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calc notes lecture 11

calc notes lecture 11 - 4 3 2 f x x x = 1 Example 2 Find...

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Lecture Sections Objectives Assignment 11 2.6 Higher-Order Derivatives 2.6: 1-39 eoo. 43, 47, 51-57 Understanding Goals: 1. To understand the definition of higher-order derivatives. 2. To understand how to use the position functions to determine the velocity and acceleration of moving objects. Notation for Higher-Order Derivatives First derivative: ', y '( ), f x dy dx , [ ] ( ) d f x dx , [ ] x D y Second derivative: ", y "( ), f x 2 2 d y dx , [ ] 2 2 ( ) d f x dx , [ ] 2 x D y Third derivative: '", y '"( ), f x 3 3 d y dx , [ ] 3 3 ( ) d f x dx , [ ] 3 x D y Fourth derivative: (4) , y (4) ( ), f x 4 4 d y dx , [ ] 4 4 ( ) d f x dx , [ ] 4 x D y . . . nth derivative: ( ) , n y ( ) ( ), n f x n n d y dx , [ ] ( ) n n d f x dx , [ ] n x D y The n th-order derivative of an n th-degree polynomial function 1 1 1 0 ( ) n n n n f x a x a x a x a - - = + + + + ��� is the constant function ( ) ( ) ! n n f x n a = . Each derivative of order higher than n is the zero function. Example 1 : Find the fourth derivative of
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Unformatted text preview: 4 3 ( ) 2 f x x x =-. 1 Example 2 : Find the value of (4) (2) f if 1 ( ) f x x = . 2 differentiate differentiate Position function ( ) Velocity function ( ) Acceleration function ( ) s t v t a t ( ) s t ( ) ds v t dt = 2 2 '( ) ( ) d s v t a t dt = = Example 3 : A ball is thrown into the air from the top of a 160-foot cliff. The initial velocity of the ball is 48 feet per second, which implies that the position function is 2 16 48 160 s t t = -+ + where the time t is measured in seconds. Find the height, the velocity, and the acceleration of the ball when 3. t = 3 Example 4 : Find the second order derivative of ( 29 2 | 4 | f x x =-. 4...
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