos (0,84 multiplicar por x más 3)
- seno de (x) más seno de (diez dividir por tres multiplicar por x) más logaritmo de (x) menos (cero , ochenta y cuatro multiplicar por x más tres)
- sin(x)+sin(10/3x)+log(x)-(0,84x+3)
- sinx+sin10/3x+logx-0,84x+3
- sin(x)+sin(10 dividir por 3*x)+log(x)-(0,84*x+3)
-
Expresiones semejantes
- sin(x)+sin(10/3*x)+log(x)+(0,84*x+3)
- sin(x)+sin(10/3*x)+log(x)-(0,84*x-3)
- sin(x)-sin(10/3*x)+log(x)-(0,84*x+3)
- sin(x)+sin(10/3*x)-log(x)-(0,84*x+3)
- sinx+sin(10/3*x)+log(x)-(0,84*x+3)
-
Expresiones con funciones
- Seno sin
- sin(x-pi/3)-2
- sin(x)+√3cos(x)
- sin(x)+x^2-1
- sin(x/2+pi/2)+4
- sin(y)^2+sin(-1)^2
- Seno sin
- sin(x-pi/3)-2
- sin(x)+√3cos(x)
- sin(x)+x^2-1
- sin(x/2+pi/2)+4
- sin(y)^2+sin(-1)^2
- Logaritmo log
- log((x^2-4),10)
- log(x*x+4*x+5)/x
- log((x^2)/(x-2))*e^x
- log(3*x)+sqrt(x)
- log(x,0.5)
Gráfico de la función y = sin(x)+sin(10/3*x)+log(x)-(0,84*x+3)
Solución
Ha introducido
[src]
/10*x\ 21*x
f(x) = sin(x) + sin|----| + log(x) + - ---- - 3
\ 3 / 25
$$f{\left(x \right)} = \left(- \frac{21 x}{25} - 3\right) + \left(\left(\sin{\left(x \right)} + \sin{\left(\frac{10 x}{3} \right)}\right) + \log{\left(x \right)}\right)$$
f = -21*x/25 - 3 + sin(x) + sin(10*x/3) + log(x)
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:
$$\left(- \frac{21 x}{25} - 3\right) + \left(\left(\sin{\left(x \right)} + \sin{\left(\frac{10 x}{3} \right)}\right) + \log{\left(x \right)}\right) = 0$$
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
o sea hay que resolver la ecuación:
$$\left(- \frac{21 x}{25} - 3\right) + \left(\left(\sin{\left(x \right)} + \sin{\left(\frac{10 x}{3} \right)}\right) + \log{\left(x \right)}\right) = 0$$
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
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
$$\cos{\left(x \right)} + \frac{10 \cos{\left(\frac{10 x}{3} \right)}}{3} - \frac{21}{25} + \frac{1}{x} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 20.3264748146376$$
$$x_{2} = 88.1071598626067$$
$$x_{3} = 52.4361080291788$$
$$x_{4} = -92.7647687432498$$
$$x_{5} = -29.7742294156023$$
$$x_{6} = -41.8070217845034$$
$$x_{7} = 98.359379170854$$
$$x_{8} = 38.1770248358623$$
$$x_{9} = 66.275733076055$$
$$x_{10} = 25.9272097510098$$
$$x_{11} = -77.6186305930536$$
$$x_{12} = 70.1835513162358$$
$$x_{13} = 61.7631772873456$$
$$x_{14} = -43.8366138043001$$
$$x_{15} = 64.4777554082914$$
$$x_{16} = 50.4072732093096$$
$$x_{17} = -39.918390002496$$
$$x_{18} = 9.73660376229262$$
$$x_{19} = 6.15442632231374$$
$$x_{20} = -7.91689597146521$$
$$x_{21} = 11.7763601051293$$
$$x_{22} = -56.0745607596557$$
$$x_{23} = 90.136079953691$$
$$x_{24} = 4.13613862668332$$
$$x_{25} = -48.622668650569$$
$$x_{26} = -88.1092315832218$$
$$x_{27} = 27.9697803776456$$
$$x_{28} = 93.7712090449464$$
$$x_{29} = 58.0288294633648$$
$$x_{30} = -27.9776669461112$$
$$x_{31} = 82.4784227417571$$
$$x_{32} = 2.2614896506161$$
$$x_{33} = 24.0619395875367$$
$$x_{34} = -61.7658931604908$$
$$x_{35} = 18.3692013149437$$
$$x_{36} = 39.9230258540148$$
