lecture25hout

lecture25hout - MATH 006 Calculus and Linear Algebra...

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Unformatted text preview: MATH 006 Calculus and Linear Algebra (Lecture 25) Albert Ku HKUST Mathematics Department Albert Ku (HKUST) MATH 006 1 / 13 Outline 1 Second Derivatives 2 Concavity 3 Inflection Points Albert Ku (HKUST) MATH 006 2 / 13 Second Derivatives Second Derivatives Definition For y = f ( x ), the second derivative of f , is d 2 y dx 2 ( x ) = f 00 ( x ) = d dx f ( x ) . Remark Similarly, we can define the n th derivative of f by differentiating f n times. The notation for the n th derivative is d n y dx n ( x ) Albert Ku (HKUST) MATH 006 3 / 13 Second Derivatives Example Find the second derivatives of (a) f ( x ) = 3 √ x 2 + 3 x 3 + 1 x 7 (b) y = p x 3 + √ x (c) y = r x + 1 x 2 + 1 Albert Ku (HKUST) MATH 006 4 / 13 Concavity Example Find the first and the second derivatives of (a) f ( x ) = x 2 , and (b) f ( x ) = √ x . Find the relationship between the shape of the graph of a function and the sign of its second derivative....
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This note was uploaded on 12/28/2010 for the course MATH MATH006 taught by Professor Forgot during the Fall '09 term at HKUST.

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lecture25hout - MATH 006 Calculus and Linear Algebra...

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