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Trazado el gráfico de la función/ sin(x*35/2)/((x*35/2))

Gráfico de la función y = sin(x*35/2)/((x*35/2))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          /x*35\
       sin|----|
          \ 2  /
f(x) = ---------
         /x*35\ 
         |----| 
         \ 2  /
f(x)=sin⁡(35x2)352xf{\left(x \right)} = \frac{\sin{\left(\frac{35 x}{2} \right)}}{\frac{35}{2} x} 
f = sin(35*x/2)/((35*x/2))
Gráfico de la función
02468-8-6-4-2-10101-1
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:
sin⁡(35x2)352x=0\frac{\sin{\left(\frac{35 x}{2} \right)}}{\frac{35}{2} x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=2π35x_{1} = \frac{2 \pi}{35}
Solución numérica
x1=72.1668712424627x_{1} = 72.1668712424627
x2=24.2351433276927x_{2} = 24.2351433276927
x3=4.12895034471801x_{3} = 4.12895034471801
x4=68.7559992185652x_{4} = 68.7559992185652
x5=−61.7547355905651x_{5} = -61.7547355905651
x6=−32.6725635973338x_{6} = -32.6725635973338
x7=52.2401978396931x_{7} = 52.2401978396931
x8=38.2376705836929x_{8} = 38.2376705836929
x9=−77.7319782288217x_{9} = -77.7319782288217
x10=92.2730642254374x_{10} = 92.2730642254374
x11=8.25790068943603x_{11} = 8.25790068943603
x12=−47.7522083345649x_{12} = -47.7522083345649
x13=−39.8533468055391x_{13} = -39.8533468055391
x14=−94.2477796076938x_{14} = -94.2477796076938
x15=831.534695510167x_{15} = 831.534695510167
x16=−11.8482922935386x_{16} = -11.8482922935386
x17=−14.0025272560002x_{17} = -14.0025272560002
x18=50.2654824574367x_{18} = 50.2654824574367
x19=−17.5929188601028x_{19} = -17.5929188601028
x20=−35.7243964608211x_{20} = -35.7243964608211
x21=56.7281873448214x_{21} = 56.7281873448214
x22=−19.7471538225644x_{22} = -19.7471538225644
x23=−60.6776181093343x_{23} = -60.6776181093343
x24=30.8773677952825x_{24} = 30.8773677952825
x25=22.2604279454362x_{25} = 22.2604279454362
x26=86.7079572390783x_{26} = 86.7079572390783
x27=−21.7218692048209x_{27} = -21.7218692048209
x28=−49.7269237168213x_{28} = -49.7269237168213
x29=94.2477796076938x_{29} = 94.2477796076938
x30=−91.734505484822x_{30} = -91.734505484822
x31=60.1390593687189x_{31} = 60.1390593687189
x32=−44.8798950512828x_{32} = -44.8798950512828
x33=67.8584013175395x_{33} = 67.8584013175395
x34=30.159289474462x_{34} = 30.159289474462
x35=−99.0948082732323x_{35} = -99.0948082732323
x36=46.1365321127187x_{36} = 46.1365321127187
x37=66.2427250956934x_{37} = 66.2427250956934
x38=−75.7572628465653x_{38} = -75.7572628465653
x39=−42.1871013482058x_{39} = -42.1871013482058
x40=−53.8558740615393x_{40} = -53.8558740615393
x41=37.6991118430775x_{41} = 37.6991118430775
x42=−23.8761041672824x_{42} = -23.8761041672824
x43=−5.74462656656419x_{43} = -5.74462656656419
x44=2.15423496246157x_{44} = 2.15423496246157
x45=−7.71934194882063x_{45} = -7.71934194882063
x46=−37.8786314232826x_{46} = -37.8786314232826
x47=−9.87357691128221x_{47} = -9.87357691128221
x48=58.1643439864625x_{48} = 58.1643439864625
x49=71.4487929216422x_{49} = 71.4487929216422
x50=16.1567622184618x_{50} = 16.1567622184618
x51=−25.8508195495389x_{51} = -25.8508195495389
x52=28.0050545120004x_{52} = 28.0050545120004
x53=−33.7496810785646x_{53} = -33.7496810785646
x54=44.7003754710776x_{54} = 44.7003754710776
x55=36.2629552014365x_{55} = 36.2629552014365
x56=90.1188292629758x_{56} = 90.1188292629758
