Sr Examen

Otras calculadoras

Trazado el gráfico de la función/ 1-cos(x)+2*sin(x)+exp(-x)

Gráfico de la función y = 1-cos(x)+2*sin(x)+exp(-x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                                -x
f(x) = 1 - cos(x) + 2*sin(x) + e
f(x)=((1−cos⁡(x))+2sin⁡(x))+e−xf{\left(x \right)} = \left(\left(1 - \cos{\left(x \right)}\right) + 2 \sin{\left(x \right)}\right) + e^{- x} 
f = 1 - cos(x) + 2*sin(x) + exp(-x)
Gráfico de la función
02468-8-6-4-2-101040000-20000
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:
((1−cos⁡(x))+2sin⁡(x))+e−x=0\left(\left(1 - \cos{\left(x \right)}\right) + 2 \sin{\left(x \right)}\right) + e^{- x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=43.9822971502571x_{1} = 43.9822971502571
x2=79.4671115577464x_{2} = 79.4671115577464
x3=92.0334821721056x_{3} = 92.0334821721056
x4=48.0511850218485x_{4} = 48.0511850218485
x5=73.1839262505669x_{5} = 73.1839262505669
x6=81.6814089933346x_{6} = 81.6814089933346
x7=230.263558930057x_{7} = 230.263558930057
x8=54.3343703290281x_{8} = 54.3343703290281
x9=106.814150222053x_{9} = 106.814150222053
x10=6.28225049394546x_{10} = 6.28225049394546
x11=35.4848144074893x_{11} = 35.4848144074893
x12=16.6352585157611x_{12} = 16.6352585157611
x13=12.5663688706842x_{13} = 12.5663688706842
x14=60.6175556362077x_{14} = 60.6175556362077
x15=50.2654824574367x_{15} = 50.2654824574367
x16=66.9007409433873x_{16} = 66.9007409433873
x17=56.5486677646163x_{17} = 56.5486677646163
x18=18.8495559182826x_{18} = 18.8495559182826
x19=100.530964914873x_{19} = 100.530964914873
x20=62.8318530717959x_{20} = 62.8318530717959
x21=94.2477796076938x_{21} = 94.2477796076938
x22=25.1327412287123x_{22} = 25.1327412287123
x23=98.3166674792852x_{23} = 98.3166674792852
x24=41.7679997146689x_{24} = 41.7679997146689
x25=29.2016291003099x_{25} = 29.2016291003099
x26=87.9645943005142x_{26} = 87.9645943005142
x27=69.1150383789755x_{27} = 69.1150383789755
x28=75.398223686155x_{28} = 75.398223686155
x29=4.07738190527114x_{29} = 4.07738190527114
x30=10.3520891418457x_{30} = 10.3520891418457
x31=22.9184437931858x_{31} = 22.9184437931858
x32=31.4159265358979x_{32} = 31.4159265358979
x33=37.6991118430775x_{33} = 37.6991118430775
x34=85.750296864926x_{34} = 85.750296864926
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 1 - cos(x) + 2*sin(x) + exp(-x).
(2sin⁡(0)+(1−cos⁡(0)))+e−0\left(2 \sin{\left(0 \right)} + \left(1 - \cos{\left(0 \right)}\right)\right) + e^{- 0}
Resultado:
f(0)=1f{\left(0 \right)} = 1
Punto:
(0, 1)
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⁡(x)+2cos⁡(x)−e−x=0\sin{\left(x \right)} + 2 \cos{\left(x \right)} - e^{- x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=55.4415190468222x_{1} = 55.4415190468222
x2=42.875148432463x_{2} = 42.875148432463
x3=52.2999263932324x_{3} = 52.2999263932324
x4=61.7247043540018x_{4} = 61.7247043540018
x5=80.5742602755405x_{5} = 80.5742602755405
x6=14.6008143462339x_{6} = 14.6008143462339
x7=−0.387672850099208x_{7} = -0.387672850099208
x8=30.3087778181039x_{8} = 30.3087778181039
x9=89.9990382363099x_{9} = 89.9990382363099
x10=86.8574455827201x_{10} = 86.8574455827201
x11=5.17855714720614x_{11} = 5.17855714720614
x12=64.8662970075916x_{12} = 64.8662970075916
x13=71.1494823147711x_{13} = 71.1494823147711
x14=20.8839998569537x_{14} = 20.8839998569537
x15=74.2910749683609x_{15} = 74.2910749683609
x16=8.31752003309144x_{16} = 8.31752003309144
x17=68.0078896611814x_{17} = 68.0078896611814
x18=27.1671851645133x_{18} = 27.1671851645133
x19=93.1406308898997x_{19} = 93.1406308898997
x20=33.4503704716936x_{20} = 33.4503704716936
x21=46.0167410860528x_{21} = 46.0167410860528
x22=58.583111700412x_{22} = 58.583111700412
x23=99.4238161970793x_{23} = 99.4238161970793
x24=11.4592266154144x_{24} = 11.4592266154144
x25=24.0255925109407x_{25} = 24.0255925109407
x26=−0.387672850099237x_{26} = -0.387672850099237
x27=96.2822235434895x_{27} = 96.2822235434895
x28=36.5919631252834x_{28} = 36.5919631252834
x29=83.7158529291303x_{29} = 83.7158529291303
x30=49.1583337396426x_{30} = 49.1583337396426
x31=1.97217200998873x_{31} = 1.97217200998873
x32=39.7335557788732x_{32} = 39.7335557788732
x33=77.4326676219507x_{33} = 77.4326676219507
x34=17.7424072125569x_{34} = 17.7424072125569
Signos de extremos en los puntos:
(55.44151904682219, -1.23606797749979)

(42.875148432463014, -1.23606797749979)

(52.2999263932324, 3.23606797749979)

(61.72470435400177, -1.23606797749979)

(80.57426027554054, -1.23606797749979)

(14.600814346233902, 3.2360684334809)

(-0.38767285009920804, 0.791686353330576)

(30.308777818103874, -1.23606797749972)

(89.99903823630991, 3.23606797749979)

(86.85744558272012, -1.23606797749979)

(5.178557147206145, -1.23042474173664)

(64.86629700759157, 3.23606797749979)

(71.14948231477115, 3.23606797749979)

(20.88399985695365, 3.23606797835131)

(74.29107496836095, -1.23606797749979)

(8.31752003309144, 3.23631216488884)

(68.00788966118137, -1.23606797749979)

(27.167185164513338, 3.23606797750138)

(93.14063088989971, -1.23606797749979)

(33.45037047169363, 3.23606797749979)

(46.01674108605281, 3.23606797749979)

(58.58311170041198, 3.23606797749979)

(99.42381619707929, -1.23606797749979)

(11.459226615414424, -1.23605742580699)

(24.02559251094071, -1.23606797746299)

(-0.3876728500992374, 0.791686353330576)

(96.2822235434895, 3.23606797749979)

(36.59196312528343, -1.23606797749979)

(83.71585292913032, 3.23606797749979)

(49.1583337396426, -1.23606797749979)

(1.9721720099887343, 3.3708881574905)

(39.733555778873225, 3.23606797749979)

(77.43266762195074, 3.23606797749979)

(17.74240721255689, -1.23606795779506)


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=55.4415190468222x_{1} = 55.4415190468222
x2=42.875148432463x_{2} = 42.875148432463
x3=61.7247043540018x_{3} = 61.7247043540018
x4=80.5742602755405x_{4} = 80.5742602755405
x5=−0.387672850099208x_{5} = -0.387672850099208
x6=30.3087778181039x_{6} = 30.3087778181039
x7=86.8574455827201x_{7} = 86.8574455827201
x8=5.17855714720614x_{8} = 5.17855714720614
x9=74.2910749683609x_{9} = 74.2910749683609
x10=68.0078896611814x_{10} = 68.0078896611814
x11=93.1406308898997x_{11} = 93.1406308898997
x12=99.4238161970793x_{12} = 99.4238161970793
x13=11.4592266154144x_{13} = 11.4592266154144
x14=24.0255925109407x_{14} = 24.0255925109407
x15=−0.387672850099237x_{15} = -0.387672850099237
x16=36.5919631252834x_{16} = 36.5919631252834
x17=49.1583337396426x_{17} = 49.1583337396426
x18=17.7424072125569x_{18} = 17.7424072125569
Puntos máximos de la función:
x18=52.2999263932324x_{18} = 52.2999263932324
x18=14.6008143462339x_{18} = 14.6008143462339
x18=89.9990382363099x_{18} = 89.9990382363099
x18=64.8662970075916x_{18} = 64.8662970075916
x18=71.1494823147711x_{18} = 71.1494823147711
x18=20.8839998569537x_{18} = 20.8839998569537
x18=8.31752003309144x_{18} = 8.31752003309144
x18=27.1671851645133x_{18} = 27.1671851645133
x18=33.4503704716936x_{18} = 33.4503704716936
x18=46.0167410860528x_{18} = 46.0167410860528
x18=58.583111700412x_{18} = 58.583111700412
x18=96.2822235434895x_{18} = 96.2822235434895
x18=83.7158529291303x_{18} = 83.7158529291303
x18=1.97217200998873x_{18} = 1.97217200998873
x18=39.7335557788732x_{18} = 39.7335557788732
x18=77.4326676219507x_{18} = 77.4326676219507
Decrece en los intervalos
[99.4238161970793,∞)\left[99.4238161970793, \infty\right)
Crece en los intervalos
(−∞,−0.387672850099237]\left(-\infty, -0.387672850099237\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
−2sin⁡(x)+cos⁡(x)+e−x=0- 2 \sin{\left(x \right)} + \cos{\left(x \right)} + e^{- x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=22.4547961840502x_{1} = 22.4547961840502
x2=31.8795741448987x_{2} = 31.8795741448987
x3=25.5963888377226x_{3} = 25.5963888377226
x4=100.994612523874x_{4} = 100.994612523874
x5=85.2866492559252x_{5} = 85.2866492559252
x6=66.4370933343865x_{6} = 66.4370933343865
x7=63.2955006807967x_{7} = 63.2955006807967
x8=16.1716108345588x_{8} = 16.1716108345588
x9=13.030019204316x_{9} = 13.030019204316
x10=97.8530198702844x_{10} = 97.8530198702844
x11=57.0123153736171x_{11} = 57.0123153736171
x12=38.1627594520783x_{12} = 38.1627594520783
x13=69.5786859879763x_{13} = 69.5786859879763
x14=9.88840286922973x_{14} = 9.88840286922973
x15=0.689905888442734x_{15} = 0.689905888442734
x16=35.0211667984885x_{16} = 35.0211667984885
x17=60.1539080272069x_{17} = 60.1539080272069
x18=53.8707227200273x_{18} = 53.8707227200273
x19=79.0034639487456x_{19} = 79.0034639487456
x20=41.3043521056681x_{20} = 41.3043521056681
x21=19.3132035323714x_{21} = 19.3132035323714
x22=82.1450566023354x_{22} = 82.1450566023354
x23=47.5875374128477x_{23} = 47.5875374128477
x24=50.7291300664375x_{24} = 50.7291300664375
x25=94.7114272166946x_{25} = 94.7114272166946
x26=91.5698345631048x_{26} = 91.5698345631048
x27=6.74735793479213x_{27} = 6.74735793479213
x28=44.4459447592579x_{28} = 44.4459447592579
x29=75.8618712951559x_{29} = 75.8618712951559
x30=28.7379814913088x_{30} = 28.7379814913088
x31=88.428241909515x_{31} = 88.428241909515
x32=3.59293375980096x_{32} = 3.59293375980096
x33=72.7202786415661x_{33} = 72.7202786415661

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
[97.8530198702844,∞)\left[97.8530198702844, \infty\right)
Convexa en los intervalos
(−∞,3.59293375980096]\left(-\infty, 3.59293375980096\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(((1−cos⁡(x))+2sin⁡(x))+e−x)=∞\lim_{x \to -\infty}\left(\left(\left(1 - \cos{\left(x \right)}\right) + 2 \sin{\left(x \right)}\right) + e^{- x}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞(((1−cos⁡(x))+2sin⁡(x))+e−x)=⟨−2,4⟩\lim_{x \to \infty}\left(\left(\left(1 - \cos{\left(x \right)}\right) + 2 \sin{\left(x \right)}\right) + e^{- x}\right) = \left\langle -2, 4\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−2,4⟩y = \left\langle -2, 4\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función 1 - cos(x) + 2*sin(x) + exp(-x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(((1−cos⁡(x))+2sin⁡(x))+e−xx)=−∞\lim_{x \to -\infty}\left(\frac{\left(\left(1 - \cos{\left(x \right)}\right) + 2 \sin{\left(x \right)}\right) + e^{- x}}{x}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
lim⁡x→∞(((1−cos⁡(x))+2sin⁡(x))+e−xx)=0\lim_{x \to \infty}\left(\frac{\left(\left(1 - \cos{\left(x \right)}\right) + 2 \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 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:
((1−cos⁡(x))+2sin⁡(x))+e−x=ex−2sin⁡(x)−cos⁡(x)+1\left(\left(1 - \cos{\left(x \right)}\right) + 2 \sin{\left(x \right)}\right) + e^{- x} = e^{x} - 2 \sin{\left(x \right)} - \cos{\left(x \right)} + 1
- No
((1−cos⁡(x))+2sin⁡(x))+e−x=−ex+2sin⁡(x)+cos⁡(x)−1\left(\left(1 - \cos{\left(x \right)}\right) + 2 \sin{\left(x \right)}\right) + e^{- x} = - e^{x} + 2 \sin{\left(x \right)} + \cos{\left(x \right)} - 1
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор