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Chapter 3 Section 05

Chapter 3 Section 05 - Chapter 3 Section 05 Higher Order...

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1 Chapter 3 Section 05 Higher Order Derivatives page 162
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2 Objectives 1. Find 2 nd and 3 rd derivatives. 2. Interpret 2 nd and 3 rd derivatives of functions. 3. Find a formula for the nth derivative of a function. 4. Work distance, velocity, acceleration applications.
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3 Higher derivatives Since the derivative of f is a function, it may also have a derivative. If so, it is called the 2 nd derivative of f. )] ( [ ) ( 4 )] ( [ ' ' ' ) ( ' ' ' 3 )] ( [ ' ' ) ( ' ' 2 4 4 4 ) 4 ( ) 4 ( 3 3 3 2 2 2 x f D dx y d y x f derivative th x f D dx y d y x f derivative rd x f D dx y d y x f derivative nd = = = = = = = = =
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4 Example: f(t) = t 4 – 7t 2 + 2 f’(t) = 4t 3 – 14t f’’(t) = 12t 2 – 14 f’’’(t) = 24t f (4) = 24 f (5) = 0 f (6) = 0 Etc. f (n) = 0 For any polynomial, f (n) = 0 for n ≥ degree + 1
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5 4 2 2 2 2 2 2 2 ) 1 ( ) 2 2 )( 2 ( ) 2 2 ( ) 1 ( ' ' ) 1 ( 2 ' ) 1 ( ) 1 ( 2 ) 1 ( ' 1 of derivative second the Find + + + - + + = + + = + - + = + = x x x x x x y x x x y x x x x y x x y
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6 Page 165, #34. Finding a formula for f (n) (x). f(x) = x -2 f’(x) = -2x -3 f’’(x) = (-3)(-2)(x) -4 f’’’(x) = (-4)(-3)(-2)(x) -5 f (n) (x) = f (n) (x) =(-1) n (n+1)!(x) -(n+2)
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7 We know that velocity is the rate of change of position. That is, given position, s, velocity, v
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