x57=47.2136495939495x_{57} = 47.2136495939495
x58=14.0025272560002x_{58} = 14.0025272560002
x59=−28.0050545120004x_{59} = -28.0050545120004
x60=11.6687727133335x_{60} = 11.6687727133335
x61=−2.15423496246157x_{61} = -2.15423496246157
x62=−65.8836859352831x_{62} = -65.8836859352831
x63=78.2705369694371x_{63} = 78.2705369694371
x64=97.8381712117964x_{64} = 97.8381712117964
x65=−89.7597901025655x_{65} = -89.7597901025655
x66=80.2452523516936x_{66} = 80.2452523516936
x67=76.1163020069756x_{67} = 76.1163020069756
x68=−95.86345582954x_{68} = -95.86345582954
x69=−81.8609285735398x_{69} = -81.8609285735398
x70=32.1340048567185x_{70} = 32.1340048567185
x71=−97.8381712117964x_{71} = -97.8381712117964
x72=−44.1618167304622x_{72} = -44.1618167304622
x73=88.1441138807193x_{73} = 88.1441138807193
x74=−63.7294509728215x_{74} = -63.7294509728215
x75=−69.833116699796x_{75} = -69.833116699796
x76=6.28318530717959x_{76} = 6.28318530717959
x77=−3.59039160410262x_{77} = -3.59039160410262
x78=64.2680097134369x_{78} = 64.2680097134369
x79=74.1415866247191x_{79} = 74.1415866247191
x80=−47.3931691741546x_{80} = -47.3931691741546
x81=29.4412111536417x_{81} = 29.4412111536417
x82=96.7610537305656x_{82} = 96.7610537305656
x83=−15.9772426382567x_{83} = -15.9772426382567
x84=−29.9797698942569x_{84} = -29.9797698942569
x85=−83.8356439557962x_{85} = -83.8356439557962
x86=89.041711781745x_{86} = 89.041711781745
x87=28.5436132526158x_{87} = 28.5436132526158
x88=−51.8811586792829x_{88} = -51.8811586792829
x89=99.8128865940529x_{89} = 99.8128865940529
x90=−67.8584013175395x_{90} = -67.8584013175395
x91=18.1314776007182x_{91} = 18.1314776007182
x92=44.1618167304622x_{92} = 44.1618167304622
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 sin(x*35/2)/((x*35/2)).
sin⁡(0⋅352)0352\frac{\sin{\left(\frac{0 \cdot 35}{2} \right)}}{0 \frac{35}{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
35235xcos⁡(35x2)2−2sin⁡(35x2)35x2=0\frac{35 \frac{2}{35 x} \cos{\left(\frac{35 x}{2} \right)}}{2} - \frac{2 \sin{\left(\frac{35 x}{2} \right)}}{35 x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=28.2742183954136x_{1} = 28.2742183954136
x2=14.2715778296947x_{2} = 14.2715778296947
x3=−67.5890736360897x_{3} = -67.5890736360897
x4=66.1529159456304x_{4} = 66.1529159456304
x5=−31.8646230121834x_{5} = -31.8646230121834
x6=46.4057411187914x_{6} = 46.4057411187914
x7=−25.7609330054571x_{7} = -25.7609330054571
x8=38.1478251975265x_{8} = 38.1478251975265
x9=78.0012157368904x_{9} = 78.0012157368904
x10=60.4082846850973x_{10} = 60.4082846850973
x11=−93.9784654921314x_{11} = -93.9784654921314
x12=86.2591204339522x_{12} = 86.2591204339522
x13=−53.766053539717x_{13} = -53.766053539717
x14=−32.7622237207337x_{14} = -32.7622237207337
x15=12.1173021911646x_{15} = 12.1173021911646
x16=−38.8659050998032x_{16} = -38.8659050998032
x17=27.7356574123217x_{17} = 27.7356574123217
x18=63.9986793216818x_{18} = 63.9986793216818
x19=84.463923827339x_{19} = 84.463923827339
x20=94.1579851385811x_{20} = 94.1579851385811
x21=−45.8671815519798x_{21} = -45.8671815519798
x22=−59.8697254582391x_{22} = -59.8697254582391
x23=8.16774112505082x_{23} = 8.16774112505082
x24=−73.8722630523613x_{24} = -73.8722630523613
x25=−43.1743834087411x_{25} = -43.1743834087411
x26=−27.7356574123217x_{26} = -27.7356574123217
x27=−55.7407710735029x_{27} = -55.7407710735029
