Gráfico de la función y = cos⁡(2π*20.4*t+6.6*sin2πt)

Ha introducido [src]

          /2*pi*102     33*sin(2*pi*t)\
f(t) = cos|--------*t + --------------|
          \   5               5       /

f{\left(t \right)} = \cos{\left(t \frac{102 \cdot 2 \pi}{5} + \frac{33 \sin{\left(2 \pi t \right)}}{5} \right)}

f = cos(t*(102*(2*pi)/5) + 33*sin((2*pi)*t)/5)

Gráfico de la función

Puntos de cruce con el eje de coordenadas X

El gráfico de la función cruce el eje T con f = 0
o sea hay que resolver la ecuación:

\cos{\left(t \frac{102 \cdot 2 \pi}{5} + \frac{33 \sin{\left(2 \pi t \right)}}{5} \right)} = 0

Resolvermos esta ecuación
Puntos de cruce con el eje T:

Solución numérica

t_{1} = 64.2499792835535

t_{2} = -15.4321341366565

t_{3} = -35.7500207164465

t_{4} = -55.7500207164465

t_{5} = 84.5678658633435

t_{6} = -65.4321341366565

t_{7} = -79.1599623532187

t_{8} = 48.5253113540508

t_{9} = -69.1599623532187

t_{10} = -85.4321341366565

t_{11} = 18.5253113540508

t_{12} = 100.840037646781

t_{13} = -17.9647445070266

t_{14} = 12.0352554929734

t_{15} = 24.2499792835535

t_{16} = -11.4746886459492

t_{17} = 34.2499792835535

t_{18} = -67.9647445070266

t_{19} = -41.4746886459492

t_{20} = 54.5678658633435

t_{21} = -71.4746886459492

t_{22} = -27.9647445070266

t_{23} = 20.8400376467813

t_{24} = 62.0352554929734

t_{25} = 92.0352554929734

t_{26} = -47.9647445070266

t_{27} = -25.7500207164465

t_{28} = 2.03525549297342

t_{29} = 14.2499792835535

t_{30} = -75.7500207164465

t_{31} = 98.5253113540508

t_{32} = -81.4746886459492

t_{33} = -15.7500207164465

t_{34} = 4.56786586334346

t_{35} = -39.1599623532188

t_{36} = 74.5678658633435

t_{37} = 88.5253113540508

t_{38} = 0.840037646781251

t_{39} = -9.15996235321875

t_{40} = -5.43213413665654

t_{41} = -57.9647445070266

t_{42} = 4.24997928355354

t_{43} = 94.5678658633435

t_{44} = -7.96474450702658

t_{45} = -5.75002071644646

t_{46} = -31.4746886459492

t_{47} = -45.7500207164465

t_{48} = 68.5253113540508

t_{49} = -51.4746886459492

t_{50} = 38.5253113540508

t_{51} = -45.4321341366565

t_{52} = -91.4746886459492

t_{53} = -25.4321341366565

t_{54} = -97.9647445070266

t_{55} = -55.4321341366565

t_{56} = -1.47468864594918

t_{57} = -95.4321341366565

t_{58} = 74.2499792835535

t_{59} = 28.5253113540508

t_{60} = -75.4321341366565

t_{61} = 80.8400376467813

t_{62} = 24.5678658633435

t_{63} = -19.1599623532187

t_{64} = 84.2499792835535

t_{65} = -85.7500207164465

t_{66} = -21.4746886459492

t_{67} = 10.8400376467813

t_{68} = 40.8400376467812

t_{69} = 44.5678658633435

t_{70} = -99.1599623532187

t_{71} = 34.5678658633435

t_{72} = -37.9647445070266

t_{73} = 22.0352554929734

t_{74} = 54.2499792835535

t_{75} = 78.5253113540508

t_{76} = 32.0352554929734

t_{77} = -61.4746886459492

t_{78} = 44.2499792835535

t_{79} = 8.52531135405082

t_{80} = 52.0352554929734

t_{81} = 82.0352554929734

t_{82} = -77.9647445070266

t_{83} = 90.8400376467813

t_{84} = 50.8400376467812

t_{85} = 70.8400376467813

t_{86} = -35.4321341366565

t_{87} = 60.8400376467812

t_{88} = 64.5678658633435

t_{89} = 58.5253113540508

t_{90} = -65.7500207164465

t_{91} = -87.9647445070266

t_{92} = 14.5678658633435

t_{93} = 42.0352554929734

t_{94} = 30.8400376467813

t_{95} = -29.1599623532187

t_{96} = 72.0352554929734

t_{97} = -49.1599623532188

t_{98} = -95.7500207164465

t_{99} = -59.1599623532188

t_{100} = 94.2499792835535

t_{101} = -89.1599623532187

Puntos de cruce con el eje de coordenadas Y

El gráfico cruce el eje Y cuando t es igual a 0:
sustituimos t = 0 en cos(((2*pi)*102/5)*t + 33*sin((2*pi)*t)/5).

\cos{\left(0 \frac{102 \cdot 2 \pi}{5} + \frac{33 \sin{\left(0 \cdot 2 \pi \right)}}{5} \right)}

Resultado:

f{\left(0 \right)} = 1

Punto:

(0, 1)

Extremos de la función

Para hallar los extremos hay que resolver la ecuación

\frac{d}{d t} f{\left(t \right)} = 0

(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:

\frac{d}{d t} f{\left(t \right)} =

primera derivada

- \left(\frac{66 \pi \cos{\left(2 \pi t \right)}}{5} + \frac{102 \cdot 2 \pi}{5}\right) \sin{\left(t \frac{102 \cdot 2 \pi}{5} + \frac{33 \sin{\left(2 \pi t \right)}}{5} \right)} = 0

Resolvermos esta ecuación
Raíces de esta ecuación

t_{1} = 13.4710903999736

t_{2} = -1.09777141126241

t_{3} = 60

t_{4} = -27.6955679346015

t_{5} = 0

t_{6} = 36.1790334700983

t_{7} = -91.0977714112624

t_{8} = -93.8209665299017

t_{9} = -80

t_{10} = 18.9022285887376

t_{11} = 28.9022285887376

t_{12} = -3.82096652990166

t_{13} = 12.3044320653985

t_{14} = 32.3044320653985

t_{15} = -33.8209665299017

t_{16} = -56.5289096000264

t_{17} = -97.6955679346015

t_{18} = 62.3044320653985

t_{19} = 93.4710903999736

t_{20} = 6.17903347009834

t_{21} = -47.6955679346015

t_{22} = -43.8209665299017

t_{23} = 83.4356156378233

t_{24} = -73.8209665299017

t_{25} = -96.5289096000264

t_{26} = -70

t_{27} = 72.3044320653985

t_{28} = -86.5289096000264

t_{29} = 10

t_{30} = 98.9022285887376

t_{31} = -50

t_{32} = 92.3044320653985

t_{33} = -67.6955679346015

t_{34} = 16.1790334700983

t_{35} = 70

t_{36} = -36.5289096000264

t_{37} = 40

t_{38} = 33.4710903999736

t_{39} = -26.5289096000264

t_{40} = -10

t_{41} = 58.9022285887376

t_{42} = 26.1790334700983

t_{43} = 22.3044320653985

t_{44} = -71.0977714112624

t_{45} = 68.9022285887376

t_{46} = 8.90222858873759

t_{47} = 52.3044320653985

t_{48} = 76.1790334700983

t_{49} = 82.3044320653985

t_{50} = 56.1790334700983

t_{51} = -60

t_{52} = -37.6955679346015

t_{53} = 53.4710903999736

t_{54} = -66.5289096000264

t_{55} = 38.9022285887376

t_{56} = -83.8209665299017

t_{57} = -63.8209665299017

t_{58} = 100

t_{59} = 88.9022285887376

t_{60} = -40

t_{61} = 3.47109039997363

t_{62} = -100

t_{63} = 43.4710903999736

t_{64} = -21.0977714112624

t_{65} = 78.9022285887376

t_{66} = 66.1790334700983

t_{67} = -57.6955679346015

t_{68} = -76.5289096000264

t_{69} = -13.8209665299017

t_{70} = -11.0977714112624

t_{71} = 23.4710903999736

t_{72} = 46.1790334700983

t_{73} = -23.8209665299017

t_{74} = -61.0977714112624

t_{75} = -51.0977714112624

t_{76} = -6.52890960002637

t_{77} = 90

t_{78} = -81.0977714112624

t_{79} = -17.6955679346015

t_{80} = -87.6955679346015

t_{81} = -90

t_{82} = 86.1790334700983

t_{83} = -77.6955679346015

t_{84} = 80

t_{85} = -16.5289096000264

t_{86} = -41.0977714112624

t_{87} = 50

t_{88} = -46.5289096000264

t_{89} = -20

t_{90} = -7.69556793460146

t_{91} = -30

t_{92} = 96.1790334700983

t_{93} = 48.9022285887376

t_{94} = 63.4710903999736

t_{95} = 20

t_{96} = 42.3044320653985

t_{97} = 30

t_{98} = -31.0977714112624

t_{99} = 2.30443206539854

t_{100} = -53.8209665299017

Signos de extremos en los puntos:

                        /33*sin(0.94218079994727*pi)                      \ 
(13.471090399973635, cos|--------------------------- + 549.620488318924*pi|)
                        \             5                                   /

                         /33*sin(0.19554282252483*pi)                      \ 
(-1.0977714112624148, cos|--------------------------- + 44.7890735795065*pi|)
                         \             5                                   /

(60, 1)

                        /33*sin(1.39113586920292*pi)                      \ 
(-27.69556793460146, cos|--------------------------- + 1129.97917173174*pi|)
                        \             5                                   /

(0, 1)

                       /33*sin(0.358066940196679*pi)                      \ 
(36.17903347009834, cos|---------------------------- + 1476.10456558001*pi|)
