График функции
0 -70 -60 -50 -40 -30 -20 -10 10 0 4
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0sin ( x ) + 1 = 0 \sin{\left(x \right)} + 1 = 0 sin ( x ) + 1 = 0 Решаем это уравнение Аналитическое решение x 1 = − π 2 x_{1} = - \frac{\pi}{2} x 1 = − 2 π x 2 = 3 π 2 x_{2} = \frac{3 \pi}{2} x 2 = 2 3 π Численное решение x 1 = − 20.420353265929 x_{1} = -20.420353265929 x 1 = − 20.420353265929 x 2 = − 45.5530935025548 x_{2} = -45.5530935025548 x 2 = − 45.5530935025548 x 3 = − 70.6858351534454 x_{3} = -70.6858351534454 x 3 = − 70.6858351534454 x 4 = − 70.6858343571487 x_{4} = -70.6858343571487 x 4 = − 70.6858343571487 x 5 = 86.3937978309099 x_{5} = 86.3937978309099 x 5 = 86.3937978309099 x 6 = 98.9601692809083 x_{6} = 98.9601692809083 x 6 = 98.9601692809083 x 7 = 23.5619451518571 x_{7} = 23.5619451518571 x 7 = 23.5619451518571 x 8 = − 14.1371674455661 x_{8} = -14.1371674455661 x 8 = − 14.1371674455661 x 9 = 61.2610571125526 x_{9} = 61.2610571125526 x 9 = 61.2610571125526 x 10 = 67.54424230971 x_{10} = 67.54424230971 x 10 = 67.54424230971 x 11 = 42.4115007274741 x_{11} = 42.4115007274741 x 11 = 42.4115007274741 x 12 = − 39.2699076683741 x_{12} = -39.2699076683741 x 12 = − 39.2699076683741 x 13 = − 1.57079639503667 x_{13} = -1.57079639503667 x 13 = − 1.57079639503667 x 14 = − 89.5353901118113 x_{14} = -89.5353901118113 x 14 = − 89.5353901118113 x 15 = 48.6946859012172 x_{15} = 48.6946859012172 x 15 = 48.6946859012172 x 16 = 4.71238874329685 x_{16} = 4.71238874329685 x 16 = 4.71238874329685 x 17 = 80.1106130902139 x_{17} = 80.1106130902139 x 17 = 80.1106130902139 x 18 = − 83.2522048211133 x_{18} = -83.2522048211133 x 18 = − 83.2522048211133 x 19 = − 51.8362783335234 x_{19} = -51.8362783335234 x 19 = − 51.8362783335234 x 20 = 48.6946866365921 x_{20} = 48.6946866365921 x 20 = 48.6946866365921 x 21 = 67.5442408278864 x_{21} = 67.5442408278864 x 21 = 67.5442408278864 x 22 = 92.6769830592094 x_{22} = 92.6769830592094 x 22 = 92.6769830592094 x 23 = 10.9955739381756 x_{23} = 10.9955739381756 x 23 = 10.9955739381756 x 24 = − 45.5530935911043 x_{24} = -45.5530935911043 x 24 = − 45.5530935911043 x 25 = − 45.5530929624673 x_{25} = -45.5530929624673 x 25 = − 45.5530929624673 x 26 = 86.3937984838325 x_{26} = 86.3937984838325 x 26 = 86.3937984838325 x 27 = 17.2787591562062 x_{27} = 17.2787591562062 x 27 = 17.2787591562062 x 28 = 98.9601690454399 x_{28} = 98.9601690454399 x 28 = 98.9601690454399 x 29 = − 58.1194639976905 x_{29} = -58.1194639976905 x 29 = − 58.1194639976905 x 30 = − 32.9867232184024 x_{30} = -32.9867232184024 x 30 = − 32.9867232184024 x 31 = − 89.535390750197 x_{31} = -89.535390750197 x 31 = − 89.535390750197 x 32 = 23.5619444059921 x_{32} = 23.5619444059921 x 32 = 23.5619444059921 x 33 = − 14.1371668370864 x_{33} = -14.1371668370864 x 33 = − 14.1371668370864 x 34 = 92.6769837888103 x_{34} = 92.6769837888103 x 34 = 92.6769837888103 x 35 = 36.1283159497235 x_{35} = 36.1283159497235 x 35 = 36.1283159497235 x 36 = − 39.2699084145515 x_{36} = -39.2699084145515 x 36 = − 39.2699084145515 x 37 = 73.8274274830848 x_{37} = 73.8274274830848 x 37 = 73.8274274830848 x 38 = − 58.1194645939029 x_{38} = -58.1194645939029 x 38 = − 58.1194645939029 x 39 = − 58.1194639046052 x_{39} = -58.1194639046052 x 39 = − 58.1194639046052 x 40 = 73.8274268520838 x_{40} = 73.8274268520838 x 40 = 73.8274268520838 x 41 = 67.5442415586719 x_{41} = 67.5442415586719 x 41 = 67.5442415586719 x 42 = − 26.7035372004893 x_{42} = -26.7035372004893 x 42 = − 26.7035372004893 x 43 = − 95.818575476176 x_{43} = -95.818575476176 x 43 = − 95.818575476176 x 44 = − 76.9690195738024 x_{44} = -76.9690195738024 x 44 = − 76.9690195738024 x 45 = 92.6769843439965 x_{45} = 92.6769843439965 x 45 = 92.6769843439965 x 46 = − 102.101761026058 x_{46} = -102.101761026058 x 46 = − 102.101761026058 x 47 = − 20.4203527465087 x_{47} = -20.4203527465087 x 47 = − 20.4203527465087 x 48 = 4.71239022926564 x_{48} = 4.71239022926564 x 48 = 4.71239022926564 x 49 = 42.4115013353669 x_{49} = 42.4115013353669 x 49 = 42.4115013353669 x 50 = − 26.7035379986821 x_{50} = -26.7035379986821 x 50 = − 26.7035379986821 x 51 = 29.8451297031011 x_{51} = 29.8451297031011 x 51 = 29.8451297031011 x 52 = − 70.6858331259916 x_{52} = -70.6858331259916 x 52 = − 70.6858331259916 x 53 = − 95.8185758680502 x_{53} = -95.8185758680502 x 53 = − 95.8185758680502 x 54 = − 64.4026498988255 x_{54} = -64.4026498988255 x 54 = − 64.4026498988255 x 55 = − 39.2699069219675 x_{55} = -39.2699069219675 x 55 = − 39.2699069219675 x 56 = 17.2787599560783 x_{56} = 17.2787599560783 x 56 = 17.2787599560783 x 57 = − 51.8362786893284 x_{57} = -51.8362786893284 x 57 = − 51.8362786893284 x 58 = − 7.85398119154045 x_{58} = -7.85398119154045 x 58 = − 7.85398119154045 x 59 = − 83.2522055723275 x_{59} = -83.2522055723275 x 59 = − 83.2522055723275 x 60 = 36.1283157235346 x_{60} = 36.1283157235346 x 60 = 36.1283157235346 x 61 = 73.8274274426229 x_{61} = 73.8274274426229 x 61 = 73.8274274426229 x 62 = − 20.4203520060805 x_{62} = -20.4203520060805 x 62 = − 20.4203520060805 x 63 = − 7.85398205280014 x_{63} = -7.85398205280014 x 63 = − 7.85398205280014 x 64 = − 7.85398149665124 x_{64} = -7.85398149665124 x 64 = − 7.85398149665124 x 65 = 48.6946873020308 x_{65} = 48.6946873020308 x 65 = 48.6946873020308 x 66 = 98.9601682515978 x_{66} = 98.9601682515978 x 66 = 98.9601682515978 x 67 = − 83.2522042893833 x_{67} = -83.2522042893833 x 67 = − 83.2522042893833 x 68 = 42.4115007162407 x_{68} = 42.4115007162407 x 68 = 42.4115007162407 x 69 = 80.1106131368654 x_{69} = 80.1106131368654 x 69 = 80.1106131368654 x 70 = 86.3937978869933 x_{70} = 86.3937978869933 x 70 = 86.3937978869933 x 71 = 10.9955747360645 x_{71} = 10.9955747360645 x 71 = 10.9955747360645 x 72 = − 1.57079643188553 x_{72} = -1.57079643188553 x 72 = − 1.57079643188553 x 73 = 23.5619437708833 x_{73} = 23.5619437708833 x 73 = 23.5619437708833 x 74 = 36.1283150875497 x_{74} = 36.1283150875497 x 74 = 36.1283150875497 x 75 = 61.2610563112167 x_{75} = 61.2610563112167 x 75 = 61.2610563112167 x 76 = 29.845130330036 x_{76} = 29.845130330036 x 76 = 29.845130330036 x 77 = − 76.9690203748894 x_{77} = -76.9690203748894 x 77 = − 76.9690203748894 x 78 = − 64.4026491641039 x_{78} = -64.4026491641039 x 78 = − 64.4026491641039 x 79 = − 64.4026502975618 x_{79} = -64.4026502975618 x 79 = − 64.4026502975618 x 80 = − 89.5353906059052 x_{80} = -89.5353906059052 x 80 = − 89.5353906059052 x 81 = 538.783139388541 x_{81} = 538.783139388541 x 81 = 538.783139388541 x 82 = 54.9778710948428 x_{82} = 54.9778710948428 x 82 = 54.9778710948428 x 83 = 4.7123894841958 x_{83} = 4.7123894841958 x 83 = 4.7123894841958 x 84 = 80.1106122287081 x_{84} = 80.1106122287081 x 84 = 80.1106122287081 x 85 = 48.6946870830469 x_{85} = 48.6946870830469 x 85 = 48.6946870830469 x 86 = − 51.8362791922783 x_{86} = -51.8362791922783 x 86 = − 51.8362791922783 x 87 = − 14.1371667858125 x_{87} = -14.1371667858125 x 87 = − 14.1371667858125 x 88 = 29.8451303231501 x_{88} = 29.8451303231501 x 88 = 29.8451303231501 x 89 = − 32.9867224188086 x_{89} = -32.9867224188086 x 89 = − 32.9867224188086 x 90 = − 1.57079581340397 x_{90} = -1.57079581340397 x 90 = − 1.57079581340397 x 91 = − 95.8185763308148 x_{91} = -95.8185763308148 x 91 = − 95.8185763308148 x 92 = 54.9778718908148 x_{92} = 54.9778718908148 x 92 = 54.9778718908148
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:sin ( 0 ) + 1 \sin{\left(0 \right)} + 1 sin ( 0 ) + 1 f ( 0 ) = 1 f{\left(0 \right)} = 1 f ( 0 ) = 1 (0, 1)
Экстремумы функции
Для того, чтобы найти экстремумы, нужно решить уравнениеd d x f ( x ) = 0 \frac{d}{d x} f{\left(x \right)} = 0 d x d f ( x ) = 0 d d x f ( x ) = \frac{d}{d x} f{\left(x \right)} = d x d f ( x ) = первая производная cos ( x ) = 0 \cos{\left(x \right)} = 0 cos ( x ) = 0 Решаем это уравнение x 1 = π 2 x_{1} = \frac{\pi}{2} x 1 = 2 π x 2 = 3 π 2 x_{2} = \frac{3 \pi}{2} x 2 = 2 3 π pi
(--, 2)
2 3*pi
(----, 0)
2 Интервалы возрастания и убывания функции: x 1 = 3 π 2 x_{1} = \frac{3 \pi}{2} x 1 = 2 3 π x 1 = π 2 x_{1} = \frac{\pi}{2} x 1 = 2 π ( − ∞ , π 2 ] ∪ [ 3 π 2 , ∞ ) \left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \infty\right) ( − ∞ , 2 π ] ∪ [ 2 3 π , ∞ ) [ π 2 , 3 π 2 ] \left[\frac{\pi}{2}, \frac{3 \pi}{2}\right] [ 2 π , 2 3 π ]
Точки перегибов
Найдем точки перегибов, для этого надо решить уравнениеd 2 d x 2 f ( x ) = 0 \frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0 d x 2 d 2 f ( x ) = 0 d 2 d x 2 f ( x ) = \frac{d^{2}}{d x^{2}} f{\left(x \right)} = d x 2 d 2 f ( x ) = вторая производная − sin ( x ) = 0 - \sin{\left(x \right)} = 0 − sin ( x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π x_{2} = \pi x 2 = π Интервалы выпуклости и вогнутости: ( − ∞ , 0 ] ∪ [ π , ∞ ) \left(-\infty, 0\right] \cup \left[\pi, \infty\right) ( − ∞ , 0 ] ∪ [ π , ∞ ) [ 0 , π ] \left[0, \pi\right] [ 0 , π ]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( sin ( x ) + 1 ) = ⟨ 0 , 2 ⟩ \lim_{x \to -\infty}\left(\sin{\left(x \right)} + 1\right) = \left\langle 0, 2\right\rangle x → − ∞ lim ( sin ( x ) + 1 ) = ⟨ 0 , 2 ⟩ Возьмём предел y = ⟨ 0 , 2 ⟩ y = \left\langle 0, 2\right\rangle y = ⟨ 0 , 2 ⟩ lim x → ∞ ( sin ( x ) + 1 ) = ⟨ 0 , 2 ⟩ \lim_{x \to \infty}\left(\sin{\left(x \right)} + 1\right) = \left\langle 0, 2\right\rangle x → ∞ lim ( sin ( x ) + 1 ) = ⟨ 0 , 2 ⟩ Возьмём предел y = ⟨ 0 , 2 ⟩ y = \left\langle 0, 2\right\rangle y = ⟨ 0 , 2 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции sin(x) + 1, делённой на x при x->+oo и x ->-oolim x → − ∞ ( sin ( x ) + 1 x ) = 0 \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} + 1}{x}\right) = 0 x → − ∞ lim ( x sin ( x ) + 1 ) = 0 Возьмём предел lim x → ∞ ( sin ( x ) + 1 x ) = 0 \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} + 1}{x}\right) = 0 x → ∞ lim ( x sin ( x ) + 1 ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).sin ( x ) + 1 = 1 − sin ( x ) \sin{\left(x \right)} + 1 = 1 - \sin{\left(x \right)} sin ( x ) + 1 = 1 − sin ( x ) sin ( x ) + 1 = sin ( x ) − 1 \sin{\left(x \right)} + 1 = \sin{\left(x \right)} - 1 sin ( x ) + 1 = sin ( x ) − 1 
