Математика / Математика / Математика / Математика / Математика / Задание 7
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Анализ графиков |
Обычная функция |
Производная функции |
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Геометрический и физический смысл производной
Геометрический смысл производной |
Физический смысл производной |
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′(0) = = |
′( ) = ( ) |
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′( ) = ( ) |
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Задачи на нахождение тангенса
Задачи на нахождение тангенса острого угла |
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Задачи на нахождение тангенса тупого угла |
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′(0) |
′(0) = . |
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′(0) |
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′(0) = . |
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6 |
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= 2 |
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6 |
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= 1 |
= −2 |
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3 |
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Ответ: 1 |
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Ответ: − 2 |
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Задачи на нахождение коэффициента к
Задачи на нахождение к для обычной функции |
Задачи на нахождение к для производной функции |
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= ( ) |
(−9; 8). |
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= 10 |
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= 6 |
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= 10 => |
= 0 |
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= ′( 0) |
′( 0) = 0. |
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= ( ) |
′( 0) = 0 |
= 6 |
=> = 6 |
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= ′( 0) |
′( 0) = 6. |
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′( ) |
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′( 0) = 6: |
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Ответ: 6 |
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4
Ответ: 5
Задачи на нахождение Значения производной
Задачи на нахождение наибольшего значения |
Задачи на нахождение наименьшего значения |
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= ( ) |
−2, −1, 1, 3 |
= ( ) |
−1, 1, 2, 4 |
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−2 |
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3 |
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Ответ: 1 |
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3 |
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Ответ: − 2
Задачи на физический смысл производной
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Задачи на нахождение скорости |
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Задачи на нахождение времени |
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( ) = |
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2 + 4 − 20, |
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( ) = |
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3 + 4 2 − 8 − 16, |
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6 |
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= 6 |
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1 м⁄с ? |
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1 |
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′( ) = ( ) |
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′( ) = ( ) |
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( ) = |
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+ 4 |
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( ) = 2 + 8 − 8 |
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= 1 |
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= 6 |
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1 = 2 + 8 − 8 |
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+ 8 − 9 = 0 |
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(6) = |
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∙ 6 + 4 = 6 |
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= 82 − 4 ∙ 1 ∙ (−9) = 100 |
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Ответ: 6 |
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−8 + √100 |
= 1 |
2 = |
−8 − √100 |
= −9 |
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2 ∙ 1 |
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2 ∙ 1 |
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Ответ: 1 |
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Задачи на построение эскиза обычной |
функции по функции производной |
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Задачи на нахождение наибольшего значения |
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Задачи на нахождение наименьшего значения |
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= ′( ) |
( ) |
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= ′( ) |
( ) |
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(−2; 9). |
[2; 6] |
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(−9; 8). |
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[−8; −4] |
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[2; 6] |
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[−8; −4] |
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[2; 6] |
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[−8; −4] |
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6 |
−8 |
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−8 |
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Ответ: 6 |
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Ответ: − 4 |
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Условие касания функции и прямой
Общая Схема |
Задачи на нахождение абсциссы точки касания |
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Задачи на нахождение параметра |
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′ = ′( 0) |
= 7 − 5 |
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= −3 − 8 |
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{ = ( 0) |
= 2 + 6 − 8 |
2 + 27 + 7 |
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′ = ′( ) |
′ = ′( ) |
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{ = ( ) |
{ = ( ) |
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7 = 2 + 6 |
−3 = 2 + 27 |
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{7 − 5 = 2 + 6 − 8 |
{−3 − 8 = 2 + 27 + 7 |
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7 = 2 + 6 |
2 = −30 |
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= −15 |
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−15 |
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2 = 7 − 6 |
{−3 − 8 = 2 + 27 + 7 |
{−3 − 8 = 2 + 27 + 7 |
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2 = 1 |
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−15 |
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−3 − 8 = 2 + 27 + 7 |
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= 0,5 |
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−15 |
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−15 |
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Ответ: 0,5 |
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{ = |
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{ = |
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−15 |
∙ 2 + 27 + 7 |
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−3 − 8 = |
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−3 − 8 = −15 + 27 + 7 |
−8 − 7 = −15 + 27 + 3 |
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{ = |
−15 |
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{ = |
−15 |
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−15 |
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= 15 |
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−1 |
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−15 = 15 |
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= −1 |
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Ответ: 15 |
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Задачи на применение формулы Ньютона-Лейбница
Формула
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фигуры под графиком = ( ) − ( )
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Задачи на нахождение разности первообразных
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= ( )
(6) − (2), |
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2 6
фигуры под графиком = ( ) − ( )
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(6) − (2) |
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трапеции = |
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трапеции = |
3 + 4 |
∙ 2 = 7 |
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Ответ: 7
Задачи на нахождение площади фигуры
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= ( ). |
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( ) = 3 + 30 2 + 305 − |
7 |
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фигуры под графиком = ( ) − ( )
(−9) − (−11) |
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(−9) = (−9)3 + 30 ∙ (−9)2 + 305 ∙ (−9) − |
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= −729 + 30 ∙ 81 − |
305 ∙ 9 − |
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(−11) = (−11)3 + 30 ∙ (−11)2 + 305 ∙ (−11) − |
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= −1331 + 30 |
∙ 121 − 305 |
∙ 11 |
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(−9) − (−11) = −729 + 30 ∙ 81 − 305 ∙ 9 − |
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+ 1331 − 30 ∙ 121 + 305 ∙ 11 |
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= 602 − 40 ∙ 30 + 2 ∙ 305 = 602 − 1200 + 610 = 12
Ответ: 12