x28=−23.6066864760059x_{28} = -23.6066864760059
x29=42.2767839019665x_{29} = 42.2767839019665
x30=35.9935851120918x_{30} = 35.9935851120918
x31=89.1314349371218x_{31} = 89.1314349371218
x32=5.11566976745241x_{32} = 5.11566976745241
x33=−489.280609175397x_{33} = -489.280609175397
x34=−81.591609183117x_{34} = -81.591609183117
x35=33.8393443743917x_{35} = 33.8393443743917
x36=−79.2578534620542x_{36} = -79.2578534620542
x37=−17.8620154237728x_{37} = -17.8620154237728
x38=100.261652976723x_{38} = 100.261652976723
x39=70.2818691901699x_{39} = 70.2818691901699
x40=54.1250931029917x_{40} = 54.1250931029917
x41=−61.8444425819835x_{41} = -61.8444425819835
x42=49.9961377759901x_{42} = 49.9961377759901
x43=44.251502730883x_{43} = 44.251502730883
x44=−19.8367490041875x_{44} = -19.8367490041875
x45=68.1276327586117x_{45} = 68.1276327586117
x46=110.135232807699x_{46} = 110.135232807699
x47=67.0505145074326x_{47} = 67.0505145074326
x48=40.1225447925763x_{48} = 40.1225447925763
x49=26.1199739082013x_{49} = 26.1199739082013
x50=−97.7483780164771x_{50} = -97.7483780164771
x51=−87.8747973517978x_{51} = -87.8747973517978
x52=2.24253899206653x_{52} = 2.24253899206653
x53=−89.8495135507275x_{53} = -89.8495135507275
x54=88.2338366634131x_{54} = 88.2338366634131
x55=80.1554518244297x_{55} = 80.1554518244297
x56=−47.8418998726837x_{56} = -47.8418998726837
x57=48.021419708037x_{57} = 48.021419708037
x58=58.2540477237206x_{58} = 58.2540477237206
x59=−32.9417438440838x_{59} = -32.9417438440838
x60=24.1452483018706x_{60} = 24.1452483018706
x61=−50.1756575898669x_{61} = -50.1756575898669
x62=91.8242297145315x_{62} = 91.8242297145315
x63=10.1425343431607x_{63} = 10.1425343431607
x64=89.490474244512x_{64} = 89.490474244512
x65=−13.7330101164599x_{65} = -13.7330101164599
x66=−83.7458451750355x_{66} = -83.7458451750355
x67=72.2565858421455x_{67} = 72.2565858421455
x68=56.279330374695x_{68} = 56.279330374695
x69=−1.883222230847x_{69} = -1.883222230847
x70=−11.2196827316064x_{70} = -11.2196827316064
x71=7.44962426887934x_{71} = 7.44962426887934
x72=16.2463210219747x_{72} = 16.2463210219747
x73=−78.0012157368904x_{73} = -78.0012157368904
x74=74.4108221128953x_{74} = 74.4108221128953
x75=82.1301681861595x_{75} = 82.1301681861595
x76=96.1327012331985x_{76} = 96.1327012331985
x77=−69.7433100907887x_{77} = -69.7433100907887
x78=−92.0037493641228x_{78} = -92.0037493641228
x79=−75.8469795854413x_{79} = -75.8469795854413
x80=51.6118160423622x_{80} = 51.6118160423622
x81=−52.68893481695x_{81} = -52.68893481695
x82=−85.7205614555066x_{82} = -85.7205614555066
x83=−95.5941423012216x_{83} = -95.5941423012216
x84=−70.8204282840947x_{84} = -70.8204282840947
x85=−63.9986793216818x_{85} = -63.9986793216818
x86=−39.7635048973264x_{86} = -39.7635048973264
x87=−5.83382665513758x_{87} = -5.83382665513758
x88=30.248941316911x_{88} = 30.248941316911
x89=−99.7230940602233x_{89} = -99.7230940602233
x90=−28.6332590039432x_{90} = -28.6332590039432
x91=−71.7180267621775x_{91} = -71.7180267621775
x92=−33.8393443743917x_{92} = -33.8393443743917
x93=52.1503754363344x_{93} = 52.1503754363344
x94=−11.7582548023256x_{94} = -11.7582548023256
x95=−41.7382241647507x_{95} = -41.7382241647507
x96=−3.85882484440824x_{96} = -3.85882484440824
Signos de extremos en los puntos:
(28.274218395413644, -0.00202101927781199)

(14.271577829694671, -0.00400393003361788)

(-67.58907363608968, 0.000845444886969719)

(66.15291594563041, 0.000863799199286591)

(-31.864623012183436, -0.00179329801697543)

(46.405741118791404, 0.00123137380080274)

(-25.760933005457083, -0.00221819281732969)

(38.14782519752653, 0.0014979305566827)

(78.0012157368904, 0.000732589117606799)

(60.408284685097314, 0.000945943621388202)

(-93.97846549213139, -0.000608041919819978)

(86.2591204339522, 0.000662455683722263)

(-53.76605353971705, -0.0010628048946843)

(-32.762223720733715, 0.00174416641288651)

(12.117302191164578, -0.00471575445232734)

(-38.86590509980318, 0.00147025510494756)

(27.73565741232171, 0.00206026253557351)

(63.998679321681756, 0.000892875211967127)

(84.46392382733902, 0.000676535513345502)

(94.15798513858108, 0.000606882640231837)

(-45.86718155197977, -0.0012458322239056)

(-59.86972545823909, -0.000954452867078907)

(8.167741125050815, -0.00699599287652265)

(-73.87226305236135, -0.000773535799308312)

(-43.17438340874111, 0.001323534989533)

(-27.73565741232171, 0.00206026253557351)

(-55.74077107350286, 0.00102515315119465)

(-23.606686476005873, -0.0024206145911235)

(42.2767839019665, -0.00135163557090012)

(35.993585112091786, 0.00158758248040339)

(89.13143493712181, 0.000641107656796582)

(5.1156697674524105, 0.0111694646341736)

(-489.2806091753967, -0.000116789538930336)

(-81.59160918311699, 0.000700351956541601)

(33.8393443743917, 0.00168864901867305)

(-79.25785346205424, -0.000720973882022797)

(-17.862015423772792, -0.00319911070377983)

(100.26165297672273, 0.000569937220916461)

(70.28186919016993, -0.000813052340724969)

(54.12509310299172, -0.00105575477141225)

(-61.84444258198345, 0.000923976841972888)

(49.99613777599006, 0.00114294468255477)

(44.25150273088296, 0.00129131907332853)

(-19.83674900418755, 0.00288064440606586)

(68.1276327586117, -0.000838761523458827)

(110.13523280769856, -0.000518842590102568)

(67.05051450743261, -0.000852235613865903)

(40.12254479257632, -0.00142420675171428)

(26.119973908201278, -0.00218770204746432)

(-97.74837801647715, 0.000584591259090004)

(-87.8747973517978, -0.00065027569659614)

(2.2425389920665255, 0.0254730530928808)

(-89.84951355072751, 0.000635983917198951)

(88.23383666341307, -0.00064762960923136)

(80.15545182442972, 0.00071290026219619)

(-47.84189987268373, 0.00119440943053084)

(48.02141970803695, -0.00118994434221327)

(58.25404772372058, 0.000980924620415334)

(-32.941743844083824, -0.00173466139013141)

(24.14524830187064, 0.00236662288175273)

(-50.17565758986693, -0.00113885542971897)

(91.82422971453147, -0.000622306838355932)

(10.142534343160731, 0.00563389270608742)

(89.49047424451199, 0.000638535508677342)

(-13.733010116459852, 0.00416094956492912)