$$x_{37} = -54.3287729735818$$
$$x_{38} = -24.068910074225$$
$$x_{39} = -92.027185754873$$
$$x_{40} = 9.11219368543285$$
$$x_{41} = 17.3734839772394$$
$$x_{42} = -75.8727329848769$$
$$x_{43} = -9.71390760670272$$
$$x_{44} = -79.5073638088917$$
$$x_{45} = -22.3226174929298$$
$$x_{46} = 71.9312325550698$$
$$x_{47} = 22.3133786277316$$
$$x_{48} = 68.3182665487933$$
$$x_{49} = -33.5918373622021$$
$$x_{50} = 12.7027849488221$$
$$x_{51} = -13.6275266737725$$
$$x_{52} = 7.9384956655144$$
$$x_{53} = -11.7599928596994$$
$$x_{54} = -53.0782995690826$$
$$x_{55} = 36.2200037715291$$
$$x_{56} = -45.6249951099989$$
$$x_{57} = -86.3210107602634$$
$$x_{58} = 100.388272620152$$
$$x_{59} = -71.9283661677536$$
$$x_{60} = 54.3253664735286$$
$$x_{61} = 56.0714773860368$$
$$x_{62} = 86.31902753419$$
$$x_{63} = -63.6315477237279$$
$$x_{64} = -25.9346368631586$$
$$x_{65} = 60.0153752136686$$
$$x_{66} = 75.8750116841133$$
$$x_{67} = -81.5366882177038$$
$$x_{68} = 76.878814699632$$
$$x_{69} = 92.0251747595862$$
$$x_{70} = -73.9149328337359$$
$$x_{71} = 41.8121934923855$$
$$x_{72} = -65.6745022641783$$
$$x_{73} = -6.1246828594817$$
$$x_{74} = -58.0323358341671$$
$$x_{75} = 43.8407776298487$$
$$x_{76} = 29.7684791004839$$
$$x_{77} = -38.1724957984287$$
$$x_{78} = -31.5624194458159$$
$$x_{79} = -93.7730528080635$$
$$x_{80} = 73.9176856251388$$
$$x_{81} = -95.7307564609111$$
$$x_{82} = -97.7172579586002$$
$$x_{83} = -60.0188105569444$$
$$x_{84} = 84.521072452578$$
$$x_{85} = -47.4221840916411$$
$$x_{86} = -90.1384787751391$$
$$x_{87} = 32.4858252883064$$
$$x_{88} = -4.0833876927505$$
$$x_{89} = 34.2336921288663$$
$$x_{90} = -70.1811611924326$$
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:
$$x_{1} = 20.3264748146376$$
$$x_{2} = 88.1071598626067$$
$$x_{3} = 52.4361080291788$$
$$x_{4} = 25.9272097510098$$
$$x_{5} = 61.7631772873456$$
$$x_{6} = 50.4072732093096$$
$$x_{7} = 90.136079953691$$
$$x_{8} = 27.9697803776456$$
$$x_{9} = 93.7712090449464$$
$$x_{10} = 58.0288294633648$$
$$x_{11} = 82.4784227417571$$
$$x_{12} = 24.0619395875367$$
$$x_{13} = 18.3692013149437$$
$$x_{14} = 9.11219368543285$$
$$x_{15} = 22.3133786277316$$
$$x_{16} = 12.7027849488221$$
$$x_{17} = 54.3253664735286$$
$$x_{18} = 56.0714773860368$$
$$x_{19} = 86.31902753419$$
$$x_{20} = 60.0153752136686$$
$$x_{21} = 76.878814699632$$
$$x_{22} = 92.0251747595862$$
$$x_{23} = 29.7684791004839$$
$$x_{24} = 84.521072452578$$
Puntos máximos de la función:
$$x_{24} = 98.359379170854$$
$$x_{24} = 38.1770248358623$$
$$x_{24} = 66.275733076055$$
$$x_{24} = 70.1835513162358$$
$$x_{24} = 64.4777554082914$$
$$x_{24} = 9.73660376229262$$
$$x_{24} = 6.15442632231374$$
$$x_{24} = 11.7763601051293$$
$$x_{24} = 4.13613862668332$$
$$x_{24} = 2.2614896506161$$
$$x_{24} = 39.9230258540148$$
$$x_{24} = 17.3734839772394$$
$$x_{24} = 71.9312325550698$$
$$x_{24} = 68.3182665487933$$
$$x_{24} = 7.9384956655144$$
$$x_{24} = 36.2200037715291$$
$$x_{24} = 100.388272620152$$
$$x_{24} = 75.8750116841133$$
$$x_{24} = 41.8121934923855$$
