tut4 - UD f(9 = " "$ f(x = 18(252 7 = 723 576...

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Unformatted text preview: UD f (9) = # "  ' "$ f (x) = 18(252) 7 = . 723 576 18(x2 − 9)(4x2 − (x2 − 9)) , (x2 − 9)4 A7  "      %   ' " $ f (x) = −  G$   @ ! H f (x) = 9(2x) , (x2 − 9)2 x2 9 =1+ 2 . 2−9 x x −9 $     ! " 5 B " 4 ( ! @ (  ! A A& 7 C 8  !    T SD (x − 3) = −1(y − 3) ⇒ x + y − 6 = 0.   " 7 "  B " $        & ! " ! $          27 + 27 dy dy dy = 18 + 18 ⇒ = −1. dx dx dx  G$   @ 5 (3, 3) " ! C     R 3x2 + 3y 2 dy dy = 6y + 6x . dx dx Q B " @!77 ! &      B !  " !  $ "  (  P '  # 7 C 8 ! ' !  '   "   I  @ 5   $ #    " ) (x0 , y0) 9 " ! C    $ G(  #    !   " 7 "  B " $     & ! "  ' $ ( B     m (  @ x − x0 = m(y − y0 ), (x0 , y0 ) A% "  G B  " ! C " G B $  $  G(  # $ !  "  B " $     & ! " !  $       5   ( ! 8  8 (  ! A    ( &  ( !     I ! H 9 7 ! ! #  A ( $ ' " ! #   "  "     G $  A 7 % $ % ! ( C  ! A   % 5   $ 7 # "  ( ! &  % " !      & ! ' " 4    '  (  G ! #  G $   @ 4 "   F " ! ' ) ED 9   " ! C A   ( ! @   ! "  8 !    B 7  B  A7 " ! 77 @ ) 9 " !  $ 8 ( !& ") 7 $ (  "  6 (  ' "  5 " % 4( ! 3 2 1 0)    (  ' "  ' "  !&  % " $ #  " !          ¨§ £¢¢¡  © ¨§ ¦ ¥ ¤ $D 9   ! '    ( ! 8  % " $ #  5   (  ! # & !   ¢ 9  7   F 7 $  C F 1 " $   7  & (  @! C  ( ! 8  ) 9 ' !    8 4 ( ! @  (   $   ! 87 $  ' !    8     $     B $ " $G ' $    9  " !  " $ C ¡  ( ! 7 A$ '  (  G! #  $    (  ! #    (   & $ ' !    8    " $ 7 C ¡  A #  ( % 7 7 @ ) x 1 x = lim − 2 + f (3) + ... 1/4 x→0 4(27) 2! 1 = −2 4(27)1/4 (3 + x)1/4 − 31/4 − 2x lim = lim x→ 0 x→ 0 x 1 x 4(27)1/4 + f (3) x − 2x + ... 2! 2  G$   @  # "  5 "      " ! # "  ( $ x2 (  & $  8 (       $     ! " (3 + x)1/4 = 31/4 + 1 x2 x + f (3) + ... 4(27)1/4 2!  G$   3 9  " ! (   $ 7    (  G! # 7 7 @   (  ! #    5  ! " &  " T 8 ! ( & 77 $ #  (   B 8  ! A & !  8 !  #  @ 5 ! %$ & ! " !  " $ C ¡  ( !7 A$           9  8 7      $  7 $ G  !  A $ @ (    ! " $   (    5 '     (   "  ( $ !  @   !   ( !! (3 + x)1/4 x=0 9  7   F 7 $  C F 1    5  8 7      $  7 $ G  !  5 A $ @ A " R ( !$7 ¤  # 7 $ # (  ! A       I 5 8 (   ¤ ' 8    " 5 A 7   ! (     % 9 9 9 9 @ ! @ 9    ¡   8 7    ! H 9  "  7 $ G     ( $ " B ( !     $   8 7 ' " $    B ( ' " $  &  7    ! H 9 "  G    " !  # "  & ' ' ! ! @ & !  "   !      @!  9 " B ( !    & ! A " # G $ " ' ' !  ( !  $ (  8  "     $    @!   " !    ( C ¡    $ 7    (3 − x)1/4 + (3 + x)1/4 ≈ 31/4 + 31/4 ⇒ (3 − x)1/4 + 2x − 31/4 ≈ −((3 + x)1/4 − 2x − 31/4 ). x 77 $ 8  ( ! &  # "  " B ( !    & ! ' ! !  (  ! %  B  " 77 $ 8  $ ( !& ' ' !  ( !  $ (  8 "    9 " B ( !      ! % $ ' ' !  ( !  $ " 8 ! "  '     $     !  