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Trazado el gráfico de la función/ (-cos(7*x)+cos(3*x))/x^2

Gráfico de la función y = (-cos(7*x)+cos(3*x))/x^2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       -cos(7*x) + cos(3*x)
f(x) = --------------------
                 2         
                x
f(x)=cos⁡(3x)−cos⁡(7x)x2f{\left(x \right)} = \frac{\cos{\left(3 x \right)} - \cos{\left(7 x \right)}}{x^{2}} 
f = (cos(3*x) - cos(7*x))/x^2
Gráfico de la función
02468-8-6-4-2-1010-2525
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
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⁡(3x)−cos⁡(7x)x2=0\frac{\cos{\left(3 x \right)} - \cos{\left(7 x \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−4π5x_{1} = - \frac{4 \pi}{5}
x2=−3π5x_{2} = - \frac{3 \pi}{5}
x3=−2π5x_{3} = - \frac{2 \pi}{5}
x4=−π5x_{4} = - \frac{\pi}{5}
x5=π5x_{5} = \frac{\pi}{5}
x6=2π5x_{6} = \frac{2 \pi}{5}
x7=3π5x_{7} = \frac{3 \pi}{5}
x8=4π5x_{8} = \frac{4 \pi}{5}
x9=πx_{9} = \pi
Solución numérica
x1=82.9380460547705x_{1} = 82.9380460547705
x2=74.1415866247191x_{2} = 74.1415866247191
x3=11.9380520836412x_{3} = 11.9380520836412
x4=−89.8495498926681x_{4} = -89.8495498926681
x5=14.1371669411541x_{5} = 14.1371669411541
x6=58.1194640914112x_{6} = 58.1194640914112
x7=62.2035345410779x_{7} = 62.2035345410779
x8=−86.0796387083603x_{8} = -86.0796387083603
x9=96.1327351998477x_{9} = 96.1327351998477
x10=−67.8584013175395x_{10} = -67.8584013175395
x11=23.8761041672824x_{11} = 23.8761041672824
x12=99.2743278534375x_{12} = 99.2743278534375
x13=36.1283155162826x_{13} = 36.1283155162826
x14=−7.85398163397448x_{14} = -7.85398163397448
x15=−79.7964534011807x_{15} = -79.7964534011807
x16=−87.9645943619239x_{16} = -87.9645943619239
x17=84.1946831162065x_{17} = 84.1946831162065
x18=33.9292006587698x_{18} = 33.9292006587698
x19=−3.76991118430775x_{19} = -3.76991118430775
x20=10.0530964914873x_{20} = 10.0530964914873
x21=−45.867252742411x_{21} = -45.867252742411
x22=1.88495559215388x_{22} = 1.88495559215388
x23=45.867252742411x_{23} = 45.867252742411
x24=82.3097275240526x_{24} = 82.3097275240526
x25=−54.0353936417444x_{25} = -54.0353936417444
x26=−42.0973415581032x_{26} = -42.0973415581032
x27=−59.6902604536577x_{27} = -59.6902604536577
x28=4.39822971502571x_{28} = 4.39822971502571
x29=65.9734457509651x_{29} = 65.9734457509651
x30=6.28318528480448x_{30} = 6.28318528480448
x31=70.3716754404114x_{31} = 70.3716754404114
x32=−43.9822971753826x_{32} = -43.9822971753826
x33=−35.8141562509236x_{33} = -35.8141562509236
x34=−57.8053048260522x_{34} = -57.8053048260522
x35=−55.9203492338983x_{35} = -55.9203492338983
x36=80.1106126665397x_{36} = 80.1106126665397
x37=−37.6991118743642x_{37} = -37.6991118743642
x38=−17.2787595947439x_{38} = -17.2787595947439
x39=−64.0884901332318x_{39} = -64.0884901332318
x40=16.3362817986669x_{40} = 16.3362817986669
x41=76.026542216873x_{41} = 76.026542216873
x42=−51.8362787842316x_{42} = -51.8362787842316
x43=−29.845130209103x_{43} = -29.845130209103
x44=−20.1061929829747x_{44} = -20.1061929829747
x45=−15.7079632945754x_{45} = -15.7079632945754
x46=−25.7610597594363x_{46} = -25.7610597594363
x47=72.2566310277481x_{47} = 72.2566310277481
x48=−95.8185759344887x_{48} = -95.8185759344887
x49=30.159289474462x_{49} = 30.159289474462
x50=−81.6814090324996x_{50} = -81.6814090324996
x51=94.2477796093554x_{51} = 94.2477796093554
x52=−69.7433569096934x_{52} = -69.7433569096934
x53=55.9203492338983x_{53} = 55.9203492338983
x54=54.0353936417444x_{54} = 54.0353936417444
x55=−47.7522083345649x_{55} = -47.7522083345649
x56=18.2212373908208x_{56} = 18.2212373908208
x57=98.0176907920015x_{57} = 98.0176907920015
x58=43.9822971685429x_{58} = 43.9822971685429
x59=86.0796387083603x_{59} = 86.0796387083603
x60=26.3893782901543x_{60} = 26.3893782901543
x61=−38.9557489045134x_{61} = -38.9557489045134
x62=−1.88495559215388x_{62} = -1.88495559215388
x63=40.2123859659494x_{63} = 40.2123859659494
x64=60.318578948924x_{64} = 60.318578948924
x65=−76.026542216873x_{65} = -76.026542216873
x66=−11.9380520836412x_{66} = -11.9380520836412
x67=−83.2522053201295x_{67} = -83.2522053201295
x68=92.3628240155399x_{68} = 92.3628240155399
x69=87.9645943323798x_{69} = 87.9645943323798
x70=−98.0176907920015x_{70} = -98.0176907920015
x71=−29.5309709437441x_{71} = -29.5309709437441
x72=67.5442420521806x_{72} = 67.5442420521806
x73=−45.238934211693x_{73} = -45.238934211693
x74=−73.8274273593601x_{74} = -73.8274273593601
x75=64.0884901332318x_{75} = 64.0884901332318
x76=20.1061929829747x_{76} = 20.1061929829747
x77=−10.0530964914873x_{77} = -10.0530964914873
x78=−60.946897479642x_{78} = -60.946897479642
x79=−23.8761041672824x_{79} = -23.8761041672824
x80=21.9911485849594x_{80} = 21.9911485849594
x81=50.2654824465185x_{81} = 50.2654824465185
x82=−13.8230076757951x_{82} = -13.8230076757951
x83=−91.734505484822x_{83} = -91.734505484822
x84=−21.9911485866377x_{84} = -21.9911485866377
x85=38.3274303737955x_{85} = 38.3274303737955
x86=42.0973415581032x_{86} = 42.0973415581032
x87=−33.9292006587698x_{87} = -33.9292006587698
x88=32.0442450666159x_{88} = 32.0442450666159
x89=48.3805268652828x_{89} = 48.3805268652828
x90=28.2743338656397x_{90} = 28.2743338656397
x91=−32.0442450666159x_{91} = -32.0442450666159
x92=8.16814089933346x_{92} = 8.16814089933346
x93=−65.9734457668509x_{93} = -65.9734457668509
x94=−99.9026463841554x_{94} = -99.9026463841554
x95=−77.9114978090269x_{95} = -77.9114978090269
x96=52.1504380495906x_{96} = 52.1504380495906
x97=89.5353906273091x_{97} = 89.5353906273091
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(7*x) + cos(3*x))/x^2.
−cos⁡(0⋅7)+cos⁡(0⋅3)02\frac{- \cos{\left(0 \cdot 7 \right)} + \cos{\left(0 \cdot 3 \right)}}{0^{2}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
−3sin⁡(3x)+7sin⁡(7x)x2−2(cos⁡(3x)−cos⁡(7x))x3=0\frac{- 3 \sin{\left(3 x \right)} + 7 \sin{\left(7 x \right)}}{x^{2}} - \frac{2 \left(\cos{\left(3 x \right)} - \cos{\left(7 x \right)}\right)}{x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=73.9967329633305x_{1} = 73.9967329633305
x2=−73.9967329633305x_{2} = -73.9967329633305
x3=−89.7047470121367x_{3} = -89.7047470121367
x4=−5.88334097907696x_{4} = -5.88334097907696
x5=10.3386869452249x_{5} = 10.3386869452249
x6=−33.1556710687763x_{6} = -33.1556710687763
x7=39.9185887685627x_{7} = 39.9185887685627
x8=−26.0505741737963x_{8} = -26.0505741737963
x9=−43.9822971502571x_{9} = -43.9822971502571
x10=28.2743338823081x_{10} = 28.2743338823081
x11=12.1711783051505x_{11} = 12.1711783051505
x12=13.4817945658253x_{12} = 13.4817945658253
x13=24.2095258592079x_{13} = 24.2095258592079
x14=−23.7306357344751x_{14} = -23.7306357344751
x15=−31.8050183397407x_{15} = -31.8050183397407
x16=−52.0054617321196x_{16} = -52.0054617321196
x17=−15.707963267949x_{17} = -15.707963267949
x18=−37.6991118430775x_{18} = -37.6991118430775
x19=−39.9185887685627x_{19} = -39.9185887685627
x20=−85.7426369606128x_{20} = -85.7426369606128
x21=20.2496992007472x_{21} = 20.2496992007472
x22=−70.0344971468579x_{22} = -70.0344971468579
x23=−102.751477358685x_{23} = -102.751477358685
x24=−4.04546347923779x_{24} = -4.04546347923779
x25=17.9253602624254x_{25} = 17.9253602624254
x26=65.9734457253857x_{26} = 65.9734457253857
x27=−11.1632443424398x_{27} = -11.1632443424398
x28=76.3177617903111x_{28} = 76.3177617903111
x29=21.9911485751286x_{29} = 21.9911485751286
x30=−93.85641703355x_{30} = -93.85641703355
x31=−65.9734457253857x_{31} = -65.9734457253857
x32=30.0140105999544x_{32} = 30.0140105999544
x33=−75.7883011951918x_{33} = -75.7883011951918
x34=70.0344971468579x_{34} = 70.0344971468579
x35=60.0801517336434x_{35} = 60.0801517336434
x36=68.1936248680213x_{36} = 68.1936248680213
x37=61.910338966879x_{37} = 61.910338966879
x38=−57.9494986979589x_{38} = -57.9494986979589
x39=64.2327201708699x_{39} = 64.2327201708699
x40=−60.0801517336434x_{40} = -60.0801517336434
x41=−41.7595109966085x_{41} = -41.7595109966085
x42=−79.9407489893177x_{42} = -79.9407489893177
x43=37.6991118430775x_{43} = 37.6991118430775
x44=−21.9911485751286x_{44} = -21.9911485751286
x45=−53.7968614690947x_{45} = -53.7968614690947
x46=−48.0429075650649x_{46} = -48.0429075650649
x47=−45.7222196961036x_{47} = -45.7222196961036
x48=34.1651562272963x_{48} = 34.1651562272963
x49=−1.72785268115637x_{49} = -1.72785268115637
x50=−27.8816169932295x_{50} = -27.8816169932295
x51=−35.9581240877896x_{51} = -35.9581240877896
x52=−61.0911103639023x_{52} = -61.0911103639023
x53=26.0505741737963x_{53} = 26.0505741737963
x54=90.18501510683x_{54} = 90.18501510683
x55=100.139638310928x_{55} = 100.139638310928
x56=−61.910338966879x_{56} = -61.910338966879
x57=8.02089510578949x_{57} = 8.02089510578949
x58=−87.9645943005142x_{58} = -87.9645943005142
x59=54.326255289094x_{59} = 54.326255289094
x60=95.9879479767112x_{60} = 95.9879479767112
x61=−9.81004212050563x_{61} = -9.81004212050563
x62=−8.02089510578949x_{62} = -8.02089510578949
x63=16.0953925366953x_{63} = 16.0953925366953
x64=6.28318530717959x_{64} = 6.28318530717959
x65=−63.7512168713818x_{65} = -63.7512168713818
x66=−97.7796098754723x_{66} = -97.7796098754723
x67=−71.8650930729889x_{67} = -71.8650930729889
x68=52.0054617321196x_{68} = 52.0054617321196
x69=86.2239538314439x_{69} = 86.2239538314439
x70=−49.8736146343732x_{70} = -49.8736146343732
x71=80.2799409683107x_{71} = 80.2799409683107
x72=−13.9660390823958x_{72} = -13.9660390823958
x73=87.9645943005142x_{73} = 87.9645943005142
x74=43.9822971502571x_{74} = 43.9822971502571
x75=32.3342623220753x_{75} = 32.3342623220753
x76=−67.7135207443646x_{76} = -67.7135207443646
x77=38.0884840641708x_{77} = 38.0884840641708
x78=−19.7665667687004x_{78} = -19.7665667687004
x79=92.0258760154351x_{79} = 92.0258760154351
x80=78.1483385546479x_{80} = 78.1483385546479
x81=98.3091082017841x_{81} = 98.3091082017841
x82=−95.9879479767112x_{82} = -95.9879479767112
x83=57.9494986979589x_{83} = 57.9494986979589
x84=−83.9017736838728x_{84} = -83.9017736838728
x85=72.2566310325652x_{85} = 72.2566310325652
x86=46.2020044431786x_{86} = 46.2020044431786
x87=94.2477796076938x_{87} = 94.2477796076938
x88=−30.0140105999544x_{88} = -30.0140105999544
x89=48.0429075650649x_{89} = 48.0429075650649
x90=82.0715410084561x_{90} = 82.0715410084561
x91=−17.9253602624254x_{91} = -17.9253602624254
x92=42.2413979761308x_{92} = 42.2413979761308
x93=2.19017794396668x_{93} = 2.19017794396668
x94=56.1569205219136x_{94} = 56.1569205219136
x95=50.2654824574367x_{95} = 50.2654824574367
x96=4.04546347923779x_{96} = 4.04546347923779
x97=−92.0258760154351x_{97} = -92.0258760154351
Signos de extremos en los puntos:
(73.99673296333052, 8.03930288521132e-5)

(-73.99673296333052, 8.03930288521132e-5)

(-89.70474701213674, -5.4703255753612e-5)

(-5.883340979076965, 0.037691714404883)

(10.338686945224943, 0.0179121518053217)

(-33.15567106877626, -0.00040042488372479)

(39.91858876856266, 0.0012022176368869)

(-26.05057417379634, -0.00282276105607107)

(-43.982297150257104, 0)

(28.274333882308138, 0)

(12.171178305150516, 0.0088173424068626)

(13.481794565825336, -0.0105364802725626)

(24.20952585920788, -0.00326835479328241)

(-23.730635734475104, 0.000781645578399185)

(-31.80501833974069, 0.00129165137305129)

(-52.005461732119564, -0.000162758554818959)

(-15.707963267948966, 0)

(-37.69911184307752, 0)

(-39.91858876856266, 0.0012022176368869)

(-85.74263696061283, 0.00026058788237464)

(20.249699200747163, 0.00107345875201028)

(-70.03449714685794, -0.000390590674268843)

(-102.75147735868458, 0.00018145667453231)

(-4.045463479237794, 0.116585382063563)

(17.925360262425393, -0.00596107574909238)

(65.97344572538566, 0)

(-11.163244342439842, 0.00353174515763499)

(76.31776179031108, -0.000328924022378928)

(21.991148575128552, 0)

(-93.85641703354997, 0.000148330070689398)

(-65.97344572538566, 0)

(30.01401059995444, 0.000488637473970616)

(-75.78830119519185, 0.000227484272732193)

(70.03449714685794, -0.000390590674268843)

(60.08015173364343, -0.000361985577253256)

(68.1936248680213, -0.000411962811520212)

(61.910338966879, -0.000499824863277712)

(-57.949498697958894, 0.000131081996001023)

(64.23272017086985, 0.000106691617603996)

(-60.08015173364343, -0.000361985577253256)

(-41.75951099660849, 0.00109856121576167)

(-79.94074898931768, -6.88822463246151e-5)

(37.69911184307752, 0)

(-21.991148575128552, 0)

(-53.79686146909471, -0.000451479279972487)

(-48.04290756506492, 0.000830004815147386)

(-45.72221969610364, -0.000210564965458014)

(34.1651562272963, -0.00111936824241238)

(-1.7278526811563655, -0.146376799874948)

(-27.88161699322951, -0.00168071423145777)

(-35.95812408778957, -0.000340442628605085)

(-61.09111036390234, -0.000117946933113421)

(26.05057417379634, -0.00282276105607107)

(90.18501510683002, 0.00023554804493505)

(100.13963831092772, 0.000130300304685561)

(-61.910338966879, -0.000499824863277712)

(8.020895105789487, -0.00683996916068079)

(-87.96459430051421, 0)

(54.32625528909404, 0.000649116134917602)

(95.98794797671118, -4.77761069264751e-5)

(-9.810042120505633, -0.0135697858716332)

(-8.020895105789487, -0.00683996916068079)

(16.095392536695318, -0.00504273203450722)

(6.283185307179586, 0)

(-63.751216871381786, -0.000471376259278609)

(-97.77960987547232, -0.000136666083829747)

(-71.86509307298887, -0.000252999282516496)

(52.005461732119564, -0.000162758554818959)

(86.22395383144388, -5.92090461286597e-5)

(-49.87361463437316, 0.000525301593330343)

(80.27994096831073, 6.83014068895433e-5)

(-13.96603908239582, 0.00225658169391437)

(87.96459430051421, 0)

(43.982297150257104, 0)

(32.33426232207532, -0.00183230572664874)

(-67.71352074436457, 9.60046704253271e-5)

(38.08848406417079, 0.00090065043125887)

(-19.766566768700443, -0.00490246455191702)

(92.02587601543506, 0.000226218695483382)

(78.14833855464792, -0.000213952060694765)

(98.30910820178411, 0.000198226325057273)

(-95.98794797671118, -4.77761069264751e-5)

(57.949498697958894, 0.000131081996001023)

(-83.90177368387278, 0.000272148173617132)

(72.25663103256524, 0)

(46.20200444317864, 0.000897462870855063)

(94.2477796076938, 0)

(-30.01401059995444, 0.000488637473970616)

(48.04290756506492, 0.000830004815147386)

(82.07154100845611, 0.000193986338777878)

(-17.925360262425393, -0.00596107574909238)

(42.24139797613083, -0.000246696736221661)

(2.190177943966682, 0.393771794426958)

(56.15692052191362, 0.000414329468010238)

(50.26548245743669, 0)

(4.045463479237794, 0.116585382063563)

(-92.02587601543506, 0.000226218695483382)


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:
x1=−89.7047470121367x_{1} = -89.7047470121367
x2=−33.1556710687763x_{2} = -33.1556710687763
x3=−26.0505741737963x_{3} = -26.0505741737963
x4=−43.9822971502571x_{4} = -43.9822971502571
x5=13.4817945658253x_{5} = 13.4817945658253
x6=24.2095258592079x_{6} = 24.2095258592079
x7=−52.0054617321196x_{7} = -52.0054617321196
x8=−37.6991118430775x_{8} = -37.6991118430775
x9=−70.0344971468579x_{9} = -70.0344971468579
x10=17.9253602624254x_{10} = 17.9253602624254
x11=76.3177617903111x_{11} = 76.3177617903111
x12=70.0344971468579x_{12} = 70.0344971468579
x13=60.0801517336434x_{13} = 60.0801517336434
x14=68.1936248680213x_{14} = 68.1936248680213
x15=61.910338966879x_{15} = 61.910338966879
x16=−60.0801517336434x_{16} = -60.0801517336434
x17=−79.9407489893177x_{17} = -79.9407489893177
x18=37.6991118430775x_{18} = 37.6991118430775
x19=−53.7968614690947x_{19} = -53.7968614690947
x20=−45.7222196961036x_{20} = -45.7222196961036
x21=34.1651562272963x_{21} = 34.1651562272963
x22=−1.72785268115637x_{22} = -1.72785268115637
x23=−27.8816169932295x_{23} = -27.8816169932295
x24=−35.9581240877896x_{24} = -35.9581240877896
x25=−61.0911103639023x_{25} = -61.0911103639023
x26=26.0505741737963x_{26} = 26.0505741737963
x27=−61.910338966879x_{27} = -61.910338966879
x28=8.02089510578949x_{28} = 8.02089510578949
x29=−87.9645943005142x_{29} = -87.9645943005142
x30=95.9879479767112x_{30} = 95.9879479767112
x31=−9.81004212050563x_{31} = -9.81004212050563
x32=−8.02089510578949x_{32} = -8.02089510578949
x33=16.0953925366953x_{33} = 16.0953925366953
x34=6.28318530717959x_{34} = 6.28318530717959
x35=−63.7512168713818x_{35} = -63.7512168713818
x36=−97.7796098754723x_{36} = -97.7796098754723
x37=−71.8650930729889x_{37} = -71.8650930729889
x38=52.0054617321196x_{38} = 52.0054617321196
x39=86.2239538314439x_{39} = 86.2239538314439
x40=87.9645943005142x_{40} = 87.9645943005142
x41=43.9822971502571x_{41} = 43.9822971502571
x42=32.3342623220753x_{42} = 32.3342623220753
x43=−19.7665667687004x_{43} = -19.7665667687004
x44=78.1483385546479x_{44} = 78.1483385546479
x45=−95.9879479767112x_{45} = -95.9879479767112
x46=94.2477796076938x_{46} = 94.2477796076938
x47=−17.9253602624254x_{47} = -17.9253602624254
x48=42.2413979761308x_{48} = 42.2413979761308
x49=50.2654824574367x_{49} = 50.2654824574367
Puntos máximos de la función:
x49=73.9967329633305x_{49} = 73.9967329633305
x49=−73.9967329633305x_{49} = -73.9967329633305
x49=−5.88334097907696x_{49} = -5.88334097907696
x49=10.3386869452249x_{49} = 10.3386869452249
x49=39.9185887685627x_{49} = 39.9185887685627
x49=28.2743338823081x_{49} = 28.2743338823081
x49=12.1711783051505x_{49} = 12.1711783051505
x49=−23.7306357344751x_{49} = -23.7306357344751
x49=−31.8050183397407x_{49} = -31.8050183397407
x49=−15.707963267949x_{49} = -15.707963267949
x49=−39.9185887685627x_{49} = -39.9185887685627
x49=−85.7426369606128x_{49} = -85.7426369606128
x49=20.2496992007472x_{49} = 20.2496992007472
x49=−102.751477358685x_{49} = -102.751477358685
x49=−4.04546347923779x_{49} = -4.04546347923779
x49=65.9734457253857x_{49} = 65.9734457253857
x49=−11.1632443424398x_{49} = -11.1632443424398
x49=21.9911485751286x_{49} = 21.9911485751286
x49=−93.85641703355x_{49} = -93.85641703355
x49=−65.9734457253857x_{49} = -65.9734457253857
x49=30.0140105999544x_{49} = 30.0140105999544
x49=−75.7883011951918x_{49} = -75.7883011951918
x49=−57.9494986979589x_{49} = -57.9494986979589
x49=64.2327201708699x_{49} = 64.2327201708699
x49=−41.7595109966085x_{49} = -41.7595109966085
x49=−21.9911485751286x_{49} = -21.9911485751286
x49=−48.0429075650649x_{49} = -48.0429075650649
x49=90.18501510683x_{49} = 90.18501510683
x49=100.139638310928x_{49} = 100.139638310928
x49=54.326255289094x_{49} = 54.326255289094
x49=−49.8736146343732x_{49} = -49.8736146343732
x49=80.2799409683107x_{49} = 80.2799409683107
x49=−13.9660390823958x_{49} = -13.9660390823958
x49=−67.7135207443646x_{49} = -67.7135207443646
x49=38.0884840641708x_{49} = 38.0884840641708
x49=92.0258760154351x_{49} = 92.0258760154351
x49=98.3091082017841x_{49} = 98.3091082017841
x49=57.9494986979589x_{49} = 57.9494986979589
x49=−83.9017736838728x_{49} = -83.9017736838728
x49=72.2566310325652x_{49} = 72.2566310325652
x49=46.2020044431786x_{49} = 46.2020044431786
x49=−30.0140105999544x_{49} = -30.0140105999544
x49=48.0429075650649x_{49} = 48.0429075650649
x49=82.0715410084561x_{49} = 82.0715410084561
x49=2.19017794396668x_{49} = 2.19017794396668
x49=56.1569205219136x_{49} = 56.1569205219136
x49=4.04546347923779x_{49} = 4.04546347923779
x49=−92.0258760154351x_{49} = -92.0258760154351
Decrece en los intervalos
[95.9879479767112,∞)\left[95.9879479767112, \infty\right)
Crece en los intervalos
(−∞,−97.7796098754723]\left(-\infty, -97.7796098754723\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(cos⁡(3x)−cos⁡(7x)x2)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(3 x \right)} - \cos{\left(7 x \right)}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(cos⁡(3x)−cos⁡(7x)x2)=0\lim_{x \to \infty}\left(\frac{\cos{\left(3 x \right)} - \cos{\left(7 x \right)}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-cos(7*x) + cos(3*x))/x^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡(3x)−cos⁡(7x)xx2)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(3 x \right)} - \cos{\left(7 x \right)}}{x x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(cos⁡(3x)−cos⁡(7x)xx2)=0\lim_{x \to \infty}\left(\frac{\cos{\left(3 x \right)} - \cos{\left(7 x \right)}}{x x^{2}}\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(-x) и f = -f(-x).
Pues, comprobamos:
cos⁡(3x)−cos⁡(7x)x2=cos⁡(3x)−cos⁡(7x)x2\frac{\cos{\left(3 x \right)} - \cos{\left(7 x \right)}}{x^{2}} = \frac{\cos{\left(3 x \right)} - \cos{\left(7 x \right)}}{x^{2}}
- Sí
cos⁡(3x)−cos⁡(7x)x2=−cos⁡(3x)−cos⁡(7x)x2\frac{\cos{\left(3 x \right)} - \cos{\left(7 x \right)}}{x^{2}} = - \frac{\cos{\left(3 x \right)} - \cos{\left(7 x \right)}}{x^{2}}
- No
es decir, función
es
par
    © Sr Examen – Калькулятор