$$x_{24} = 43.8407776298487$$
$$x_{24} = 73.9176856251388$$
$$x_{24} = 32.4858252883064$$
$$x_{24} = 34.2336921288663$$
Decrece en los intervalos
$$\left[93.7712090449464, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, 9.11219368543285\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\cos{\left(x \right)} + \frac{10 \cos{\left(\frac{10 x}{3} \right)}}{3} - \frac{21}{25} + \frac{1}{x} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 20.3264748146376$$
$$x_{2} = 88.1071598626067$$
$$x_{3} = 52.4361080291788$$
$$x_{4} = -92.7647687432498$$
$$x_{5} = -29.7742294156023$$
$$x_{6} = -41.8070217845034$$
$$x_{7} = 98.359379170854$$
$$x_{8} = 38.1770248358623$$
$$x_{9} = 66.275733076055$$
$$x_{10} = 25.9272097510098$$
$$x_{11} = -77.6186305930536$$
$$x_{12} = 70.1835513162358$$
$$x_{13} = 61.7631772873456$$
$$x_{14} = -43.8366138043001$$
$$x_{15} = 64.4777554082914$$
$$x_{16} = 50.4072732093096$$
$$x_{17} = -39.918390002496$$
$$x_{18} = 9.73660376229262$$
$$x_{19} = 6.15442632231374$$
$$x_{20} = -7.91689597146521$$
$$x_{21} = 11.7763601051293$$
$$x_{22} = -56.0745607596557$$
$$x_{23} = 90.136079953691$$
$$x_{24} = 4.13613862668332$$
$$x_{25} = -48.622668650569$$
$$x_{26} = -88.1092315832218$$
$$x_{27} = 27.9697803776456$$
$$x_{28} = 93.7712090449464$$
$$x_{29} = 58.0288294633648$$
$$x_{30} = -27.9776669461112$$
$$x_{31} = 82.4784227417571$$
$$x_{32} = 2.2614896506161$$
$$x_{33} = 24.0619395875367$$
$$x_{34} = -61.7658931604908$$
$$x_{35} = 18.3692013149437$$
$$x_{36} = 39.9230258540148$$
$$x_{37} = -54.3287729735818$$
$$x_{38} = -24.068910074225$$
$$x_{39} = -92.027185754873$$
$$x_{40} = 9.11219368543285$$
$$x_{41} = 17.3734839772394$$
$$x_{42} = -75.8727329848769$$
$$x_{43} = -9.71390760670272$$
$$x_{44} = -79.5073638088917$$
$$x_{45} = -22.3226174929298$$
$$x_{46} = 71.9312325550698$$
$$x_{47} = 22.3133786277316$$
$$x_{48} = 68.3182665487933$$
$$x_{49} = -33.5918373622021$$
$$x_{50} = 12.7027849488221$$
$$x_{51} = -13.6275266737725$$
$$x_{52} = 7.9384956655144$$
$$x_{53} = -11.7599928596994$$
$$x_{54} = -53.0782995690826$$
$$x_{55} = 36.2200037715291$$
$$x_{56} = -45.6249951099989$$
$$x_{57} = -86.3210107602634$$
$$x_{58} = 100.388272620152$$
$$x_{59} = -71.9283661677536$$
$$x_{60} = 54.3253664735286$$
$$x_{61} = 56.0714773860368$$
$$x_{62} = 86.31902753419$$
$$x_{63} = -63.6315477237279$$
$$x_{64} = -25.9346368631586$$
$$x_{65} = 60.0153752136686$$
$$x_{66} = 75.8750116841133$$
$$x_{67} = -81.5366882177038$$
$$x_{68} = 76.878814699632$$
$$x_{69} = 92.0251747595862$$
$$x_{70} = -73.9149328337359$$
$$x_{71} = 41.8121934923855$$
$$x_{72} = -65.6745022641783$$
$$x_{73} = -6.1246828594817$$
$$x_{74} = -58.0323358341671$$
$$x_{75} = 43.8407776298487$$
$$x_{76} = 29.7684791004839$$
$$x_{77} = -38.1724957984287$$
$$x_{78} = -31.5624194458159$$
$$x_{79} = -93.7730528080635$$
$$x_{80} = 73.9176856251388$$
$$x_{81} = -95.7307564609111$$
$$x_{82} = -97.7172579586002$$
$$x_{83} = -60.0188105569444$$
$$x_{84} = 84.521072452578$$
$$x_{85} = -47.4221840916411$$
$$x_{86} = -90.1384787751391$$
$$x_{87} = 32.4858252883064$$
$$x_{88} = -4.0833876927505$$
$$x_{89} = 34.2336921288663$$
$$x_{90} = -70.1811611924326$$
Signos de extremos en los puntos:
(20.32647481463759, -17.0446080722536)
(88.10715986260668, -73.3882072766648)
(52.43610802917885, -43.1708374309715)
(-92.76476874324977, 79.4751796154569 + pi*I)
(-29.774229415602296, 27.3605065292784 + pi*I)
(-41.80702178450338, 35.7707777105377 + pi*I)
(98.35937917085398, -80.9499409822209)
(38.17702483586229, -29.9667874190284)
(66.275733076055, -53.9299305693658)
(25.927209751009848, -21.8096148571052)
(-77.61863059305364, 64.8558356762974 + pi*I)
(70.18355131623584, -55.8319330224968)
(61.763177287345584, -52.6290319957312)
(-43.83661380430008, 36.7491150091499 + pi*I)
(64.47775540829143, -51.0349132173157)
(50.4072732093096, -42.2793595709589)
(-39.91839000249605, 32.523632775985 + pi*I)
(9.736603762292622, -8.34754155306323)
(6.154426322313743, -5.4854026360725)
(-7.916895971465205, 3.77002883247696 + pi*I)
(11.77636010512928, -10.1365272015442)
(-56.07456075965573, 49.5858117164858 + pi*I)
(90.13607995369095, -74.296379261265)
(4.136138626683319, -4.95374583502061)
(-48.622668650569004, 43.6846103609719 + pi*I)
(-88.1092315832218, 76.3453383895366 + pi*I)
(27.96978037764564, -23.7131982437132)
(93.77120904494645, -78.6855343795832)
(58.02882946336478, -47.6629539746136)
(-27.97766694611117, 24.375728745064 + pi*I)
(82.47842274175714, -68.1533165605613)
(2.2614896506160975, -2.36223845461116)
(24.06193958753672, -21.9043926946554)
(-61.76589316049081, 54.875690660878 + pi*I)
(18.369201314943687, -16.9810857549881)
(39.92302585401478, -31.1498422743441)
(-54.32877297358182, 48.3264065890837 + pi*I)
(-24.068910074225002, 22.2659449020395 + pi*I)
(-92.02718575487302, 80.520700824583 + pi*I)
(9.112193685432853, -9.00049900430161)
(17.37348397723945, -14.7558217349856)
(-75.87273298487693, 63.6053104818516 + pi*I)
(-9.713907606702717, 6.89701379027883 + pi*I)
(-79.50736380889172, 68.0807945406636 + pi*I)
(-22.322617492929776, 20.0177822533006 + pi*I)
(71.93123255506983, -57.9801483296808)
(22.313378627731584, -19.8069965715063)
(68.31826654879333, -55.8789885760179)
(-33.591837362202064, 28.8110605395262 + pi*I)
(12.702784948822053, -11.990154871166)
(-13.627526673772486, 9.19447551278607 + pi*I)
(7.938495665514403, -5.62929365084291)
(-11.759992859699434, 9.06732647714134 + pi*I)
(-53.07829956908257, 44.3940084871237 + pi*I)
(36.220003771529065, -29.8546530150086)
(-45.62499510999886, 37.1880725303613 + pi*I)
(-86.32101076026342, 75.9258598585117 + pi*I)
(100.38827262015187, -81.8604619206263)
(-71.92836616775358, 60.5315296362215 + pi*I)
(54.32536647352862, -46.336361390362)
(56.0714773860368, -47.5325022097812)
(86.31902753419003, -73.0097367793979)
(-63.63154772372795, 54.8853308664576 + pi*I)
(-25.934636863158644, 22.3204872131055 + pi*I)
(60.01537521366864, -50.4849248115257)
(75.87501168411332, -60.9471657070446)
(-81.53668821770384, 69.0369357569637 + pi*I)
(76.87881469963196, -63.2152970397162)
(92.02517475958616, -77.4765546257793)
(-73.91493283373589, 63.4144039996088 + pi*I)
(41.81219349238551, -34.3045252112697)
(-65.67450226417832, 56.8963449026919 + pi*I)
(-6.1246828594816956, 3.11491142969842 + pi*I)
(-58.03233583416715, 49.7848942624748 + pi*I)
(43.84077762984872, -35.1880812285114)
(29.768479100483873, -26.5734133502736)
(-38.17249579842873, 31.2511365703267 + pi*I)
(-31.562419445815923, 27.8178073831611 + pi*I)
(-93.7730528080635, 81.7672697799159 + pi*I)
(73.91768562513876, -60.8085369909345)
(-95.73075646091108, 81.9526970379258 + pi*I)
(-97.71725795860019, 84.8288666017865 + pi*I)
(-60.01881055694437, 52.6741835546042 + pi*I)
(84.52107245257803, -70.10826534941)
(-47.42218409164112, 40.1492296240703 + pi*I)
(-90.13847877513908, 77.2990469130665 + pi*I)
(32.485825288306444, -24.9350203983231)
(-4.083387692750495, 1.7807028802751 + pi*I)
(34.23369212886632, -27.0554284999452)
(-70.18116119243255, 58.3341269159225 + pi*I)
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:
$$x_{1} = 20.3264748146376$$
$$x_{2} = 88.1071598626067$$
$$x_{3} = 52.4361080291788$$
$$x_{4} = 25.9272097510098$$
$$x_{5} = 61.7631772873456$$
$$x_{6} = 50.4072732093096$$
$$x_{7} = 90.136079953691$$
$$x_{8} = 27.9697803776456$$
$$x_{9} = 93.7712090449464$$
$$x_{10} = 58.0288294633648$$
$$x_{11} = 82.4784227417571$$
$$x_{12} = 24.0619395875367$$
$$x_{13} = 18.3692013149437$$
$$x_{14} = 9.11219368543285$$
$$x_{15} = 22.3133786277316$$
$$x_{16} = 12.7027849488221$$
$$x_{17} = 54.3253664735286$$
$$x_{18} = 56.0714773860368$$
$$x_{19} = 86.31902753419$$
$$x_{20} = 60.0153752136686$$
$$x_{21} = 76.878814699632$$
$$x_{22} = 92.0251747595862$$
$$x_{23} = 29.7684791004839$$
$$x_{24} = 84.521072452578$$
Puntos máximos de la función:
$$x_{24} = 98.359379170854$$
$$x_{24} = 38.1770248358623$$
$$x_{24} = 66.275733076055$$
$$x_{24} = 70.1835513162358$$
$$x_{24} = 64.4777554082914$$
$$x_{24} = 9.73660376229262$$
$$x_{24} = 6.15442632231374$$
$$x_{24} = 11.7763601051293$$
$$x_{24} = 4.13613862668332$$
$$x_{24} = 2.2614896506161$$
$$x_{24} = 39.9230258540148$$
$$x_{24} = 17.3734839772394$$
$$x_{24} = 71.9312325550698$$
$$x_{24} = 68.3182665487933$$
$$x_{24} = 7.9384956655144$$
$$x_{24} = 36.2200037715291$$
$$x_{24} = 100.388272620152$$
$$x_{24} = 75.8750116841133$$
$$x_{24} = 41.8121934923855$$
$$x_{24} = 43.8407776298487$$
$$x_{24} = 73.9176856251388$$
$$x_{24} = 32.4858252883064$$
$$x_{24} = 34.2336921288663$$
Decrece en los intervalos
$$\left[93.7712090449464, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, 9.11219368543285\right]$$
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
$$- (\sin{\left(x \right)} + \frac{100 \sin{\left(\frac{10 x}{3} \right)}}{9} + \frac{1}{x^{2}}) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 10.3459891535317$$
$$x_{2} = 46.202939898987$$
$$x_{3} = 69.7278155833422$$
$$x_{4} = 26.3638514509402$$
$$x_{5} = 48.0448637230843$$
$$x_{6} = 33.9129460846312$$
$$x_{7} = -39.5581665318107$$
$$x_{8} = -78.2337880522672$$
$$x_{9} = 44.3050347680876$$
$$x_{10} = -46.202914946311$$
$$x_{11} = 76.3629170154809$$
$$x_{12} = 28.2742991713$$
$$x_{13} = -71.6120873471365$$
$$x_{14} = 65.9734393499088$$
$$x_{15} = 41.4852375128148$$
$$x_{16} = -69.7278264642108$$
$$x_{17} = -12.2434621894089$$
$$x_{18} = 99.9181796206136$$
$$x_{19} = 74.4335400357424$$
$$x_{20} = -30.1848071507631$$
$$x_{21} = -57.5133480651066$$
$$x_{22} = -19.8141744945723$$
$$x_{23} = 17.8849532736588$$
$$x_{24} = -62.2190772597939$$
$$x_{25} = 52.7625161090086$$
$$x_{26} = -65.97345210086$$
$$x_{27} = 62.2190635941936$$
$$x_{28} = 73.5391814003115$$
$$x_{29} = -35.8400958696204$$
$$x_{30} = -33.9129941451064$$
$$x_{31} = -73.53919150069$$
$$x_{32} = -16.0138845726635$$
$$x_{33} = 60.3348019007869$$
$$x_{34} = 12.2438320340066$$
$$x_{35} = 14.1643286529706$$
$$x_{36} = -40.5346644853485$$
$$x_{37} = -83.902035482328$$
$$x_{38} = -55.5839702505857$$
$$x_{39} = 16.0140899303021$$
$$x_{40} = 90.4616345023349$$
$$x_{41} = -23.5348693098294$$
$$x_{42} = -32.0287349655944$$
$$x_{43} = -29.1952650169188$$
$$x_{44} = 49.9427700757633$$
$$x_{45} = -99.9181849195256$$
$$x_{46} = 85.743967650719$$
$$x_{47} = 35.8400533452899$$
$$x_{48} = -1.86686645640636$$
$$x_{49} = -7.51480968625305$$
$$x_{50} = 30.1847482007348$$
$$x_{51} = -21.6850685938247$$
$$x_{52} = 19.8143145760038$$
$$x_{53} = 78.2337966566762$$
$$x_{54} = 82.0041366111785$$
$$x_{55} = 98.0339184601643$$
$$x_{56} = -3.78806804379283$$
$$x_{57} = -17.8847813387092$$
$$x_{58} = -96.1068198002513$$
$$x_{59} = 96.1068138864476$$
$$x_{60} = -5.67122513222173$$
$$x_{61} = 8.50418385952301$$
$$x_{62} = 67.8838836910847$$
$$x_{63} = -87.641867195962$$
$$x_{64} = -53.7130899694754$$
$$x_{65} = 3.78421210948768$$
$$x_{66} = -98.0339242114768$$
$$x_{67} = 94.2477766479706$$
$$x_{68} = 31.0932317137121$$
$$x_{69} = 83.9020430491075$$
$$x_{70} = -89.5624139330734$$
$$x_{71} = 21.6851805855588$$
$$x_{72} = 51.8633155386264$$
$$x_{73} = 80.0835902075532$$
$$x_{74} = -37.6991303413281$$
$$x_{75} = -94.2477825674166$$
$$x_{76} = -80.0835817595622$$
$$x_{77} = 58.4076969944574$$
$$x_{78} = 24.5199145886044$$
$$x_{79} = -67.8838953464676$$
$$x_{80} = -82.0041283665825$$
$$x_{81} = 64.0629953884117$$
$$x_{82} = 32.0286833956245$$
$$x_{83} = -85.7439604055421$$
$$x_{84} = -48.0448406469855$$
$$x_{85} = -14.1640585951392$$
$$x_{86} = -75.3982283107216$$
$$x_{87} = -64.0630084755717$$
$$x_{88} = -51.863295395815$$
$$x_{89} = 92.3887391723532$$
$$x_{90} = 42.3844892202862$$
$$x_{91} = 1.85105698812704$$
$$x_{92} = -49.9427478479982$$
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
$$\left[85.743967650719, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -89.5624139330734\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- (\sin{\left(x \right)} + \frac{100 \sin{\left(\frac{10 x}{3} \right)}}{9} + \frac{1}{x^{2}}) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 10.3459891535317$$
$$x_{2} = 46.202939898987$$
$$x_{3} = 69.7278155833422$$
$$x_{4} = 26.3638514509402$$
$$x_{5} = 48.0448637230843$$
$$x_{6} = 33.9129460846312$$
$$x_{7} = -39.5581665318107$$
$$x_{8} = -78.2337880522672$$
$$x_{9} = 44.3050347680876$$
$$x_{10} = -46.202914946311$$
$$x_{11} = 76.3629170154809$$
$$x_{12} = 28.2742991713$$
$$x_{13} = -71.6120873471365$$
$$x_{14} = 65.9734393499088$$
$$x_{15} = 41.4852375128148$$
$$x_{16} = -69.7278264642108$$
$$x_{17} = -12.2434621894089$$
$$x_{18} = 99.9181796206136$$
$$x_{19} = 74.4335400357424$$
$$x_{20} = -30.1848071507631$$
$$x_{21} = -57.5133480651066$$
$$x_{22} = -19.8141744945723$$
$$x_{23} = 17.8849532736588$$
$$x_{24} = -62.2190772597939$$
$$x_{25} = 52.7625161090086$$
$$x_{26} = -65.97345210086$$
$$x_{27} = 62.2190635941936$$
$$x_{28} = 73.5391814003115$$
$$x_{29} = -35.8400958696204$$
$$x_{30} = -33.9129941451064$$
$$x_{31} = -73.53919150069$$
$$x_{32} = -16.0138845726635$$
$$x_{33} = 60.3348019007869$$
$$x_{34} = 12.2438320340066$$
$$x_{35} = 14.1643286529706$$
$$x_{36} = -40.5346644853485$$
$$x_{37} = -83.902035482328$$
$$x_{38} = -55.5839702505857$$
$$x_{39} = 16.0140899303021$$
$$x_{40} = 90.4616345023349$$
$$x_{41} = -23.5348693098294$$
$$x_{42} = -32.0287349655944$$
$$x_{43} = -29.1952650169188$$
$$x_{44} = 49.9427700757633$$
$$x_{45} = -99.9181849195256$$
$$x_{46} = 85.743967650719$$
$$x_{47} = 35.8400533452899$$
$$x_{48} = -1.86686645640636$$
$$x_{49} = -7.51480968625305$$
$$x_{50} = 30.1847482007348$$
$$x_{51} = -21.6850685938247$$
$$x_{52} = 19.8143145760038$$
$$x_{53} = 78.2337966566762$$
$$x_{54} = 82.0041366111785$$
$$x_{55} = 98.0339184601643$$
$$x_{56} = -3.78806804379283$$
$$x_{57} = -17.8847813387092$$
$$x_{58} = -96.1068198002513$$
$$x_{59} = 96.1068138864476$$
$$x_{60} = -5.67122513222173$$
$$x_{61} = 8.50418385952301$$
$$x_{62} = 67.8838836910847$$
$$x_{63} = -87.641867195962$$
$$x_{64} = -53.7130899694754$$
$$x_{65} = 3.78421210948768$$
$$x_{66} = -98.0339242114768$$
$$x_{67} = 94.2477766479706$$
$$x_{68} = 31.0932317137121$$
$$x_{69} = 83.9020430491075$$
$$x_{70} = -89.5624139330734$$
$$x_{71} = 21.6851805855588$$
$$x_{72} = 51.8633155386264$$
$$x_{73} = 80.0835902075532$$
$$x_{74} = -37.6991303413281$$
$$x_{75} = -94.2477825674166$$
$$x_{76} = -80.0835817595622$$
$$x_{77} = 58.4076969944574$$
$$x_{78} = 24.5199145886044$$
$$x_{79} = -67.8838953464676$$
$$x_{80} = -82.0041283665825$$
$$x_{81} = 64.0629953884117$$
$$x_{82} = 32.0286833956245$$
$$x_{83} = -85.7439604055421$$
$$x_{84} = -48.0448406469855$$
$$x_{85} = -14.1640585951392$$
$$x_{86} = -75.3982283107216$$
$$x_{87} = -64.0630084755717$$
$$x_{88} = -51.863295395815$$
$$x_{89} = 92.3887391723532$$
$$x_{90} = 42.3844892202862$$
$$x_{91} = 1.85105698812704$$
$$x_{92} = -49.9427478479982$$
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
$$\left[85.743967650719, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -89.5624139330734\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\left(- \frac{21 x}{25} - 3\right) + \left(\left(\sin{\left(x \right)} + \sin{\left(\frac{10 x}{3} \right)}\right) + \log{\left(x \right)}\right)\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(\left(- \frac{21 x}{25} - 3\right) + \left(\left(\sin{\left(x \right)} + \sin{\left(\frac{10 x}{3} \right)}\right) + \log{\left(x \right)}\right)\right)$$
$$\lim_{x \to -\infty}\left(\left(- \frac{21 x}{25} - 3\right) + \left(\left(\sin{\left(x \right)} + \sin{\left(\frac{10 x}{3} \right)}\right) + \log{\left(x \right)}\right)\right) = \infty$$
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
True
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(\left(- \frac{21 x}{25} - 3\right) + \left(\left(\sin{\left(x \right)} + \sin{\left(\frac{10 x}{3} \right)}\right) + \log{\left(x \right)}\right)\right)$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x) + sin(10*x/3) + log(x) - 21*x/25 - 3, dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(- \frac{21 x}{25} - 3\right) + \left(\left(\sin{\left(x \right)} + \sin{\left(\frac{10 x}{3} \right)}\right) + \log{\left(x \right)}\right)}{x}\right) = - \frac{21}{25}$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = - \frac{21 x}{25}$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\left(- \frac{21 x}{25} - 3\right) + \left(\left(\sin{\left(x \right)} + \sin{\left(\frac{10 x}{3} \right)}\right) + \log{\left(x \right)}\right)}{x}\right)$$
$$\lim_{x \to -\infty}\left(\frac{\left(- \frac{21 x}{25} - 3\right) + \left(\left(\sin{\left(x \right)} + \sin{\left(\frac{10 x}{3} \right)}\right) + \log{\left(x \right)}\right)}{x}\right) = - \frac{21}{25}$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = - \frac{21 x}{25}$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\left(- \frac{21 x}{25} - 3\right) + \left(\left(\sin{\left(x \right)} + \sin{\left(\frac{10 x}{3} \right)}\right) + \log{\left(x \right)}\right)}{x}\right)$$
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:
$$\left(- \frac{21 x}{25} - 3\right) + \left(\left(\sin{\left(x \right)} + \sin{\left(\frac{10 x}{3} \right)}\right) + \log{\left(x \right)}\right) = \frac{21 x}{25} + \log{\left(- x \right)} - \sin{\left(x \right)} - \sin{\left(\frac{10 x}{3} \right)} - 3$$
- No
$$\left(- \frac{21 x}{25} - 3\right) + \left(\left(\sin{\left(x \right)} + \sin{\left(\frac{10 x}{3} \right)}\right) + \log{\left(x \right)}\right) = - \frac{21 x}{25} - \log{\left(- x \right)} + \sin{\left(x \right)} + \sin{\left(\frac{10 x}{3} \right)} + 3$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\left(- \frac{21 x}{25} - 3\right) + \left(\left(\sin{\left(x \right)} + \sin{\left(\frac{10 x}{3} \right)}\right) + \log{\left(x \right)}\right) = \frac{21 x}{25} + \log{\left(- x \right)} - \sin{\left(x \right)} - \sin{\left(\frac{10 x}{3} \right)} - 3$$
- No
$$\left(- \frac{21 x}{25} - 3\right) + \left(\left(\sin{\left(x \right)} + \sin{\left(\frac{10 x}{3} \right)}\right) + \log{\left(x \right)}\right) = - \frac{21 x}{25} - \log{\left(- x \right)} + \sin{\left(x \right)} + \sin{\left(\frac{10 x}{3} \right)} + 3$$
- No
es decir, función
no es
par ni impar