c c 9   ¡  77 @  8 7        ' " $ 5  8 $      % 77 @  $  8 7 ' " $   B ( ' " $ ' " $  & 7    "    5 & ! ' ! !  (  ! %  B  " 77 $ 8  $   ! % $ "  G   B " (  '  " ! #  ( $  @  $   " !  # "  &     $   @!   " $ #  @ & A77 $ #  $ ¢ 9 77 $ ( ! & & " ! C $   ! % $     " !  # " & $  $   ' " R 9 77 $ ( !& & " ! C $   ! % $      " ! # " & $ $   77 $ #   9 7 $    ( $   8 7 ' " $   B ( ' " $ ' " $  & 7     $   @!   !   G$   @ 5    ¡  8 7     $   @!   !  " $ @  @ &) x c c f (c + x) = f (c − x) f (c + x) = −f (c − x) x (3 + x)1/4 − 31/4 − 2x . lim x→ 0 x B " @!77 ! &    !  " !    ( C ¡     A& 7 C 8  A77 $   # $ " $ #  @  $     ! " 5   7  %    % $     A 7 (  C ! ( C    ! ' !    ¢ 9 ( !  $ 7  # 7 $ # $ " ! !      ! 7 #   #  @ 5 © ¨ § ¦ ¥9 ¤ !  7 $     " !    ( C ¡      $   ' " £ 7 7 @  ! A ' " $ 5 4 $  5   $ #    ") 9 ( !  $7  #7 $ # (  !A     I " $ #  ! A 5  ! " ( !   ¡  8 7    (     @   @ 7 % $  ( ! & 8 ! # "   ( $  ! A (  G  & 5 (  % 8  8  (   ¢ 9 " !    ( C ¡  A7 B  " $  4 7  4 ! !7    ±10 −1.890327166 x= (E ) = −5 0 = 2 3/2 sin x 3 0 π /2 0 = π /2 √ 0 = π /2 sin x − sin3 xdx = 0 π π cos x cos x ≥ 0 x ∈ [0, π/2] √ √ 2 3/2 − sin x 3 π/2 sin x cos xdx + π sin x| cos x|dx + π/2 = π √ 4 3 sin x(− cos x)dx π/2 π √ sin x| cos x|dx sin x| cos x|dx cos x ≤ 0 |·| x ∈ [π/2, π ]  G$   @ ! H 9 (!& '"$ ( !&  G$   3 9     G! 8  ( !   $ !  5  G  $ B  "   G   ! C  (  @ !  7 $ ( B   "     7 C  !   5 " B   7  ' ! 8 $  G$   @  # "  5 ! ' !  B "    ¡   9 ¤   ¦  ¦      ¨ ¤ ©  ¦ ¥  ¦      ¨ ¤ ¤      ¤      ¨        ©  ¨ ¤ ¤  § ¦  ¥ ¤£    5 " $ B $  # " ¢ 0 = 0 sin x − sin3 xdx = 0 π √ sin x| cos x|dx 0 sin x(1 − sin2 x)dx = π π π √ sin x cos2 xdx $  !  ¡D #"  √ x2 ≥ 0 √ 9 |·|     G! 8  ( '7  ! #  @  (   @ f (g (x))g (x) = 4x3 sin f (t) = sin x4 = 4x3 sin |x2 | = 4x3 sin x2 , t, g (x) = x4  G$   @     ' " $ 5   $ #    "  G$   @ 5 $ 7  8 ( ! &    (  % 8  8  (    T 9 9 9  )  ( $ C   7  # 7 $ # & ! 8  ( !    7 $  "  8 $ ' "  & (  ! A " $ B R D M = f (4) = 37, m = f (2) = −15. 6(x2 − x − 2) = 0 ⇒ x = 2, x = −1. f (2) = 16 − 12 − 24+5 = −15 f (−1) = −2 − 3+12+5 = 12 f (−2) = −16 − 12 + 24 + 5 = 1, f (4) = 128 − 48 − 48 + 5 = 37  G$   @ ! H 9 ' "$ 5  G$   3 9   " ! C ! @      4#  # !   G$   @ !  5 " $ 8 ! '    " B " ! 7  %   " ! C ! @      9 '"$   " ! C ' "  ! @    4#  # !  7 $  @ 5 7 $ G (   " '   ! 7 # $  " $ 8 ! '  (   @ 4#  #  @ 5 " 8   7 !  % $ ' " $ ¡ $ 8   7 !  % $ 4#  # !  " $ B R −2 f (x) = 0 4 f (x) = 6x2 −6x−12 =  # "  ' "$ 5 ¨ SD x→ 0 f (0) = lim f (x) = 2.  G$     8  @ !  tan 8x lim = lim x→0 tan 4x x→ 0 sin 8x 8x sin 4x 4x cos 4x 2 = 2. cos 8x  G$   @   ¢ x→ 0 lim f (x) = f (0).  G$     8  @ 5 " !  " £  ' A % 5 x=0 $   ! " "!#  % ! "! # " &   ( !