Derivative by za187cY

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									Derivatives
                                                    Advanced Level Pure Mathematics




Advanced Level Pure Mathematics



     Calculus I                                     4
Chapter 4               Derivatives

          4.1   Introduction                                                 2

          4.2   Differentiability                                            4

                Continuity and Differentiability                             6

          4.3   Rules of Differentiation                                     10

          4.5   Higher Derivatives                                           17

          4.6   Mean Value Theorem                                           22




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Derivatives
                                                                                                  Advanced Level Pure Mathematics


4.1       INTRODUCTION




Let A( x0 , y 0 ) be a fixed point and P( x, y ) be a variable point on the curve y  f (x) as shown on about
                                                               y  y0    f ( x)  f ( x 0 )
figure. Then the slope of the line AP is given by                     or                    . When the variable point P
                                                               x  x0         x  x0
moves closer and closer to A along the curve y  f (x) , i.e. x  x0 . the line AP becomes the tangent line of
                                                                                                              f ( x)  f ( x 0 )
the curve at the point A. Hence, the slope of the tangent line at the point A is equal to lim                                    .
                                                                                                     x  x0        x  x0
This term is defined to be the derivative of f (x) at x  x0 and is usually denoted by f ' ( x0 ) . The
definition of derivative at any point x may be defined as follows.

Definition Let y  f (x) be a function defined on the interval a, b and x0  a, b  .
            f (x) is said to be differentiable at x 0 ( or have a derivative at x 0 ) if the limit

                       f ( x)  f ( x 0 )                                                      dy
              lim                         exists. This lime value is denoted by f ' ( x 0 ) or          and is called the
              x  x0        x  x0                                                             dx x x0

              derivative of f (x) at x 0 .
              If f (x) has a derivative at every point x in a, b  , then f (x) is said to be differentiable on
              a, b .

Remark        As x  x0 , the difference between x and x 0 is very small, i.e. x  x0 tends to zero. Usually,
              this difference is denoted by h or x . Then the derivative at x 0 may be rewritten as
                       f ( x 0  h)  f ( x 0 )
              lim                               . ( First Principle )
              h 0                h

Example       Let f ( x)  x 2  1 . Find f ' (2) .




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                                                                                      Advanced Level Pure Mathematics
                            2    1
                            x cos , x  0
Example       Let f ( x)        x        . Find f ' (0) .
                            0,
                                    x0




Example       If f ( x)  ln x , find f ' (x) .




Example       Let f be a real-valued function defined on R such that for all real numbers x and y,
               f ( x  y)  f ( x)  f ( y) . Suppose f is differentiable at x 0 , where x0  R .
                                              f ( h)
              (a)    Find the value of lim           .
                                         h 0   h
              (b)    Show that f is differentiable on R and express f ' (x) in terms of x 0 .




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Derivatives
                                                                                         Advanced Level Pure Mathematics


4.2        DIFFERENTIABILITY

                                 1
                            x sin , x  0
Example       Let f ( x)        x        . Show that x  0 , f is continuous but not differentiable.
                            0,
                                    x0
Solution



                                                      f (0  h)  f (0)              f (h)  f( 0 )
              By definition, f ' (0)      =    lim                        =   lim
                                               h 0           h               h 0         h
                                                                                              1
                                                                                     h sin
                                                                          =   lim             h
                                                                              h 0      h
                                                                                        1
                                                                          =   lim sin     .
                                                                              h 0      h
                                1
              Since lim sin       does not exist, f is not differentiable at x  0 .
                     h 0       h

                            1

Example       If f ( x )  x , show that x  0 , f is continuous but not differentiable.
                            3




                            x2     , if x  1
Example       Let f ( x)                     .
                           2 x  1 , if x  1

              Find f '  ( 1 ) and f '  ( 1 ) by definition. Does f ' (1) exist? Why?




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Derivatives
                                                                                     Advanced Level Pure Mathematics

Example       Show that f ( x)  x      is not differentiable at x  0 . Find also the derivative of f (x) when

              x  0.




                           x 2  x   x0
Example       Let f ( x)                  , Find f ' (0) .
                            x        x0




Example       Let f ( x)  x 3 . Find f ' (0) .




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                                                                                          Advanced Level Pure Mathematics

                                                   x2    ,x  c
Example       A function f is defined as f ( x)                 .
                                                  ax  b , x  c

              Find a, b ( in term of c ) if f ' (c) exists.