                       \             5                                    /

                        /33*sin(0.195542822524828*pi)                      \ 
(-91.09777141126241, cos|---------------------------- + 3716.78907357951*pi|)
                        \             5                                    /

                        /33*sin(1.64193305980334*pi)                      \ 
(-93.82096652990167, cos|--------------------------- + 3827.89543441999*pi|)
                        \             5                                   /

(-80, 1)

                        /33*sin(1.80445717747517*pi)                      \ 
(18.902228588737586, cos|--------------------------- + 771.210926420493*pi|)
                        \             5                                   /

                        /33*sin(1.80445717747517*pi)                      \ 
(28.902228588737586, cos|--------------------------- + 1179.21092642049*pi|)
                        \             5                                   /

                         /33*sin(1.64193305980333*pi)                      \ 
(-3.8209665299016633, cos|--------------------------- + 155.895434419988*pi|)
                         \             5                                   /

                        /33*sin(0.608864130797084*pi)                     \ 
(12.304432065398542, cos|---------------------------- + 502.02082826826*pi|)
                        \             5                                   /

                        /33*sin(0.608864130797087*pi)                      \ 
(32.304432065398544, cos|---------------------------- + 1318.02082826826*pi|)
                        \             5                                    /

                        /33*sin(1.64193305980332*pi)                      \ 
(-33.82096652990166, cos|--------------------------- + 1379.89543441999*pi|)
                        \             5                                   /

                        /33*sin(1.05781920005273*pi)                      \ 
(-56.52890960002637, cos|--------------------------- + 2306.37951168108*pi|)
                        \             5                                   /

                        /33*sin(1.39113586920291*pi)                      \ 
(-97.69556793460146, cos|--------------------------- + 3985.97917173174*pi|)
                        \             5                                   /

                        /33*sin(0.608864130797087*pi)                      \ 
(62.304432065398544, cos|---------------------------- + 2542.02082826826*pi|)
                        \             5                                    /

                       /33*sin(0.942180799947266*pi)                      \ 
(93.47109039997363, cos|---------------------------- + 3813.62048831892*pi|)
                       \             5                                    /

                       /33*sin(0.358066940196673*pi)                      \ 
(6.179033470098337, cos|---------------------------- + 252.104565580012*pi|)
                       \             5                                    /

                         /33*sin(1.39113586920291*pi)                      \ 
(-47.695567934601456, cos|--------------------------- + 1945.97917173174*pi|)
                         \             5                                   /

                        /33*sin(1.64193305980332*pi)                      \ 
(-43.82096652990166, cos|--------------------------- + 1787.89543441999*pi|)
                        \             5                                   /

                       /33*sin(0.871231275646693*pi)                      \ 
(83.43561563782335, cos|---------------------------- + 3404.17311802319*pi|)
                       \             5                                    /

                        /33*sin(1.64193305980334*pi)                      \ 
(-73.82096652990167, cos|--------------------------- + 3011.89543441999*pi|)
                        \             5                                   /

                        /33*sin(1.05781920005273*pi)                      \ 
(-96.52890960002637, cos|--------------------------- + 3938.37951168108*pi|)
                        \             5                                   /

(-70, 1)

                       /33*sin(0.608864130797087*pi)                      \ 
(72.30443206539854, cos|---------------------------- + 2950.02082826826*pi|)
                       \             5                                    /

                        /33*sin(1.05781920005273*pi)                      \ 
(-86.52890960002637, cos|--------------------------- + 3530.37951168108*pi|)
                        \             5                                   /

(10, 1)

                       /33*sin(1.80445717747517*pi)                      \ 
(98.90222858873759, cos|--------------------------- + 4035.21092642049*pi|)
                       \             5                                   /

(-50, cos(1.99999999999977*pi))

                       /33*sin(0.608864130797087*pi)                      \ 
(92.30443206539854, cos|---------------------------- + 3766.02082826826*pi|)
                       \             5                                    /

                        /33*sin(1.39113586920291*pi)                      \ 
(-67.69556793460146, cos|--------------------------- + 2761.97917173174*pi|)
                        \             5                                   /

                        /33*sin(0.358066940196672*pi)                      \ 
(16.179033470098336, cos|---------------------------- + 660.104565580012*pi|)
                        \             5                                    /

(70, 1)

                        /33*sin(1.05781920005273*pi)                      \ 
(-36.52890960002637, cos|--------------------------- + 1490.37951168108*pi|)
                        \             5                                   /

(40, 1)

                       /33*sin(0.942180799947266*pi)                      \ 
(33.47109039997363, cos|---------------------------- + 1365.62048831892*pi|)
                       \             5                                    /

                         /33*sin(1.05781920005273*pi)                      \ 
(-26.528909600026363, cos|--------------------------- + 1082.37951168108*pi|)
                         \             5                                   /

(-10, 1)

                        /33*sin(1.80445717747517*pi)                      \ 
(58.902228588737586, cos|--------------------------- + 2403.21092642049*pi|)
                        \             5                                   /

                        /33*sin(0.358066940196672*pi)                      \ 
(26.179033470098336, cos|---------------------------- + 1068.10456558001*pi|)
                        \             5                                    /

                       /33*sin(0.60886413079708*pi)                     \ 
(22.30443206539854, cos|--------------------------- + 910.02082826826*pi|)
                       \             5                                  /

                        /33*sin(0.195542822524828*pi)                      \ 
(-71.09777141126241, cos|---------------------------- + 2900.78907357951*pi|)
                        \             5                                    /

                       /33*sin(1.80445717747517*pi)                      \ 
(68.90222858873759, cos|--------------------------- + 2811.21092642049*pi|)
                       \             5                                   /

                       /33*sin(1.80445717747517*pi)                      \ 
(8.902228588737586, cos|--------------------------- + 363.210926420493*pi|)
                       \             5                                   /

                        /33*sin(0.608864130797087*pi)                      \ 
(52.304432065398544, cos|---------------------------- + 2134.02082826826*pi|)
                        \             5                                    /

                       /33*sin(0.358066940196665*pi)                      \ 
(76.17903347009833, cos|---------------------------- + 3108.10456558001*pi|)
                       \             5                                    /

                       /33*sin(0.608864130797087*pi)                      \ 
(82.30443206539854, cos|---------------------------- + 3358.02082826826*pi|)
                       \             5                                    /

                       /33*sin(0.358066940196679*pi)                      \ 
(56.17903347009834, cos|---------------------------- + 2292.10456558001*pi|)
                       \             5                                    /