& 5    x→ c lim f (x) = f (c). c & " ! C $  $   ! " " ! #  " ! # " & $  $   77 $ #   ¢D 9 "B  G $ B " 7$ "!  ' '$ "$  %     G! 8  (  ! A "   @  $   !  5 8 !(&  8 7   B " 4$ ($  @ # "  $  ! "  T 9 B " G7 ! G "  " !     & ! 4#   ( $  A  B  ! A 4 "   ) |·| − x+3 < 0 |·| '7  !    (    5 $ #   " x→3 ED 9 sin x x2 ( ! &   @  4 7 lim | x→∞ 1 x2 cos x 1 cos x | ≤ lim 2 = 0 ⇒ lim = 0. x→∞ x x→∞ x2 x2  G$   @ 5 | cos2x | ≤ x # "  $  !  21092002 − 275 cos2x − 50 21092002x − 275 cos x − 50x2 x x = lim x→∞ x→∞ 2x2 + 398x + 96 sin x 2 + 398 + 96 sin2x x x = −50/2 = −25 lim " $%! ! ( !$ " 8 ! " ' ' "$ (!$( 8 "   ! % "! A% " !    ( C ¡     ' G ' ! H 9   # "   8 (       ( $ #  3 9 "  @ ( !$ " 8 ! " '   ' "$ ( !$( 8 "      ! % ( ! &  8 (    " $ " 8 ! '     ( $  $  @ &7   (  ! A 4  $ '7  !    ! A ! H 9 (  @  8 7    ' " £ !  '  (    ( A7 " !  ( $  @ 5    " $ #  @  $   ¢ 9 " !    ( C ¡  '   $ # 7 C 8 ! # $ " $ B R 1, | sin x| ≤ 1 x→∞ x2 x→∞ | cos x| ≤ x2 ¡D 9 A ( ! @   ! "  ( $ ' " $    8 (   ¤ ' 8 (  ! ( ! &  C ! #     (  ' "   ( $  $    " !          B  ! (   !     I 77 @ ) 9   B " $ #   % $ 77 A  !    '  % '7  ! " 9 § § ' " $ ¨ § § & !  $   " $   (  ' ( $  A7  " $ # £ " B      8 (   ¤ ' 8     $     !    ¨§  © ¨§ ¦ ¥ ¤ ¡ ¢¢¡ 9    #  ( A C C $  ' " $     8 (   ¤ ' 8 (  ! A ( ! & 4# 7 ' ! ! B 5  ! " &) 9   %  ! ' B " " $ 8  ( (  !A & ! A" $ A& ( $7 # " $ # )  $   ! C    8 " $ #  @  8   $  @    !  7 $ 8  " $ ' "   5 C 7   '   " A7 7 $  ( !  @   !   ( ! & 5 A$ '   " '  3 ' " $ A$ '    " !   C 8 $ #    " !  % 77 @ ) ' " $ 5  8 '  4  $  ! A & !  8 !   # "    % 5 4 $  ( % 8 (      B " (  '  (  !  " !  $ 7   " ! # ! "  ( $  (    A 7 7 $ 8 ( !  9 8 ! #9 7 $ 8 B ¡  $ % # " $ ( & !  7 $ 8  " $ ' "     $  7 C 5  " !      A " R 1 f (t) = − 2 . t  G$   @ 5 t = e−x      %    @ &     ' " R f (e−x ) = −e2x = − e−2x 1 . f (g (x))g (x) = f (e−x )(−e−x ) = ex ,   ' "$ g (x) = e−x !H 9  G $   @ 5   $ #    " ) 9 $ 7  8 ( ! &     G (  ' !  @ !  ' ! !   (  ' "   G $   ! A  $   (     % F    (  ! # & ¢ 9 # " $ C F " ! '    T 9 B " ! ( @ ! B F " $ #  ! A 5 $ 7  8 ( ! &    (  % 8  8  (  ! A & 5 ' $  )  4 7 5 " $ B $   ¢ 9   7  # 7 $ " & ! 8  ( !    7 $  "  8 $ ' " !    B " A 7 C C $ & ! 8 ( ! &  "  (  P ' $     @ 5 A 7 7 $ " ! ¡¡ D 9 4a = b = c  G$   @  " !  $ 7  ( ! @    B " " % 8 ! # ! H • 2a(2) = b • a(2)2 = 2b − c  G$     8  @     •  $   ! " " ! # • x=c  $   ! " " ! # ! 7 $  % !  G  $G (  '   ' " $ x=c x=c  % ! " ! # " &   $  (  (  @ 5 " ! C $  $ 7 % $ "  (  P '  % !  " !  # "  &    ( !! 9 " !        ! G  ( C    & ! " !  $  "  " ! # $ & !  ( !   8 !   4 7     ...
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