CONTINUITY AND DIFFERENTIABILITY




Theorem       Let y  f (x) . If f ' ( x) exist, then x  0 implies that y  0 .
                             y                                 y
Proof         Since y  (      )x , we have lim y      (       )x
                             x               x 0             x
                                                            
                                                            lim
                                                                    y 
                                                                         
                                                                        lim x
                                                             x 0 x  x 0
                                                                                      ( Since both limit exist )

                                                                    
                                                           f ' ( x) lim x .
                                                                    x 0
                                                                              
              As f ' ( x) exists, f ' ( x) is bounded. Futhermore, lim x  0 and so lim y  0 .
                                                                            x0               x 0


Theorem       If y  f (x) is differentiable at x0 , then f (x) is continuous at x 0 .

Proof         By Theorem, lim y  0 , i.e. lim [ f ( x0  x)  f ( x0 )]  0 .
                              x 0             x0




              Hence, f (x) is continuous at x 0 .

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                                                                                       Advanced Level Pure Mathematics
Remark        We should have a clear concept about the difference between
              (a) f (x) is well-defined at x 0 .
              (b) the limit of f (x) at x 0 exists.
              (c) f (x) is continuous at x 0 .
              (d) f (x) is differentiable at x 0 .


                     D      C         L

                            x2  a2
                                       ,0  x  a
                                  a
                           
Example       Let f ( x)         0      ,x  a .
                           
                            a( x  a )
                                 2   2
                                          ,x  a
                           
                                 x2
              Show f (x) is continuous at x  a and discuss the continuity of f ' ( x) at x  a .




Example       Prove that if f satisfy f ( x  y)  f ( x) f ( y) x, y  R and f ( x)  1  xg ( x) where

              lim g ( x)  1 , then f ' ( x) exist x  R and f ' ( x)  f ( x) . Find f (x).
              x 0




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                                                                                        Advanced Level Pure Mathematics
                                                                                    f '(1 )
Example       Prove that if f satisfies f ( xy)  f ( x)  f ( y) , then f '(x)            , where x  0 and
                                                                                      x
              find f (x) .




                                                                         1
                                                                         x           if x  c, c  0
                                                                        
Example       Find a, b in terms of c for f ' (c) exists where f ( x)  
                                                                        a  bx 2        if x  c
                                                                        
                                                                        




Example       Let f be a real-valued function such that

                          f ( x)  f (a)  ( x  a) 2            (x, a  R)

              Show that f ' ( x)  0 for all x  R .




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Derivatives
                                                                                            Advanced Level Pure Mathematics


Exercise 4A                                                        Name:                  ________________________
1. Determine which of following functions are differentiable at x  0 .
                                                                          1
                                                                    x sin
                           2
                                                                               ( x  0)
     (a)      f ( x)  x   3
                                                   (b)    f ( x)         x
                                                                    0
                                                                              ( x  0)




2.   Find f ' ( x) of the following functions. Also, locate the points at which the function is not differentiable.
                        4x     ( x  3)
     (a)      f ( x)   2                         (b)    f ( x)  x 3  8
                       2 x  6 ( x  3)




                   x3    ,x 1
3.   Let f ( x)                 .
                  ax  b , x  1

     Find the values of a and b so that f is differentiable at x  1 .




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Derivatives
                                                                                        Advanced Level Pure Mathematics


4.3        RULES OF DIFFERENTIATION

Composite functions

d         du                      d            du dv              d        dv   du
   ku  k                            (u  v)                       uv  u  v
dx        dx                      dx           dx dx              dx       dx   dx

         du    dv
            uv
d u                             dy dy du                        dy   1
      dx 2 dx                                                   
dx  v     v                     dx du dx                        dx dx
                                                                       dy


Algebraic functions


d k
   x  kxk 1           where k must be independent of x (usually a constant)
dx


Inverse functions (esp.: inverse of trigo func)


                                 dy   1
 If     y  f 1 ( x)    the n      
                                 dx df ( y )
                                      dy


Trigonometric functions

d                                 d                               d
   sin x  cos x                     cos x   sin x                 tan x  sec2 x
dx                                dx                              dx

d                                 d                               d
   sec x  sec x tan x               csc x   csc x cot x           cot x   csc2 x
dx                                dx                              dx

Logarithmic functions

d x                                               d         1
   e  ex                                            ln x 
dx                                                dx        x

d x                                               d               1
   a  a x ln a                                      log a x 
dx                                                dx           x ln a

Parametric functions (commonly use in Rate of change)
                  dy
              dy dt
                
              dx dx
                  dt


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Derivatives
                                                                                                           Advanced Level Pure Mathematics
Theorem       Chain Rule
              If h  g  f , i.e. h( x)  g ( f ( x)) and f, g are differentiable, then h' ( x)  g ' ( f ( x)) f ' ( x) .


