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Trazado el gráfico de la función/ (16*cos(x)+(cos(3*x)+sin(3*x))*exp(x)+sin(x))*exp(x)

Gráfico de la función y = (16*cos(x)+(cos(3*x)+sin(3*x))*exp(x)+sin(x))*exp(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       /                                   x         \  x
f(x) = \16*cos(x) + (cos(3*x) + sin(3*x))*e  + sin(x)/*e
f(x)=(((sin⁡(3x)+cos⁡(3x))ex+16cos⁡(x))+sin⁡(x))exf{\left(x \right)} = \left(\left(\left(\sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{x} + 16 \cos{\left(x \right)}\right) + \sin{\left(x \right)}\right) e^{x} 
f = ((sin(3*x) + cos(3*x))*exp(x) + 16*cos(x) + sin(x))*exp(x)
Gráfico de la función
02468-8-6-4-2-1010-500000000500000000
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⁡(3x)+cos⁡(3x))ex+16cos⁡(x))+sin⁡(x))ex=0\left(\left(\left(\sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{x} + 16 \cos{\left(x \right)}\right) + \sin{\left(x \right)}\right) e^{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=12.3045549976198x_{1} = 12.3045549976198
x2=−54.9154526278254x_{2} = -54.9154526278254
x3=−51.7738599742356x_{3} = -51.7738599742356
x4=−95.7561571244927x_{4} = -95.7561571244927
x5=−45.490674667056x_{5} = -45.490674667056
x6=−39.2074893598765x_{6} = -39.2074893598765
x7=−4.65044592747542x_{7} = -4.65044592747542
x8=−1.51978392515159x_{8} = -1.51978392515159
x9=−64.3402305885948x_{9} = -64.3402305885948
x10=−92.6145644709029x_{10} = -92.6145644709029
x11=−67.4818232421846x_{11} = -67.4818232421846
x12=−42.3490820134663x_{12} = -42.3490820134663
x13=−48.6322673206458x_{13} = -48.6322673206458
x14=14.3989667465345x_{14} = 14.3989667465345
x15=−130.31367631398x_{15} = -130.31367631398
x16=−70.6234158957744x_{16} = -70.6234158957744
x17=−23.4995260919306x_{17} = -23.4995260919306
x18=−61.198637935005x_{18} = -61.198637935005
x19=−17.2163407864044x_{19} = -17.2163407864044
x20=−20.3579334384093x_{20} = -20.3579334384093
x21=6.01264433821948x_{21} = 6.01264433821948
x22=−76.906601202954x_{22} = -76.906601202954
x23=−108.322527738852x_{23} = -108.322527738852
x24=−7.79158335155013x_{24} = -7.79158335155013
x25=−58.0570452814152x_{25} = -58.0570452814152
x26=2.04021421938726x_{26} = 2.04021421938726
x27=−10.9331563645858x_{27} = -10.9331563645858
x28=−73.7650085493642x_{28} = -73.7650085493642
x29=−80.0481938565438x_{29} = -80.0481938565438
x30=−86.3313791637234x_{30} = -86.3313791637234
x31=−32.9243040526969x_{31} = -32.9243040526969
x32=−83.1897865101336x_{32} = -83.1897865101336
x33=−26.6411187455174x_{33} = -26.6411187455174
x34=−36.0658967062867x_{34} = -36.0658967062867
x35=10.2102803533459x_{35} = 10.2102803533459
x36=3.97780037124102x_{36} = 3.97780037124102
x37=13.3517732827534x_{37} = 13.3517732827534
x38=−89.4729718173132x_{38} = -89.4729718173132
x39=8.11600483029408x_{39} = 8.11600483029408
x40=−98.8977497780825x_{40} = -98.8977497780825
x41=−29.7827113991071x_{41} = -29.7827113991071
x42=−14.0747481694895x_{42} = -14.0747481694895
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 (16*cos(x) + (cos(3*x) + sin(3*x))*exp(x) + sin(x))*exp(x).
(sin⁡(0)+((sin⁡(0⋅3)+cos⁡(0⋅3))e0+16cos⁡(0)))e0\left(\sin{\left(0 \right)} + \left(\left(\sin{\left(0 \cdot 3 \right)} + \cos{\left(0 \cdot 3 \right)}\right) e^{0} + 16 \cos{\left(0 \right)}\right)\right) e^{0}
Resultado:
f(0)=17f{\left(0 \right)} = 17
Punto:
(0, 17)
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
(((sin⁡(3x)+cos⁡(3x))ex+16cos⁡(x))+sin⁡(x))ex+((−3sin⁡(3x)+3cos⁡(3x))ex+(sin⁡(3x)+cos⁡(3x))ex−16sin⁡(x)+cos⁡(x))ex=0\left(\left(\left(\sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{x} + 16 \cos{\left(x \right)}\right) + \sin{\left(x \right)}\right) e^{x} + \left(\left(- 3 \sin{\left(3 x \right)} + 3 \cos{\left(3 x \right)}\right) e^{x} + \left(\sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{x} - 16 \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−80.8335920199412x_{1} = -80.8335920199412
x2=−55.7008507912229x_{2} = -55.7008507912229
x3=−39.9928875232739x_{3} = -39.9928875232739
x4=−30.5681095625045x_{4} = -30.5681095625045
x5=−99.68314794148x_{5} = -99.68314794148
x6=−96.5415552878902x_{6} = -96.5415552878902
x7=−21.1433316018712x_{7} = -21.1433316018712
x8=−52.5592581376331x_{8} = -52.5592581376331
x9=−11.7185553269422x_{9} = -11.7185553269422
x10=−71.4088140591718x_{10} = -71.4088140591718
x11=−18.0017389512938x_{11} = -18.0017389512938
x12=−90.2583699807106x_{12} = -90.2583699807106
x13=7.78855853458055x_{13} = 7.78855853458055
x14=−46.2760728304535x_{14} = -46.2760728304535
x15=−5.43626931437669x_{15} = -5.43626931437669
x16=−27.426516908915x_{16} = -27.426516908915
x17=−8.57699999873181x_{17} = -8.57699999873181
x18=−24.2849242553308x_{18} = -24.2849242553308
x19=−61.9840360984025x_{19} = -61.9840360984025
x20=−43.1344801768637x_{20} = -43.1344801768637
x21=−65.1256287519923x_{21} = -65.1256287519923
x22=−36.8512948696841x_{22} = -36.8512948696841
x23=−77.6919993663514x_{23} = -77.6919993663514
x24=5.68881445086257x_{24} = 5.68881445086257
x25=14.0713691222027x_{25} = 14.0713691222027
x26=−14.8601463674135x_{26} = -14.8601463674135
x27=−74.5504067127616x_{27} = -74.5504067127616
x28=−83.975184673531x_{28} = -83.975184673531
x29=−93.3999626343004x_{29} = -93.3999626343004
x30=0.633286585380397x_{30} = 0.633286585380397
x31=−68.267221405582x_{31} = -68.267221405582
x32=−87.1167773271208x_{32} = -87.1167773271208
x33=11.9769640833544x_{33} = 11.9769640833544
x34=−2.31370196214823x_{34} = -2.31370196214823
x35=3.61403731599096x_{35} = 3.61403731599096
x36=−58.8424434448127x_{36} = -58.8424434448127
x37=−49.4176654840433x_{37} = -49.4176654840433
x38=9.8826069836513x_{38} = 9.8826069836513
x39=−33.7097022160943x_{39} = -33.7097022160943
Signos de extremos en los puntos:
(-80.83359201994122, 8.88932116322487e-35)

