Slides MH1810 2016 Part 3 6 Linearization

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  MH1810 Math 1 Part 3 Di¤erentiation LinearizationTang Wee Kee Nanyang Technological University Tang Wee Kee (Nanyang Technological University)   MH1810 Math 1 Part 3 Di¤erentiation 1 / 13  Linearization Aim: To approximate  f   ( x  )  near  x   =  a  by a linear function  L ( x  )  through x   =  a . De…nition The linearization of   f   at  a  is the linear function L ( x  ) =  f   ( a ) + ( x   a ) f  0 ( a ) Diagram to illustrate: Tang Wee Kee (Nanyang Technological University)   MH1810 Math 1 Part 3 Di¤erentiation 2 / 13  Remark (a) Note that the equation  y   =  L ( x  ) , i.e., y   =  f   ( a ) +  f  0 ( a )( x   a ) is the equation of tangent of the curve  y   =  f   ( x  )  at  x   =  a .(b) Linearization is a local approximation for  f   at  a  via the tangent of  y   =  f   ( x  )  at  x   =  a .We use the linearization  L ( x  )  to approximate value of   f   ( x  )  for  x   near a , i.e.,  f   ( x  )  L ( x  ) . Tang Wee Kee (Nanyang Technological University)   MH1810 Math 1 Part 3 Di¤erentiation 3 / 13  Remark (c) (OPTIONAL.) There is also quadratic approximation of   f   at  x   =  a .More generally, if   f   can be di¤erentiated  n  times at  x   =  a , we havethe Taylor’s polynomial  P  n ( x  )  (degree  n ) of   f   ( x  )  at  x   =  a , . P  n ( x  ) =  f   ( a ) + ( x   a ) f  0 ( a ) + ( x   a ) 2 2 !  f  00 ( a )+  + ( x   a ) k  k  !  f   ( k  ) ( a )+  + ( x   a ) n n !  f   ( n ) ( a ) . Tang Wee Kee (Nanyang Technological University)   MH1810 Math 1 Part 3 Di¤erentiation 4 / 13
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