Example       h( x)  tan e sin x ,   h' ( x ) =


                                              =

                                              =      sec2 e sin x (e sin x cos x)


Example       Find the derivatives of the following functions:
                                                                                     x
              (a)    y  x ln(1  x)                              (b)    y
                                                                                    1 x2




                                                                                               1
                                                                                         sin
              (c)    y  (3x  100 ) 60                                  (d)    ye            x
                                                                                                   , where x  0




                                              dy
Example       Let sin( xy)  xy, find            .
                                              dx




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                                                                                                 Advanced Level Pure Mathematics
Example       ( Derivatives of inverse function )
                         d               1
              Prove        (sin 1 x) 
                        dx              1 x2
Solution      Let y  sin 1 x .         x        =   sin y
                                                                  dy
                                      1           =   cos y
                                                                  dx
                                         dy             1
                                            =
                                         dx           cos y
                              d                          1
                                (sin 1 x) =
                              dx                       1 x2

                            d                            -1
Example       Prove            cos1 x            =
                            dx                         1 x2
Solution




              d                               1                        d                  -1
Remark           (sin 1 x)     =                             ,           cos1 x   =
              dx                             1 x2                     dx                1 x2
              d                            1                           d                  -1
                 tan 1 x       =                             ,           cot1 x   =
              dx                         1 x2                         dx               1 x2

              d                                1                       d                   -1
                 sec1 x        =                             ,           csc1 x   =
              dx                         x x2 1                       dx               x x2 1

                            d                                          d x
Example*      (a) Find         ln x                   (b) Find            e
                            dx                                         dx




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                                                                        Advanced Level Pure Mathematics
                      dy
Example       Find         if
                      dx
              (a)    y  a x 1 , where a is a constant.
              (b)    y  x x  sin x .




Example       Find the derivative of following functions
              (a) x 2 sin 3 (1  3x)             (b) (ln (cos 2 x)) 3




                                                     dy
Example       If x  cost and y  sin t , find
                                                     dx




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                                                                                        Advanced Level Pure Mathematics


Exercise 4B                                                                 Name     ________________________

1.   Find the derivatives of the following functions.

     (a)      y  e sin x               (b)   y  tan e x                      (c)       
                                                                                     y  ln( x 2  1)   
                                                                                                        3




                                                                  sin x
     (d)      y  ln(tan e x )                     (e)      y
                                                                 sin 1 x




                          2x 
     (f)      y  tan 1     2 
                                                   (g)      y  (cos x) x
                         1 x 




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                                                                                Advanced Level Pure Mathematics
                                                                    1
                                                              tan
     (h)      y  3x e x                         (i)    y3         x




                    1                                           x

     (j)      yx   x
                                                 (k)    y4   ln x




                                                       d                1
2.   By considering the inverse of cos x , show that      (cos 1 x) 
                                                       dx              1  x2




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                                                                              Advanced Level Pure Mathematics
               dy
3.   Find         if
               dx
                                      1
     (a)      tan 1 y  xy  x 2           (b)    x 2  y 2  x cos y  1
                                      y




               dy
4.   Find         if
               dx

               x  t  cost                      x  cos3 t
     (a)                                    (b) 
               y  1  sin t                     y  sin t
                                                          3




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                                                                                           Advanced Level Pure Mathematics


4.5       HIGHER DERIVATIVES

Definition    If y  f (x) is a function of x , then the nth derivatives of y w.r.t. x is defined as

              d n y d  d n1 y  d n 1 y
                     n 1  if           is differentiable.
              dx n dx  dx 
                                dx n 1

                                                                                                          dny
              Symbolically, the nth derivatives of y w.r.t. x is denoted by y ( n ) , f   (n)
                                                                                                ( x) or        .
                                                                                                          dx n
                           dy      1             d2y     1
Remark             1.                    but       2
                                                      
                           dx      dx            dx     d 2x
                                   dy                   dy 2


                               d 2 y d  dy 
                                       
                               dx 2 dx  dx 


                                                                          dy dy dz    dy             dz
                   2.    If y is function of z , z  f (x) , then                (.     f ' ( x)  ).
                                                                          dx dz dx    dx             dx
                         d 2 y d  dy                            dy           d dy
                              =                =   f ' ' ( x)       f ' ( x) ( )
                         dx 2 dx  dx                            dz           dx dz
                                                               dy         dz d  dy 
                                                 =   f ''(x)       f '(x)   
                                                               dz         dx dz  dz 
                                                                            2
                                                            dy           2 d y
                                                 =   f ''(x)  ( f '(x))
                                                            dz             dz 2


                                        d2y
Example       Let y  sin 1 x . Find        .
                                        dx 2




Example       Prove that y  e  kx (a cos x  b sin x) satisfies the equation y ' '2ky'(k 2  1) y  0 .




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                                                                                                    ( n 1)
Example       If y  xu , where u is a function of x, Prove y             (n)
                                                                                 xu   (n)
                                                                                              nu             where y (n ) and u (n ) are
              the nth derivative of y and u respectively.
                                       1                                    1

              Hence, if y  x n 1e x , prove y ( n )  (1) n x ( n 1) e x .




Example       Find a general formula for the n th derivative of
              (a) y  e ax ( a  R )
              (b) y  sin x
              (c)    y  xm      ( m  R)




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                                                                                                                Advanced Level Pure Mathematics
Theorem       Let f (x) and g (x) be two functions which are both differentiable up to nth order. Then
                     dn              dn
              (a)         kf ( x)  k n f ( x)
                     dx n            dx
                     dn                       dn         dn
              (b)         [ f ( x)  g ( x)]  n f ( x)  n g ( x)
                     dx n                     dx         dx

Theorem       Leibniz's Theorem
              Let f and g be two functions with nth derivative. Then
                           dn                        n
                                [ f ( x) g ( x)]   C rn f   (r )
                                                                     ( x)g ( n  r ) ( x) where f   (0)
                                                                                                          ( x)  f ( x) .
                           dx n                    r 0



Example       Let y  xe ax , where a is a real constant. Find y ( 20) .




                     dn 3
Example       Find        ( x ln x) , x  0 .
                     dx n




Example       Let y  tan 1 x . Show that for n  1 , (1  x 2 ) y ( n  2)  2(n  1) xy ( n 1)  n(n  1) y ( n )  0 .




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                                                                                             Advanced Level Pure Mathematics
                                                              2
                                                            d y       dy
Example       Prove that if y  x 2 cos x, then x 2            2
                                                                  4 x  ( x 2  6) y  0.
                                                            dx        dx
                                                    dny          d n2 y
              Deduce that when x  0 (n  2)( n  3) n  n(n  1) n  2  0
                                                    dx           dx




              (a) Prove if y  e  x , y ( n  2)  2 xy ( n 1)  2(n  1) y ( n )  0 .
                                        2
Example

                          d n  x2      x2           d2                d
              (b) Show if    n
                               (e )  e f ( x) , then    2
                                                           f ( x)  2 x    f ( x)  2nf ( x)  0 .
                          dx                          dx                dx




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                                                                                                  Advanced Level Pure Mathematics


Exercise 4D                                                                                Name    ____________________

                                         x  1
                                                               r

1.   Let r be a real number. Define y                            for x  1 .
                                         x 1

                     dy  2ry
     (a) Show that              .
                     dx x 2  1
     (b) For n  1, 2, 3, , show that
                     ( x 2  1) y ( n 1)  2(nx  r ) y ( n )  (n 2  n) y ( n 1)  0
                                               dky
          where y ( 0)  y and y ( k )             for k  1 .
                                               dx k




2.   Let f ( x)  x n e ax , where a is real and n is a positive integer.
     Evaluate f ( 2 n ) (0) .




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4.6           MEAN VALUE THEOREM

Definition    Let y  f (x) be a function defined on an interval I. f is said to have an absolute maximum at
              c if f (c)  f ( x), x  I and f (c) is called the absolute maximum value.
              Similarly, f is said to have an absolute minimum at d if f (d )  f ( x), x  I and f(d) is
              called the absolute minimum value.


Theorem       Fermat's Theorem
              Let y  f (x) be defined and differentiable on an open interval (a, b). If f (x) attains its
              absolute maximum or absolute minimum (both are called absolute extremum) at x  c ,
              where c  (a, b) , then f ' (c)  0 .


Proof         For any x  (a, b), there exists a real number h such that ( x  h)  (a, b) and ( x  h)  (a, b) .
              Now, suppose f (x) attains its absolute maximum at x  c . Then we have (c  h)  (a,b)
              and (c  h)  (a,b) , and so f (c  h)  f (c) and f (c  h)  f (c) . Now, the left and right
              hand derivatives are given by
                                              f(c  h)  f(c)
                              f'(c)  lim                   0,         ( since f(c  h)  f(c)  0 )
                                      h 0           h
                                          f(c  h)  f(c)
                   and        f'(c)  lim                0.         ( since f(c  h)  f(c)  0 )
                                     h 0        h
              Since f (x) is differentiable at x  c , the left and right hand derivatives must be equal,
              i.e. f '  (c)  f '  (c) . This is possible only if f ' (c)  0 .
              The proof for f (x) attaining its absolute minimum at x  c is similar and is left as an exercise.


Remark        1.    f ' ( p)  0 NOT IMPLIES            absolute max. or min. at x  p .
                   e.g. f ( x)  x 3 at x  0 , not max. and min.




                                                figure
              2.   Fermat's Theorem can't apply to function in closed interval. absolute max. or min may be
                   attained at the end-points. As a result, one of the left and right hand derivatives at c may
                   not exist.
                   e.g. f ( x)  ( x  2) 2  1defined on [ 0, 5] attains its absolute max. at x  5 but its right
                   hand derivative does not exist.
              3.   Fermat' s Theorem can't apply to function which are not differentiable.

                   e.g. f ( x)  x . Not differentiable at x  0 but min. at x  0 .




                                               figure
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Derivatives
                                                                                        Advanced Level Pure Mathematics


Theorem       Rolle's Theorem
              If a function f (x) satisfies all the following three conditions:
              (1) f (x) is continuous on the closed interval [a, b] ,
              (2) f (x) is differentiable in the open interval (a, b) ,
              (3) f (a)  f (b) ;
                    then there exists at least a point   (a, b) such that f ' ( )  0 .


Proof         Since f (x) is continuous on [a, b]  f (x) is bounded
              (i)   m  M , where m (min), M (Max) are constant.
                            m  f ( x)  M , x  [a, b]
                              f ( x)  M , x  [a, b]
                              f ' ( x)  0, x  (a, b)
              (ii) m  M , the max. and min. cannot both occur at the end points a, b.
                    p  (a, b) such that f ( p)  M
                        i.e. f ( p)  f ( x) x sufficiently closed to p.
                        By Fermat's Theorem, f ' ( p) exist and equal to 0.


Example       Define f ( x)  ( x  2) 2  1 on [0,4]. Note that f (0)  f (4)  5 .
              We have f ' ( x)  2( x  2) and so f ' (2)  0 . Since 2  (0,4) , Rolle's Theorem is verified.
              The geometric significance of Rolle's theorem is illustrated in the following diagram.




              If the line joining the end points (a, f (a)) and (b, f (b)) is horizontal (i.e. parallel to the x-axis)
              then there must be at least a point  (or more than one point) lying between a and b such that the
              tangent at this point is horizontal.




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Derivatives
                                                                                    Advanced Level Pure Mathematics
Theorem       Mean Value Theorem
              If a function f (x) is
              (1) continuous on the closed interval [a, b] and
              (2) differentiable in the open interval (a, b) ,
                    then there exists at least a point   (a, b) such that
                          f (b)  f (a)
                                         f ' ( ) .
                              ba


Proof         Consider the function g defined by
                      f (b)  f (a)
              g ( x)               ( x  a)  f (a)  g (x) is differentiable and continuous on (a, b) .
                          ba
              Let h( x)  g ( x)  f ( x)
               h(x) is also differentiable and continuous on (a, b) .
                    We have h(a)  0, h(b)  0.
                   By Rolle's Theorem,       (a, b) such that h ' ( )  0
                                        g ' ( )  f ' ( )  0
                                                               f (b)  f (a)
                                                 f ' ( ) 
                                                                   ba
                                                                              f (b)  f (a)
Remark:       1.    The Mean Value Theorem still holds for a  b . f ' ( p)                .
                                                                                  ba
              2.    Another form of Mean Value Theorem f (b)  f (a)  f ' ( p)(b  a)
              3.    The value of p can be expressed as p  a   (b  a) , 0    1.
                     f (b)  f (a)  f ' (a   (b  a))(b  a)


Example       Use the Mean Value Theorem. show a  b

              (a)   sin a  sin b  a  b

                    cos ax  cos bx
              (b)                    a b , x  0
                           x
                    sin px
              (c)           p,      p  0, x  0 .
                       x




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Derivatives
                                                                                   Advanced Level Pure Mathematics
Example       By using Mean Value Theorem, show that
                                            e y  e a  e a ( y  a)
              for all real values y and a .


Solution      Let f ( x)  e x .
              Case (i) y  a




              Case (ii) y  a


              Case (iii) y  a




Example       Let a, b  R such that a  b and f (x) be a differentiable function on R such that
               f (a)  0 , f (b)  0 and f ' ( x) is strictly decreasing. Show that f ' (b)  0 .




Example       Let f (x) be a continuous function defined on [ 3, 6 ]. If f (x) is differentiable on ( 3, 6 ) and

               f ' ( x)  9  3 . Show that 18  f (6)  f (3)  36 .




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Derivatives
                                                                                          Advanced Level Pure Mathematics

Example       Let P ( x)  a n x n  a n 1 x n 1    a0 be a polynomial with real coefficients.

                   an     a
              If         n1    a0  0, by using Mean Value Theorem, show that the equation P( x)  0 has
                  n 1      n
              at least one real root between 0 and 1.




Example       Let f be a real-valued function defined on (0, ) . If f ' (t ) is an increasing function,
              show that     f (n)  f ' (n)(t  n)  f (t )  f (n)  f ' (n  1)(t  n) (t (n, n  1))




Example       Let f be a real-valued function such that

                                     f ( x)  f ( y)  ( x  y) 2 ,   (x, y  R)

              Show that f is a differentiable function.
              Hence deduce that f ( x)  k for all x  R , where k is a real constant.




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Derivatives
                                                                                                   Advanced Level Pure Mathematics
Example       Let f (x) be a function such that f ' ( x) is strictly increasing for x  0 .
              (a) Using Mean Value Theorem, or otherwise, show that
                                f ' (k )  f (k  1)  f (k )  f ' (k  1) ,                k 1
              (b) Hence, deduce that
                  f ' (1)  f ' (2)    f ' (n  1)  f (n)  f (1)  f ' (2)  f ' (3)   f ' (n) , n  2




Theorem       Generalized Mean Value Theorem
              Let f (x) and g (x) such that
              (i) f (x) and g (x) are continuous on [ a, b ].
              (ii) f (x) and g (x) are differentiable on ( a, b ).
              Then there is at least one points p  (a, b) such that
                                         [ f (b)  f (a)] g ' ( p)  [ g (b)  g (a)] f ' ( p) .


Proof         Let h( x)  [ f (b)  f (a)] g ( x)  [ g (b)  g (a)] f ( x) , a  x  b .
                       (i) h(x) is continuous on [ a, b ].
                        (ii) h(x) is differentiable on ( a, b ).
                                                                                    h(b)  h(a)
              By Mean Value Theorem, p  (a, b) such that h ' ( p)                            , hence the result is obtained.
                                                                                       ba
              ( Why ? )

Remark:       Suppose that f (x) and g (x) are differentiable on ( a, b ) and that f ' ( x)  g ' ( x)  0 ,


                                    f (b)  f (a)   f ' ( p)
              x  (a, b) then                              .
                                    g (b)  g (a)   g ' ( p)

              This is useful to establish an inequality by using generalized mean value theorem.




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Derivatives
                                                                                          Advanced Level Pure Mathematics
Example       (a) Let f and g be real-valued functions continuous on [a, b] and differentiable in (a, b) .

                   (i)   By considering the function

                         h( x)  f ( x)[g (b)  g (a)]  g ( x)[ f (b)  f (a)] on [a, b] , or otherwise,

                         show that there is c  (a, b) such that f ' (c)[g (b)  g (a)]  g ' (c)[ f (b)  f (a)]

                   (ii) Suppose g ' ( x)  0 for all x  (a, b) . Show that g ( x)  g (a)  0 for any x  (a, b) .
                                            f ' ( x)                                              f ( x)  f (a)
                         If, in addition,            is increasing on (a, b) , show that P ( x)                  is also
                                            g ' ( x)                                              g ( x)  g (a )
                         Increasing on (a, b) .
                               e x cos x  1                   
                                                 if       x   0, 
                                                                4
                               sin x  cos x  1                  
              (b) Let Q( x)  
                                       1         if          x0
                              
                              
                                                                             
                   Show that Q is continuous at x  0 and increasing on 0,  .
                                                                             4
                                                                  x
                   Hence or otherwise, deduce that for x  0,  ,  Q (t )dt  x .
                                                            4     0




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