(-55.70085079122287, 7.30936127309036e-24)

(-39.99288752327391, -4.85021730835731e-17)

(-30.56810956250454, 6.01021846770254e-13)

(-99.68314794147997, 5.78909230248975e-43)

(-96.54155528789019, -1.33963605594704e-41)

(-21.143331601871203, -7.44765104992703e-9)

(-52.55925813763308, -1.69143682562524e-22)

(-11.718555326942232, 9.22886513582704e-5)

(-71.40881405917185, -1.10153337106143e-30)

(-18.00173895129383, 1.72343803724332e-7)

(-90.2583699807106, -7.17363929399746e-39)

(7.7885585345805515, -6840393.55260736)

(-46.276072830453494, -9.05750305969614e-20)

(-5.436269314376695, 0.0494148483212453)

(-27.426516908914987, -1.39080618216954e-11)

(-8.576999998731813, -0.00213561437711268)

(-24.28492425533082, 3.21842183733343e-10)

(-61.984036098402456, 1.36498135828605e-26)

(-43.1344801768637, 2.09596894324886e-18)

(-65.12562875199225, -5.8986192848546e-28)

(-36.85129486968411, 1.12237387934882e-15)

(-77.69199936635142, -2.05705048752247e-33)

(5.688814450862571, -100038.045586924)

(14.071369122202722, -1962905116030.03)

(-14.860146367413476, -3.98815495796951e-6)

(-74.55040671276163, 4.76015730618661e-32)

(-83.97518467353102, -3.8414239816802e-36)

(-93.3999626343004, 3.10001062104591e-40)

(0.6332865853803971, 27.6223095196631)

(-68.26722140558205, 2.54902451644817e-29)

(-87.1167773271208, 1.66002981960823e-37)

(11.976964083354392, -29764723059.854)

(-2.3137019621482326, -1.14140221205263)

(3.614037315990961, -2117.64241195028)

(-58.842443444812666, -3.15866140615709e-25)

(-49.41766548404328, 3.91410196895576e-21)

(9.882606983651302, -451691217.278745)

(-33.70970221609432, -2.59725089610712e-14)


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=−39.9928875232739x_{1} = -39.9928875232739
x2=−96.5415552878902x_{2} = -96.5415552878902
x3=−21.1433316018712x_{3} = -21.1433316018712
x4=−52.5592581376331x_{4} = -52.5592581376331
x5=−71.4088140591718x_{5} = -71.4088140591718
x6=−90.2583699807106x_{6} = -90.2583699807106
x7=7.78855853458055x_{7} = 7.78855853458055
x8=−46.2760728304535x_{8} = -46.2760728304535
x9=−27.426516908915x_{9} = -27.426516908915
x10=−8.57699999873181x_{10} = -8.57699999873181
x11=−65.1256287519923x_{11} = -65.1256287519923
x12=−77.6919993663514x_{12} = -77.6919993663514
x13=5.68881445086257x_{13} = 5.68881445086257
x14=14.0713691222027x_{14} = 14.0713691222027
x15=−14.8601463674135x_{15} = -14.8601463674135
x16=−83.975184673531x_{16} = -83.975184673531
x17=11.9769640833544x_{17} = 11.9769640833544
x18=−2.31370196214823x_{18} = -2.31370196214823
x19=3.61403731599096x_{19} = 3.61403731599096
x20=−58.8424434448127x_{20} = -58.8424434448127
x21=9.8826069836513x_{21} = 9.8826069836513
x22=−33.7097022160943x_{22} = -33.7097022160943
Puntos máximos de la función:
x22=−80.8335920199412x_{22} = -80.8335920199412
x22=−55.7008507912229x_{22} = -55.7008507912229
x22=−30.5681095625045x_{22} = -30.5681095625045
x22=−99.68314794148x_{22} = -99.68314794148
x22=−11.7185553269422x_{22} = -11.7185553269422
x22=−18.0017389512938x_{22} = -18.0017389512938
x22=−5.43626931437669x_{22} = -5.43626931437669
x22=−24.2849242553308x_{22} = -24.2849242553308
x22=−61.9840360984025x_{22} = -61.9840360984025
x22=−43.1344801768637x_{22} = -43.1344801768637
x22=−36.8512948696841x_{22} = -36.8512948696841
x22=−74.5504067127616x_{22} = -74.5504067127616
x22=−93.3999626343004x_{22} = -93.3999626343004
x22=0.633286585380397x_{22} = 0.633286585380397
x22=−68.267221405582x_{22} = -68.267221405582
x22=−87.1167773271208x_{22} = -87.1167773271208
x22=−49.4176654840433x_{22} = -49.4176654840433
Decrece en los intervalos
[14.0713691222027,∞)\left[14.0713691222027, \infty\right)
Crece en los intervalos
(−∞,−96.5415552878902]\left(-\infty, -96.5415552878902\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
(−12(sin⁡(3x)−cos⁡(3x))ex−5(sin⁡(3x)+cos⁡(3x))ex−32sin⁡(x)+2cos⁡(x))ex=0\left(- 12 \left(\sin{\left(3 x \right)} - \cos{\left(3 x \right)}\right) e^{x} - 5 \left(\sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{x} - 32 \sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) e^{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−53.3446563010305x_{1} = -53.3446563010305
x2=−94.1853607976978x_{2} = -94.1853607976978
x3=−65.9110269153897x_{3} = -65.9110269153897
x4=−9.36234920518276x_{4} = -9.36234920518276
x5=−109.893324065647x_{5} = -109.893324065647
x6=−97.3269534512876x_{6} = -97.3269534512876
x7=−6.22053703415353x_{7} = -6.22053703415353
x8=−50.2030636474407x_{8} = -50.2030636474407
x9=−56.4862489546203x_{9} = -56.4862489546203
x10=−21.9287297650979x_{10} = -21.9287297650979
x11=2.16699892901357x_{11} = 2.16699892901357
x12=−12.5039513745199x_{12} = -12.5039513745199
x13=−3.07420812213691x_{13} = -3.07420812213691
x14=−31.353507725902x_{14} = -31.353507725902
x15=−28.2119150723121x_{15} = -28.2119150723121
x16=−59.6278416082101x_{16} = -59.6278416082101
x17=−47.0614709938509x_{17} = -47.0614709938509
x18=−43.9198783402611x_{18} = -43.9198783402611
x19=−78.4773975297489x_{19} = -78.4773975297489
x20=11.6493712006166x_{20} = 11.6493712006166
x21=0.106936572241941x_{21} = 0.106936572241941
x22=−62.7694342617999x_{22} = -62.7694342617999
x23=−37.6366930330816x_{23} = -37.6366930330816
x24=6.41332316024597x_{24} = 6.41332316024597
x25=−84.7605828369285x_{25} = -84.7605828369285
x26=−91.043768144108x_{26} = -91.043768144108
x27=9.55497751923063x_{27} = 9.55497751923063
x28=−75.3358048761591x_{28} = -75.3358048761591
x29=−81.6189901833387x_{29} = -81.6189901833387
x30=−18.7871371107401x_{30} = -18.7871371107401
x31=−40.7782856866714x_{31} = -40.7782856866714
x32=4.32591750474832x_{32} = 4.32591750474832
x33=−25.0703224187209x_{33} = -25.0703224187209
x34=−69.0526195689795x_{34} = -69.0526195689795
x35=−15.6455444393777x_{35} = -15.6455444393777
x36=−100.468546104877x_{36} = -100.468546104877
x37=−87.9021754905183x_{37} = -87.9021754905183
x38=−34.4951003794918x_{38} = -34.4951003794918
x39=−72.1942122225693x_{39} = -72.1942122225693
x40=13.7437710742476x_{40} = 13.7437710742476

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
[13.7437710742476,∞)\left[13.7437710742476, \infty\right)
Convexa en los intervalos
(−∞,−109.893324065647]\left(-\infty, -109.893324065647\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((((sin⁡(3x)+cos⁡(3x))ex+16cos⁡(x))+sin⁡(x))ex)=0\lim_{x \to -\infty}\left(\left(\left(\left(\sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{x} + 16 \cos{\left(x \right)}\right) + \sin{\left(x \right)}\right) e^{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞((((sin⁡(3x)+cos⁡(3x))ex+16cos⁡(x))+sin⁡(x))ex)y = \lim_{x \to \infty}\left(\left(\left(\left(\sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{x} + 16 \cos{\left(x \right)}\right) + \sin{\left(x \right)}\right) e^{x}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (16*cos(x) + (cos(3*x) + sin(3*x))*exp(x) + sin(x))*exp(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((((sin⁡(3x)+cos⁡(3x))ex+16cos⁡(x))+sin⁡(x))exx)=0\lim_{x \to -\infty}\left(\frac{\left(\left(\left(\sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{x} + 16 \cos{\left(x \right)}\right) + \sin{\left(x \right)}\right) e^{x}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞((((sin⁡(3x)+cos⁡(3x))ex+16cos⁡(x))+sin⁡(x))exx)y = x \lim_{x \to \infty}\left(\frac{\left(\left(\left(\sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{x} + 16 \cos{\left(x \right)}\right) + \sin{\left(x \right)}\right) e^{x}}{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:
(((sin⁡(3x)+cos⁡(3x))ex+16cos⁡(x))+sin⁡(x))ex=((−sin⁡(3x)+cos⁡(3x))e−x−sin⁡(x)+16cos⁡(x))e−x\left(\left(\left(\sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{x} + 16 \cos{\left(x \right)}\right) + \sin{\left(x \right)}\right) e^{x} = \left(\left(- \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{- x} - \sin{\left(x \right)} + 16 \cos{\left(x \right)}\right) e^{- x}
- No
(((sin⁡(3x)+cos⁡(3x))ex+16cos⁡(x))+sin⁡(x))ex=−((−sin⁡(3x)+cos⁡(3x))e−x−sin⁡(x)+16cos⁡(x))e−x\left(\left(\left(\sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{x} + 16 \cos{\left(x \right)}\right) + \sin{\left(x \right)}\right) e^{x} = - \left(\left(- \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{- x} - \sin{\left(x \right)} + 16 \cos{\left(x \right)}\right) e^{- x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор