# 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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## This note was uploaded on 11/30/2011 for the course EEE 1001 taught by Professor Phoon during the Spring '11 term at National University of Singapore.

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