(-83.74584517503554, 0.000682336463630519)

(72.25658584214553, 0.000790832262662835)

(56.279330374695014, -0.0010153430630305)

(-1.8832222308469986, 0.0303291711863103)

(-11.2196827316064, 0.00509302423193926)

(7.4496242688793375, -0.00767034337297047)

(16.24632102197467, 0.00351725806749418)

(-78.0012157368904, 0.000732589117606799)

(74.41082211289532, 0.000767937225673957)

(82.13016818615954, -0.000695759482464959)

(96.13270123319849, -0.000594416325710795)

(-69.74331009078867, 0.000819330741377003)

(-92.00374936412277, 0.000621092580641494)

(-75.84697958544128, 0.000753396394132818)

(51.61181604236224, -0.00110716550009878)

(-52.68893481695003, -0.0010845317662123)

(-85.72056145550664, -0.000666617710804053)

(-95.5941423012216, 0.000597765151274257)

(-70.82042828409467, 0.000806869429150205)

(-63.998679321681756, 0.000892875211967127)

(-39.763504897326385, -0.00143706643279575)

(-5.83382665513758, 0.00979462014674114)

(30.248941316911008, 0.0018890828139447)

(-99.72309406022332, -0.000573015190714214)

(-28.633259003943177, -0.00199567724172285)

(-71.7180267621775, -0.00079677093172051)

(-33.8393443743917, 0.00168864901867305)

(52.150375436334414, 0.00109573176340691)

(-11.758254802325558, -0.0048597503050537)

(-41.738224164750676, 0.00136907606236116)

(-3.858824844408235, -0.0148067339465492)