(-60, 1)

                         /33*sin(1.39113586920291*pi)                      \ 
(-37.695567934601456, cos|--------------------------- + 1537.97917173174*pi|)
                         \             5                                   /

                       /33*sin(0.942180799947266*pi)                      \ 
(53.47109039997363, cos|---------------------------- + 2181.62048831892*pi|)
                       \             5                                    /

                        /33*sin(1.05781920005273*pi)                      \ 
(-66.52890960002637, cos|--------------------------- + 2714.37951168108*pi|)
                        \             5                                   /

                        /33*sin(1.80445717747517*pi)                      \ 
(38.902228588737586, cos|--------------------------- + 1587.21092642049*pi|)
                        \             5                                   /

                        /33*sin(1.64193305980334*pi)                      \ 
(-83.82096652990167, cos|--------------------------- + 3419.89543441999*pi|)
                        \             5                                   /

                        /33*sin(1.64193305980332*pi)                      \ 
(-63.82096652990166, cos|--------------------------- + 2603.89543441999*pi|)
                        \             5                                   /

(100, cos(1.99999999999955*pi))

                       /33*sin(1.80445717747517*pi)                      \ 
(88.90222858873759, cos|--------------------------- + 3627.21092642049*pi|)
                       \             5                                   /

(-40, 1)

                       /33*sin(0.94218079994727*pi)                      \ 
(3.471090399973635, cos|--------------------------- + 141.620488318924*pi|)
                       \             5                                   /

(-100, cos(1.99999999999955*pi))

                       /33*sin(0.942180799947266*pi)                      \ 
(43.47109039997363, cos|---------------------------- + 1773.62048831892*pi|)
                       \             5                                    /

                         /33*sin(0.195542822524828*pi)                      \ 
(-21.097771411262414, cos|---------------------------- + 860.789073579506*pi|)
                         \             5                                    /

                       /33*sin(1.80445717747517*pi)                      \ 
(78.90222858873759, cos|--------------------------- + 3219.21092642049*pi|)
                       \             5                                   /

                       /33*sin(0.358066940196665*pi)                      \ 
(66.17903347009833, cos|---------------------------- + 2700.10456558001*pi|)
                       \             5                                    /

                         /33*sin(1.39113586920291*pi)                      \ 
(-57.695567934601456, cos|--------------------------- + 2353.97917173174*pi|)
                         \             5                                   /

                        /33*sin(1.05781920005273*pi)                      \ 
(-76.52890960002637, cos|--------------------------- + 3122.37951168108*pi|)
                        \             5                                   /

                         /33*sin(1.64193305980333*pi)                      \ 
(-13.820966529901664, cos|--------------------------- + 563.895434419988*pi|)
                         \             5                                   /

                         /33*sin(0.195542822524828*pi)                      \ 
(-11.097771411262414, cos|---------------------------- + 452.789073579506*pi|)
                         \             5                                    /

                        /33*sin(0.942180799947273*pi)                      \ 
(23.471090399973637, cos|---------------------------- + 957.620488318924*pi|)
                        \             5                                    /

                       /33*sin(0.358066940196679*pi)                      \ 
(46.17903347009834, cos|---------------------------- + 1884.10456558001*pi|)
                       \             5                                    /

                         /33*sin(1.64193305980333*pi)                      \ 
(-23.820966529901664, cos|--------------------------- + 971.895434419988*pi|)
                         \             5                                   /

                         /33*sin(0.195542822524828*pi)                      \ 
(-61.097771411262414, cos|---------------------------- + 2492.78907357951*pi|)
                         \             5                                    /

                         /33*sin(0.195542822524828*pi)                      \ 
(-51.097771411262414, cos|---------------------------- + 2084.78907357951*pi|)
                         \             5                                    /

                        /33*sin(1.05781920005273*pi)                      \ 
(-6.528909600026365, cos|--------------------------- + 266.379511681076*pi|)
                        \             5                                   /

(90, cos(1.99999999999955*pi))

                        /33*sin(0.195542822524828*pi)                      \ 
(-81.09777141126241, cos|---------------------------- + 3308.78907357951*pi|)
                        \             5                                    /

                        /33*sin(1.39113586920292*pi)                      \ 
(-17.69556793460146, cos|--------------------------- + 721.979171731739*pi|)
                        \             5                                   /

                        /33*sin(1.39113586920291*pi)                      \ 
(-87.69556793460146, cos|--------------------------- + 3577.97917173174*pi|)
                        \             5                                   /

(-90, cos(1.99999999999955*pi))

                       /33*sin(0.358066940196665*pi)                      \ 
(86.17903347009833, cos|---------------------------- + 3516.10456558001*pi|)
                       \             5                                    /

                        /33*sin(1.39113586920291*pi)                      \ 
(-77.69556793460146, cos|--------------------------- + 3169.97917173174*pi|)
                        \             5                                   /

(80, 1)

                         /33*sin(1.05781920005273*pi)                      \ 
(-16.528909600026363, cos|--------------------------- + 674.379511681076*pi|)
                         \             5                                   /

                         /33*sin(0.195542822524828*pi)                      \ 
(-41.097771411262414, cos|---------------------------- + 1676.78907357951*pi|)
                         \             5                                    /

(50, cos(1.99999999999977*pi))

                        /33*sin(1.05781920005273*pi)                      \ 
(-46.52890960002637, cos|--------------------------- + 1898.37951168108*pi|)
                        \             5                                   /

(-20, 1)

                        /33*sin(1.39113586920292*pi)                      \ 
(-7.695567934601459, cos|--------------------------- + 313.979171731739*pi|)
                        \             5                                   /

(-30, 1)

                       /33*sin(0.358066940196665*pi)                      \ 
(96.17903347009833, cos|---------------------------- + 3924.10456558001*pi|)
                       \             5                                    /

                        /33*sin(1.80445717747517*pi)                      \ 
(48.902228588737586, cos|--------------------------- + 1995.21092642049*pi|)
                        \             5                                   /

                       /33*sin(0.942180799947266*pi)                      \ 
(63.47109039997363, cos|---------------------------- + 2589.62048831892*pi|)
                       \             5                                    /

(20, 1)

                        /33*sin(0.608864130797087*pi)                      \ 
(42.304432065398544, cos|---------------------------- + 1726.02082826826*pi|)
                        \             5                                    /

(30, 1)

                         /33*sin(0.195542822524828*pi)                      \ 
(-31.097771411262414, cos|---------------------------- + 1268.78907357951*pi|)
                         \             5                                    /

                        /33*sin(0.608864130797083*pi)                      \ 
(2.3044320653985415, cos|---------------------------- + 94.0208282682605*pi|)
                        \             5                                    /

                        /33*sin(1.64193305980332*pi)                      \ 
(-53.82096652990166, cos|--------------------------- + 2195.89543441999*pi|)
                        \             5                                   /

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:

t_{1} = 83.4356156378233

Puntos máximos de la función:

t_{1} = 13.4710903999736

t_{1} = -1.09777141126241

t_{1} = 60

t_{1} = -27.6955679346015

t_{1} = 0

t_{1} = 36.1790334700983

t_{1} = -91.0977714112624

t_{1} = -93.8209665299017

t_{1} = -80

t_{1} = 18.9022285887376

t_{1} = 28.9022285887376

t_{1} = -3.82096652990166

t_{1} = 12.3044320653985

t_{1} = 32.3044320653985

t_{1} = -33.8209665299017

t_{1} = -56.5289096000264

t_{1} = -97.6955679346015

t_{1} = 62.3044320653985

t_{1} = 93.4710903999736

t_{1} = 6.17903347009834

t_{1} = -47.6955679346015

t_{1} = -43.8209665299017

t_{1} = -73.8209665299017

t_{1} = -96.5289096000264

t_{1} = -70

t_{1} = 72.3044320653985

t_{1} = -86.5289096000264

t_{1} = 10

t_{1} = 98.9022285887376

t_{1} = -50

t_{1} = 92.3044320653985

t_{1} = -67.6955679346015

t_{1} = 16.1790334700983

t_{1} = 70

t_{1} = -36.5289096000264

t_{1} = 40

t_{1} = 33.4710903999736

t_{1} = -26.5289096000264

t_{1} = -10

t_{1} = 58.9022285887376

t_{1} = 26.1790334700983

t_{1} = 22.3044320653985

t_{1} = -71.0977714112624

t_{1} = 68.9022285887376

t_{1} = 8.90222858873759

t_{1} = 52.3044320653985

t_{1} = 76.1790334700983

t_{1} = 82.3044320653985

t_{1} = 56.1790334700983

t_{1} = -60

t_{1} = -37.6955679346015

t_{1} = 53.4710903999736

t_{1} = -66.5289096000264

t_{1} = 38.9022285887376

t_{1} = -83.8209665299017

t_{1} = -63.8209665299017

t_{1} = 100

t_{1} = 88.9022285887376

t_{1} = -40

t_{1} = 3.47109039997363

t_{1} = -100

t_{1} = 43.4710903999736

t_{1} = -21.0977714112624

t_{1} = 78.9022285887376

t_{1} = 66.1790334700983

t_{1} = -57.6955679346015

t_{1} = -76.5289096000264

t_{1} = -13.8209665299017

t_{1} = -11.0977714112624

t_{1} = 23.4710903999736

t_{1} = 46.1790334700983

t_{1} = -23.8209665299017

t_{1} = -61.0977714112624

t_{1} = -51.0977714112624

t_{1} = -6.52890960002637

t_{1} = 90

t_{1} = -81.0977714112624

t_{1} = -17.6955679346015

t_{1} = -87.6955679346015

t_{1} = -90

t_{1} = 86.1790334700983

t_{1} = -77.6955679346015

t_{1} = 80

t_{1} = -16.5289096000264

t_{1} = -41.0977714112624

t_{1} = 50

t_{1} = -46.5289096000264

t_{1} = -20

t_{1} = -7.69556793460146

t_{1} = -30

t_{1} = 96.1790334700983

t_{1} = 48.9022285887376

t_{1} = 63.4710903999736

t_{1} = 20

t_{1} = 42.3044320653985

t_{1} = 30

t_{1} = -31.0977714112624

t_{1} = 2.30443206539854

t_{1} = -53.8209665299017

Decrece en los intervalos

\left(-\infty, -100\right] \cup \left[83.4356156378233, \infty\right)

Crece en los intervalos

\left(-\infty, 83.4356156378233\right] \cup \left[100, \infty\right)

Puntos de flexiones

Hallemos los puntos de flexiones, para eso hay que resolver la ecuación

\frac{d^{2}}{d t^{2}} f{\left(t \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 t^{2}} f{\left(t \right)} =

segunda derivada

\frac{12 \pi^{2} \left(- 3 \left(11 \cos{\left(2 \pi t \right)} + 34\right)^{2} \cos{\left(\frac{3 \left(68 \pi t + 11 \sin{\left(2 \pi t \right)}\right)}{5} \right)} + 55 \sin{\left(2 \pi t \right)} \sin{\left(\frac{3 \left(68 \pi t + 11 \sin{\left(2 \pi t \right)}\right)}{5} \right)}\right)}{25} = 0

Resolvermos esta ecuación
Raíces de esta ecuación

t_{1} = -59.6929767964378

t_{2} = -8.35346744755798

t_{3} = 81.646532552442

t_{4} = -90.5036321561601

t_{5} = -35.7501443423123

t_{6} = 21.646532552442

t_{7} = 84.2498556576877

t_{8} = -68.353467447558

t_{9} = -49.6929767964378

t_{10} = 11.646532552442

t_{11} = 71.4746263338456

t_{12} = 79.1393218681413

t_{13} = -5.75014434231228

t_{14} = -16.6151014900722

t_{15} = 54.2498556576877

t_{16} = 14.2498556576877

t_{17} = -88.353467447558

t_{18} = -79.6929767964378

t_{19} = -59.4963678438399

t_{20} = 69.8359141422877

t_{21} = -24.9907425698983

t_{22} = 91.646532552442

t_{23} = 71.646532552442

t_{24} = -29.6929767964378

t_{25} = 90.3070232035622

t_{26} = 60.3070232035622

t_{27} = -45.7501443423123

t_{28} = 41.646532552442

t_{29} = -18.353467447558

t_{30} = 64.2498556576877

t_{31} = -40.6703248088377

t_{32} = -50.2801760108047

t_{33} = -75.7501443423123

t_{34} = -9.69297679643779

t_{35} = -85.7501443423123

t_{36} = 20.3070232035622

t_{37} = 24.2498556576877

t_{38} = -0.431988140220151

t_{39} = 39.7198239891953

t_{40} = 100.539753915327

t_{41} = 1.64653255244202

t_{42} = 74.2498556576877

t_{43} = -15.7501443423123

t_{44} = 40.3070232035622

t_{45} = -28.353467447558

t_{46} = 44.2498556576877

t_{47} = 61.646532552442

t_{48} = -86.5108927720504

t_{49} = -95.7501443423123

t_{50} = -25.7501443423123

t_{51} = 99.3596016919414

t_{52} = -78.353467447558

t_{53} = 60.9572970100533

t_{54} = 29.4602460846731

t_{55} = 94.2498556576877

t_{56} = -55.7501443423123

t_{57} = 50.3070232035622

t_{58} = 30.3070232035622

t_{59} = 4.24985565768772

t_{60} = -58.353467447558

t_{61} = 0.307023203562212

t_{62} = 51.646532552442

t_{63} = 70.3070232035622

t_{64} = -48.353467447558

t_{65} = -98.353467447558

t_{66} = 80.3070232035622

t_{67} = 10.3070232035622

t_{68} = -39.6929767964378

t_{69} = -65.7501443423123

t_{70} = 34.2498556576877

t_{71} = -70.6984716353416

t_{72} = -89.6929767964378

t_{73} = 31.646532552442

t_{74} = -38.353467447558

t_{75} = -20.467420249636

t_{76} = -99.6929767964378

t_{77} = -69.6929767964378

t_{78} = -19.6929767964378

t_{79} = 100.307023203562

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[100.307023203562, \infty\right)

Convexa en los intervalos

\left(-\infty, -99.6929767964378\right]

Asíntotas horizontales

Hallemos las asíntotas horizontales mediante los límites de esta función con t->+oo y t->-oo

\lim_{t \to -\infty} \cos{\left(t \frac{102 \cdot 2 \pi}{5} + \frac{33 \sin{\left(2 \pi t \right)}}{5} \right)} = \left\langle -1, 1\right\rangle

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:

y = \left\langle -1, 1\right\rangle

\lim_{t \to \infty} \cos{\left(t \frac{102 \cdot 2 \pi}{5} + \frac{33 \sin{\left(2 \pi t \right)}}{5} \right)} = \left\langle -1, 1\right\rangle

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:

y = \left\langle -1, 1\right\rangle

Asíntotas inclinadas

Se puede hallar la asíntota inclinada calculando el límite de la función cos(((2*pi)*102/5)*t + 33*sin((2*pi)*t)/5), dividida por t con t->+oo y t ->-oo

\lim_{t \to -\infty}\left(\frac{\cos{\left(t \frac{102 \cdot 2 \pi}{5} + \frac{33 \sin{\left(2 \pi t \right)}}{5} \right)}}{t}\right) = 0

Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha

\lim_{t \to \infty}\left(\frac{\cos{\left(t \frac{102 \cdot 2 \pi}{5} + \frac{33 \sin{\left(2 \pi t \right)}}{5} \right)}}{t}\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(-t) и f = -f(-t).
Pues, comprobamos:

\cos{\left(t \frac{102 \cdot 2 \pi}{5} + \frac{33 \sin{\left(2 \pi t \right)}}{5} \right)} = \cos{\left(\frac{204 \pi t}{5} + \frac{33 \sin{\left(2 \pi t \right)}}{5} \right)}

- No

\cos{\left(t \frac{102 \cdot 2 \pi}{5} + \frac{33 \sin{\left(2 \pi t \right)}}{5} \right)} = - \cos{\left(\frac{204 \pi t}{5} + \frac{33 \sin{\left(2 \pi t \right)}}{5} \right)}

- No
es decir, función
no es
par ni impar

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