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=28.2742183954136x_{1} = 28.2742183954136
x2=14.2715778296947x_{2} = 14.2715778296947
x3=−31.8646230121834x_{3} = -31.8646230121834
x4=−25.7609330054571x_{4} = -25.7609330054571
x5=−93.9784654921314x_{5} = -93.9784654921314
x6=−53.766053539717x_{6} = -53.766053539717
x7=12.1173021911646x_{7} = 12.1173021911646
x8=−45.8671815519798x_{8} = -45.8671815519798
x9=−59.8697254582391x_{9} = -59.8697254582391
x10=8.16774112505082x_{10} = 8.16774112505082
x11=−73.8722630523613x_{11} = -73.8722630523613
x12=−23.6066864760059x_{12} = -23.6066864760059
x13=42.2767839019665x_{13} = 42.2767839019665
x14=−489.280609175397x_{14} = -489.280609175397
x15=−79.2578534620542x_{15} = -79.2578534620542
x16=−17.8620154237728x_{16} = -17.8620154237728
x17=70.2818691901699x_{17} = 70.2818691901699
x18=54.1250931029917x_{18} = 54.1250931029917
x19=68.1276327586117x_{19} = 68.1276327586117
x20=110.135232807699x_{20} = 110.135232807699
x21=67.0505145074326x_{21} = 67.0505145074326
x22=40.1225447925763x_{22} = 40.1225447925763
x23=26.1199739082013x_{23} = 26.1199739082013
x24=−87.8747973517978x_{24} = -87.8747973517978
x25=88.2338366634131x_{25} = 88.2338366634131
x26=48.021419708037x_{26} = 48.021419708037
x27=−32.9417438440838x_{27} = -32.9417438440838
x28=−50.1756575898669x_{28} = -50.1756575898669
x29=91.8242297145315x_{29} = 91.8242297145315
x30=56.279330374695x_{30} = 56.279330374695
x31=7.44962426887934x_{31} = 7.44962426887934
x32=82.1301681861595x_{32} = 82.1301681861595
x33=96.1327012331985x_{33} = 96.1327012331985
x34=51.6118160423622x_{34} = 51.6118160423622
x35=−52.68893481695x_{35} = -52.68893481695
x36=−85.7205614555066x_{36} = -85.7205614555066
x37=−39.7635048973264x_{37} = -39.7635048973264
x38=−99.7230940602233x_{38} = -99.7230940602233
x39=−28.6332590039432x_{39} = -28.6332590039432
x40=−71.7180267621775x_{40} = -71.7180267621775
x41=−11.7582548023256x_{41} = -11.7582548023256
x42=−3.85882484440824x_{42} = -3.85882484440824
Puntos máximos de la función:
x42=−67.5890736360897x_{42} = -67.5890736360897
x42=66.1529159456304x_{42} = 66.1529159456304
x42=46.4057411187914x_{42} = 46.4057411187914
x42=38.1478251975265x_{42} = 38.1478251975265
x42=78.0012157368904x_{42} = 78.0012157368904
x42=60.4082846850973x_{42} = 60.4082846850973
x42=86.2591204339522x_{42} = 86.2591204339522
x42=−32.7622237207337x_{42} = -32.7622237207337
x42=−38.8659050998032x_{42} = -38.8659050998032
x42=27.7356574123217x_{42} = 27.7356574123217
x42=63.9986793216818x_{42} = 63.9986793216818
x42=84.463923827339x_{42} = 84.463923827339
x42=94.1579851385811x_{42} = 94.1579851385811
x42=−43.1743834087411x_{42} = -43.1743834087411
x42=−27.7356574123217x_{42} = -27.7356574123217
x42=−55.7407710735029x_{42} = -55.7407710735029
x42=35.9935851120918x_{42} = 35.9935851120918
x42=89.1314349371218x_{42} = 89.1314349371218
x42=5.11566976745241x_{42} = 5.11566976745241
x42=−81.591609183117x_{42} = -81.591609183117
x42=33.8393443743917x_{42} = 33.8393443743917
x42=100.261652976723x_{42} = 100.261652976723
x42=−61.8444425819835x_{42} = -61.8444425819835
x42=49.9961377759901x_{42} = 49.9961377759901
x42=44.251502730883x_{42} = 44.251502730883
x42=−19.8367490041875x_{42} = -19.8367490041875
x42=−97.7483780164771x_{42} = -97.7483780164771
x42=2.24253899206653x_{42} = 2.24253899206653
x42=−89.8495135507275x_{42} = -89.8495135507275
x42=80.1554518244297x_{42} = 80.1554518244297
x42=−47.8418998726837x_{42} = -47.8418998726837
x42=58.2540477237206x_{42} = 58.2540477237206
x42=24.1452483018706x_{42} = 24.1452483018706
x42=10.1425343431607x_{42} = 10.1425343431607
x42=89.490474244512x_{42} = 89.490474244512
x42=−13.7330101164599x_{42} = -13.7330101164599
x42=−83.7458451750355x_{42} = -83.7458451750355
x42=72.2565858421455x_{42} = 72.2565858421455
x42=−1.883222230847x_{42} = -1.883222230847
x42=−11.2196827316064x_{42} = -11.2196827316064
x42=16.2463210219747x_{42} = 16.2463210219747
x42=−78.0012157368904x_{42} = -78.0012157368904
x42=74.4108221128953x_{42} = 74.4108221128953
x42=−69.7433100907887x_{42} = -69.7433100907887
x42=−92.0037493641228x_{42} = -92.0037493641228
x42=−75.8469795854413x_{42} = -75.8469795854413
x42=−95.5941423012216x_{42} = -95.5941423012216
x42=−70.8204282840947x_{42} = -70.8204282840947
x42=−63.9986793216818x_{42} = -63.9986793216818
x42=−5.83382665513758x_{42} = -5.83382665513758
x42=30.248941316911x_{42} = 30.248941316911
x42=−33.8393443743917x_{42} = -33.8393443743917
x42=52.1503754363344x_{42} = 52.1503754363344
x42=−41.7382241647507x_{42} = -41.7382241647507
Decrece en los intervalos
[110.135232807699,∞)\left[110.135232807699, \infty\right)
Crece en los intervalos
(−∞,−489.280609175397]\left(-\infty, -489.280609175397\right]
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
−35sin⁡(35x2)2−2cos⁡(35x2)x+4sin⁡(35x2)35x2x=0\frac{- \frac{35 \sin{\left(\frac{35 x}{2} \right)}}{2} - \frac{2 \cos{\left(\frac{35 x}{2} \right)}}{x} + \frac{4 \sin{\left(\frac{35 x}{2} \right)}}{35 x^{2}}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=52.2400728280478x_{1} = 52.2400728280478
x2=−55.2919125915689x_{2} = -55.2919125915689
x3=−91.7344342943908x_{3} = -91.7344342943908
x4=70.3715826385321x_{4} = 70.3715826385321
x5=99.8128211654476x_{5} = 99.8128211654476
x6=−69.8330231822184x_{6} = -69.8330231822184
x7=−95.8633877053692x_{7} = -95.8633877053692
x8=38.2374997926631x_{8} = 38.2374997926631
x9=−5.74348944544407x_{9} = -5.74348944544407
x10=78.2704535329133x_{10} = 78.2704535329133
x11=−29.9795520581782x_{11} = -29.9795520581782
x12=66.2426265093928x_{12} = 66.2426265093928
x13=−77.7318942142141x_{13} = -77.7318942142141
x14=44.1616688506446x_{14} = 44.1616688506446
x15=88.8621187099988x_{15} = 88.8621187099988
x16=−74.8595777073401x_{16} = -74.8595777073401
x17=−147.744570306773x_{17} = -147.744570306773
x18=−44.1616688506446x_{18} = -44.1616688506446
x19=825.969580617206x_{19} = 825.969580617206
x20=−74.5005381265034x_{20} = -74.5005381265034
x21=76.1162162090336x_{21} = 76.1162162090336
x22=4.12736787211517x_{22} = 4.12736787211517
x23=58.164231707554x_{23} = 58.164231707554
x24=11.1296271851374x_{24} = 11.1296271851374
x25=1516.04285052466x_{25} = 1516.04285052466
x26=62.472709375886x_{26} = 62.472709375886
x27=−97.8381044626115x_{27} = -97.8381044626115
x28=6.28214569862151x_{28} = 6.28214569862151
x29=−21.7215685523906x_{29} = -21.7215685523906
x30=32.1338016244309x_{30} = 32.1338016244309
x31=14.0020608471747x_{31} = 14.0020608471747
x32=1376.01757752631x_{32} = 1376.01757752631
x33=−9.87291542905516x_{33} = -9.87291542905516
x34=74.1414985415948x_{34} = 74.1414985415948
x35=10.4115083899157x_{35} = 10.4115083899157
x36=60.1389507765986x_{36} = 60.1389507765986
x37=40.0327032535901x_{37} = 40.0327032535901
x38=−75.7571766419968x_{38} = -75.7571766419968
x39=64.2679080979358x_{39} = 64.2679080979358
x40=−15.9768338796762x_{40} = -15.9768338796762
x41=97.8381044626115x_{41} = 97.8381044626115
x42=−39.8531829385474x_{42} = -39.8531829385474
x43=−28.0048213153555x_{43} = -28.0048213153555
x44=−7.71849581882194x_{44} = -7.71849581882194
x45=376.093503165418x_{45} = 376.093503165418
x46=80.2451709684231x_{46} = 80.2451709684231
x47=86.3488424480751x_{47} = 86.3488424480751
x48=−45.2387898528538x_{48} = -45.2387898528538
x49=−138.589068796225x_{49} = -138.589068796225
x50=−89.7597173459415x_{50} = -89.7597173459415
x51=50.2653525345864x_{51} = 50.2653525345864
x52=−51.8810328024995x_{52} = -51.8810328024995
x53=−11.8477410733964x_{53} = -11.8477410733964
x54=−35.7242136541833x_{54} = -35.7242136541833
x55=72.1667807490957x_{55} = 72.1667807490957
x56=36.262775109785x_{56} = 36.262775109785
x57=−94.6067497390564x_{57} = -94.6067497390564
x58=90.1187567962194x_{58} = 90.1187567962194
x59=22.2601345671139x_{59} = 22.2601345671139
x60=−14.0020608471747x_{60} = -14.0020608471747
x61=−53.8557528002663x_{61} = -53.8557528002663
x62=−23.875830642257x_{62} = -23.875830642257
x63=96.5814665326901x_{63} = 96.5814665326901
x64=−78.9885326122498x_{64} = -78.9885326122498
x65=−25.8505669193533x_{65} = -25.8505669193533
x66=−63.7293484985942x_{66} = -63.7293484985942
x67=81.6813290409826x_{67} = 81.6813290409826
x68=88.1440397904679x_{68} = 88.1440397904679
x69=18.1311174102599x_{69} = 18.1311174102599
x70=35.0061315834303x_{70} = 35.0061315834303
x71=24.2348738550109x_{71} = 24.2348738550109
x72=46.1363905624404x_{72} = 46.1363905624404
x73=−3.58857146037256x_{73} = -3.58857146037256
x74=−33.7494875756348x_{74} = -33.7494875756348
x75=16.156358001948x_{75} = 16.156358001948
x76=−65.8835868117254x_{76} = -65.8835868117254
x77=−61.7546298395309x_{77} = -61.7546298395309
x78=−83.8355660578995x_{78} = -83.8355660578995
x79=−19.7468231036094x_{79} = -19.7468231036094
x80=92.2729934505154x_{80} = 92.2729934505154
x81=30.5181146430322x_{81} = 30.5181146430322
x82=30.1590729350515x_{82} = 30.1590729350515
x83=49.9063124394729x_{83} = 49.9063124394729
x84=8.25710975641742x_{84} = 8.25710975641742
x85=−17.5925476427948x_{85} = -17.5925476427948
x86=94.2477103156779x_{86} = 94.2477103156779
x87=−47.7520715736189x_{87} = -47.7520715736189
x88=−81.8608487965223x_{88} = -81.8608487965223
x89=−49.7267923868545x_{89} = -49.7267923868545
x90=2.15119773274779x_{90} = 2.15119773274779
x91=−37.8784590133609x_{91} = -37.8784590133609
x92=−67.8583050785409x_{92} = -67.8583050785409
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0

lim⁡x→0−(−35sin⁡(35x2)2−2cos⁡(35x2)x+4sin⁡(35x2)35x2x)=−122512\lim_{x \to 0^-}\left(\frac{- \frac{35 \sin{\left(\frac{35 x}{2} \right)}}{2} - \frac{2 \cos{\left(\frac{35 x}{2} \right)}}{x} + \frac{4 \sin{\left(\frac{35 x}{2} \right)}}{35 x^{2}}}{x}\right) = - \frac{1225}{12}
lim⁡x→0+(−35sin⁡(35x2)2−2cos⁡(35x2)x+4sin⁡(35x2)35x2x)=−122512\lim_{x \to 0^+}\left(\frac{- \frac{35 \sin{\left(\frac{35 x}{2} \right)}}{2} - \frac{2 \cos{\left(\frac{35 x}{2} \right)}}{x} + \frac{4 \sin{\left(\frac{35 x}{2} \right)}}{35 x^{2}}}{x}\right) = - \frac{1225}{12}
- los límites son iguales, es decir omitimos el punto correspondiente

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
[1516.04285052466,∞)\left[1516.04285052466, \infty\right)
Convexa en los intervalos
(−∞,−138.589068796225]\left(-\infty, -138.589068796225\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→−∞(sin⁡(35x2)352x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(\frac{35 x}{2} \right)}}{\frac{35}{2} x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(sin⁡(35x2)352x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(\frac{35 x}{2} \right)}}{\frac{35}{2} x}\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 sin(x*35/2)/((x*35/2)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(235xsin⁡(35x2)x)=0\lim_{x \to -\infty}\left(\frac{\frac{2}{35 x} \sin{\left(\frac{35 x}{2} \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(235xsin⁡(35x2)x)=0\lim_{x \to \infty}\left(\frac{\frac{2}{35 x} \sin{\left(\frac{35 x}{2} \right)}}{x}\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:
sin⁡(35x2)352x=2sin⁡(35x2)35x\frac{\sin{\left(\frac{35 x}{2} \right)}}{\frac{35}{2} x} = \frac{2 \sin{\left(\frac{35 x}{2} \right)}}{35 x}
- No
sin⁡(35x2)352x=−2sin⁡(35x2)35x\frac{\sin{\left(\frac{35 x}{2} \right)}}{\frac{35}{2} x} = - \frac{2 \sin{\left(\frac{35 x}{2} \right)}}{35 x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор