Applications of Differential Calculus
Advanced Level Pure Mathematics
Advanced Level Pure Mathematics
5
Calculus I
Chapter 5
Applications of Differential Calculus
5.1
L’Hospital’s Rule
2
5.4
Monotonic Functions
5
Proving Inequalities by Using Differential Calculus
6
5.5
Maxima and Minima
9
5.7
Curve Sketching
22
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Page 1
Applications of Differential Calculus
Advanced Level Pure Mathematics
5.1
L' Hospital's Rule
Theorem
The limit of a non-constant function f ( x ) as x tends to x0 is said to be an indeterminate of
the form .
(i)
g (x)
1 x
0
.
, where lim g1( x )  lim g2 ( x )  0, e.g. lim
if f ( x )  1
x 1 2  2 x 2
x  x0
x  x0
0
g2( x )
(ii)
g (x)

tan 3x
, where lim g1 ( x)  lim g 2 ( x)  , e.g. lim
if f ( x )  1
.

x  x0
x  x0

g2( x )
x  tan x
2
(iii)
0 0 if f ( x )  [ g1( x )] g 2 ( x ) , where lim g1 ( x )  lim g 2 ( x )  0, e.g. lim x x .
(iv)
0
(v)
1
(vi)
0
x x0
x x0
x 0
Remark
   ,    and   are not indeterminate forms because none of limits of these forms can
exist.
Theorem
L'Hospital's Rule
g (x)
g ( x)
g '( x)
0

If lim 1
is an indeterminate of the form
or
, then lim 1
 lim 1
.
x  x0 g ( x )
x  x0 g ( x)
x  x0 g ' ( x )
0

2
2
2
Example
(a) Evaluate lim
x 1
ln x
x 1
(b) Evaluate lim
x 0
ln x
.
cot x
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Page 2
Applications of Differential Calculus
Advanced Level Pure Mathematics
x 1
xe
.
x 1 ( x  1 ) 2
Example
Evaluate lim
Example
(ln x ) 2
Evaluate lim
x 
x
Example
ln ( 1  xe2 x )
Evaluate lim
x 
x2
Example
Evaluate lim
x 0
x  sin x
x3
Prepared by K. F. Ngai(2003)
Page 3
Applications of Differential Calculus
Example
Advanced Level Pure Mathematics
( Other Indeterminate Forms )
Evaluate lim ( x 
x

2

) tan x.
2
Example
Evaluate lim (tan x ln x ).
Example
Evaluate lim x x .
Example
Evaluate lim ( 1 
Example
Evaluate lim (
x 0
x 0
x 
x 1
1 1 2x

)
x x2
x
1

)
x  1 ln x
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Page 4
Applications of Differential Calculus
Advanced Level Pure Mathematics
It is important that before applying L'Hospital's rule, we should check each step whether the limit under
consideration is an indeterminate or not, if the limit is not an indeterminate or is an indeterminate but not of
the form
0

or
, L'Hospital's rule is not applicable.
0

In the following, which step is wrong?
e x  ex
lim
x 0
sin x
=
e x  ex
lim
x 0
cos x
=
e x  ex
lim
x 0  sin x
=
e x  ex
lim
x 0  cos x
=
e0  e0
 cos 0
=
2
5.4
Monotonic Functions
Theorem
Let f ( x ) be continuous on [a, b] and differentiable on (a, b). f ( x ) is a constant function if
and only if f ' ( x )  0 for all x  ( a ,b )
Definition
A function f ( x ) is said to be monotonic increasing ( resp. monotonic decreasing ) or simply
increasing ( resp. decreasing ) on an interval I if and only if x1 , x2  I , if x1  x2 then
f ( x1 )  f ( x2 ) (resp. x1 , x2  I , if x1  x2 , then f ( x1 )  f ( x2 ) ).
Definition
A function f ( x ) is said to be strictly increasing ( resp. strictly decreasing ) on an interval I
if and only if x1 , x2  I , if x1  x2 then f ( x1 )  f ( x2 ) (resp. x1 , x2  I , if x1  x2 , then
f ( x1 )  f ( x2 ) ).
Theorem
Let f ( x ) be continuous on [a, b] and differentiable on (a, b). Then
(a) if f ' ( x)  0, x  ( a ,b ), f ( x ) is strictly increasing on [a, b]; and
(b) if f ' ( x)  0, x  ( a ,b ), f ( x ) is strictly decreasing on [a, b].
Example
Prove that f ( x )  x 3 is strictly increasing on R .
Prepared by K. F. Ngai(2003)
Page 5
Applications of Differential Calculus
Advanced Level Pure Mathematics
Proving Inequalities by Using Differential Calculus
In Practical problems, we always encounter inequalities within a certain range such as:
f ( x )  g( x )
a  x  b
Usually, it is transformed to
F ( x )  f ( x )  g( x )  0
a  x  b
Based on the properties of increasing function and decreasing function, we can establish inequalities and
the method is outlined in the following
Making Use of Strictly Increasing or Decreasing Functions
Want to prove that
f ( x )  g( x )
(1) Consider
F ( x )  f ( x )  g( x )
Try to prove that
Hence, we have
F' ( x )  0
a  x  b
F ( x ) is strictly increasing on a ,b
(2) Try to prove that
Then we have
F ( x ) is continuous on a ,b
F ( x )  f ( x )  g( x )  F ( a )
(3) Try to prove that
F( a )  0
Then we can conclude that
f ( x )  g( x )
a  x  b
Making Use of The Greatest and Least Values of a Function
Want to prove that
f ( x )  g( x )
(1) Consider
F ( x )  f ( x )  g( x )
Try to prove that
F ( c ) is the least value in a ,b
(2) Try to prove that
Then we have
F ( x ) is continuous on a ,b
F ( x )  f ( x )  g( x )  F ( c )
(3) Try to prove that
F( c )  0
Then we can conclude that
Example
f ( x )  g( x )
x  c 
a  x  b, x  c
Show that sin x  x for all x  0 .
Prepared by K. F. Ngai(2003)
Page 6
Applications of Differential Calculus
Example
Advanced Level Pure Mathematics
Let k be an integer greater than 1 . Show that
x k  k  1  kx
 x  0
When does the equality hold?
Example

1
1
Prove that 0  x sin x  sin 2 x  (   1 ) x  ( 0 , ) .
2
2
2
Example
Prove e y  e a  e a ( y  a ) .
Example
Prove if x  0, x 
x2
x 2 x3
 ln ( 1  x )  x 

2
2
3
Prepared by K. F. Ngai(2003)
Page 7
Applications of Differential Calculus
Example
Advanced Level Pure Mathematics
e
x
for x  0 .
ex
Find the greatest value of f (x) .
Let f ( x) 
Hence show that e    e .
Example
Suppose f ( x ) satisfy (i) f ( x ) is continuous for x  0 .
(ii) f ' ( x ) exists for x  0 .
(iii) f ( 0 )  0
(iv) f ' ( x ) is increasing on 0,
Let g ( x ) 
f(x)
, show g is increasing.
x
Prepared by K. F. Ngai(2003)
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Applications of Differential Calculus
Advanced Level Pure Mathematics
5.5
Maxima and Minima
Definition
A neighborhood of a point x 0 is an open interval containing x 0 , i.e. ( x 0  δ , x 0  δ ) is
a neighborhood of x 0 for some δ  0 .
Definition
A function f ( x ) is said to attain a relative maximum ( minimum ) at a point x 0 if
f (x)  f (x 0 ) ( f (x)  f (x 0 ) ) in a certain neighborhood of x 0 , i.e. δ  0 such that
f (x)  f (x 0 ) ( f (x)  f (x 0 ) ) for x  x 0  δ
Theorem
Fermat Theorem
Given f ( x ) is a point defined on (a, b ) and differentiable at a point x 0 if f ( x ) has an
extreme value ( max. or min ) x 0 , then f ' ( x 0 )  0 .
Note
f ' ( x 0 )  0  f ( x ) has maximum or minimum at x 0 .
Definition
(a) A turning point is a maximum or minimum point.
(b) If f ' ( x)  0 , then x is called a critical or stationary value and its corresponding point
on the graph y  f ( x ) is called stationary point.
Notes
1.
2.
turning point
stationary point
turning point + differentiable  stationary point
3.
stationary point
turning point

Therefore, in searching extreme value of a function, we have to investigate not only the
stationary points < f ' ( x)  0 >, but also the points where the functions is not differentiable.
Theorem
Suppose that the function f ( x ) has a continuous derivatives f ' ( x)  0 which vanishes only
at a finite no. of points, then the function has maximum (minimum) at point x 0 if and only if
f ' ( x ) is  ve ( ve) at points immediately to the left of x 0 and  ve (+ve) immediately
to the right of x 0 .
Theorem
f ( x ) is a function of x , if f ' ( x 0 )  0 and f ' ' ( x 0 ) exists such that
(i) f ' ' (x 0 )  0 , then f ( x ) attains minimum at x  x 0 .
(ii) f ' ' (x 0 )  0 , then f ( x ) attains maximum at x  x 0 .
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Applications of Differential Calculus
Example
Example
Solution
Advanced Level Pure Mathematics
2
3
Find the maximum or minimum points of y  ( x  1 )x  3
2
3
Find the local extreme of the function f ( x )  x ( x  1)
f ' ( x) 
1
3
3x  2
1
3
3x ( x  1)
2
3
2
, 1 , f ' ( x)  0 or f (x ) is not differentiable.
3
These three points divide (,) into four intervals.
2
2
2
0 x
x
 x 1
x0
x0
3
3
3
When x  0,
x
x 1
x 1
f ' ( x)
f (x )
change sign
Maximum point (
Example
Solution
x
change sign
,
)
Minimum point (
,
)
Find the maximum or minimum points of y  ( x  2) 2 ( x  1) 3
f ' ( x)  ( x  2)( x  1) 2 (5x  4)
x
x
x
x
x
x
x
f ' ( x)
f (x )
Maximum point (
,
)
Minimum point (
,
)
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Applications of Differential Calculus
Advanced Level Pure Mathematics
Given that f (x) is continuous on [a, b] , if any x1 , x2  (a, b) such that
Definition
(i)
 x  x 2  f ( x1 )  f ( x 2 )
f 1

2
 2 
Concave Downward
 x  x 2  f ( x1 )  f ( x 2 )
(ii) f  1

2
 2 
Concave Upward
*Theorem
If f (x) is a function on [a, b] such that f (x) is second differentiable on (a, b) then
(i) f ' ' ( x)  0 iff f (x) is concave upward on (a, b)
(ii) f ' ' ( x)  0 iff f (x) is concave downward on (a, b) .
Example
If f ( x )  x 3  3x 2  4 x  2 , determine intervals on which the graph of f is concave upward
or is concave downward.
Example
1
3
If f ( x)  x , determine intervals on which the graph of f is concave upward or is concave
downward.
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Applications of Differential Calculus
Advanced Level Pure Mathematics
Point of Inflection
Definition
Let f ( x ) be a continuous function. A point (c, f (c)) on the graph of f is a point of
inflexion (point of inflection) if the graph on one side of this point is concave downward and
concave upward on the other side. That is, the graph changes concavity at x  c .
A point of inflexion of a curve y  f ( x ) must be a continuous point but need not be
Note
differentiable there. In Figure (c), R is a point of inflexion of the curve but the function is
not differentiable at x 0 .
Theorem
If f ( x ) is second differentiable function and attains a point of inflexion at x  c , then f '' (c)  0 .
Note:
(i)
max. or min. point but not derivative.
(ii)
point of inflexion may not be obtained by solving f '' ( x)  0 where f ' (c)  
and f ' (c)   such that f ' (c)f ' (c)  0 .
(iii)
Let f ( x ) be a function which is second differentiable in a neighborhood of a point
of inflexion iff f ' ( x ) does not change sign as x increases through (sign gradient
test)
 if f ' (c)  0 and f ' (c)f ' (c)  0 , then f ( x ) attains a relative max. or
relative min.
 if f ' (c)  0 and f ' (c)f ' (c)  0 , then f ( x ) attains an inflexion
point at c .
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Applications of Differential Calculus
Example
Advanced Level Pure Mathematics
Find the points of inflexion of the curve y  x  6 x  8 x  10 .
4
2
y'  4 x 3  12 x  8
y' '  12 x 2  12  12( x  1)( x  1)
When
x  1 , y ' '  0 .
x
x  1
x  1
1  x  1
x 1
x 1
y''
y
change sign
change sign
The curve has points of inflexion at x  1 . These two points are (1,3) , (1,13) .
Example
Find the points of inflexion of the curve y  3x 5  5 x 4  4 .
y'  15 x 4  20 x 3
y' '  60 x 3  60 x 2  60 x 2 ( x  1) .
When x  0 or 1 , y ' '  0 .
x
x0
x0
0 x1
x1
x1
y''
y
The curve has a point of inflexion at x  1 . This point is (1,2)
At the point of inflexion, the first derivative may not be zero.
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Page 13
Applications of Differential Calculus
Example
Advanced Level Pure Mathematics
Find the points of inflexion of the curve y  3  5 ( x  2 )7 .
y'  
75
( x  2) 2 ,
5
y' '  
14
25 ( x  2) 3
.
5
When x  2 , y ' ' does not exist
(i.e. y' is not differentiable there.)
x
x  2
x  2
x  2
y''
y
The curve has a point of inflexion at x  2 , which is (2,3)
Example
2
3
Find the points of inflexion of the curve f ( x )  x ( 5  x ) .
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Page 14
Applications of Differential Calculus
Example
Advanced Level Pure Mathematics
Find the points of inflexion of the curve
2
2
(a)
f ( x)  ( x  2) 3  ( x  2) 3
(c)
f ( x) 
(a)
(0,0)
f ( x) 
ln x
x
;x  0
x
(1  x 2 ) 2
8
Ans:
(b)

4
(b) (e 3 ,8e 3 )
(c)
1
1
(1, ) , (0,0) and (1, ) .
4
4
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Page 15
Applications of Differential Calculus
Example
Advanced Level Pure Mathematics
(1) Determine the max. and min. points of f ( x ) 
x.
(2) Determine whether there is min., max. and point of inflexion f ( x ) 
xx
, x  1,1 .
1 x2
(3) Find the max., min. and point of inflexion of the curve y  f ( x )  x 4  6 x 2  8 x  10
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Applications of Differential Calculus
Advanced Level Pure Mathematics
Asymptotes to a Curve
Definition
A straight line is an ASYMPTOTE to a curve if and only if the perpendicular distance from a
variable point on the curve to the line approaches to zero as a limit when the point tends to
infinity along the curve on both sides or one side of the curve. (see figure below.)
Definition
(i)
the line x  c is said to be vertical asymptote of the curve y  f (x) .
lim f ( x)   or lim f ( x)   .
x c 
x c
(ii) the line y  ax  b is said to be an oblique asymptote of the curve y  f (x) if
lim  f ( x)  (ax  b)  0 .
x 
Example
(1) The curve y 
1
has two asymptotes x  0 or y  0 .
x
(2) The curve y  e x has an asymptote y  0 .
(3) The curve y 
Definition
1
sin x has an asymptote y  0 .
x
An asymptote parallel to the x-axis is called a horizontal asymptote.
An asymptote parallel to the y-axis is called a vertical asymptote.
An asymptote not parallel either coordinate axes is called an oblique asymptote.
A curve may have MORE
THAN ONE asymptote.
A curve may CROSS its own
asymptote.
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Page 17
Applications of Differential Calculus
Advanced Level Pure Mathematics
1
x 1
Example
Consider the curve y 
Example
x2 1
Consider the curve y 
.
x2
Theorem
The line y  ax  b is an asymptote to the curve y  f (x) if and only if
a  lim
x 
f ( x)
x
and
b  lim  f ( x)  ax
x 
where both limits are taken under x   , or x   or x   .
Note
Equation of an asymptote (other than the vertical asymptote) to a given curve can be
found by using Theorem
Prepared by K. F. Ngai(2003)
Page 18
Applications of Differential Calculus
Example
Advanced Level Pure Mathematics
Find the equations of all the asymptotes to each of the following curves.
(a)
y
(d)
x 2  6x  8
x 2  4x  3
y
y  xe x
(e)
y  x 3 ( x  1) 3
(a)
x  3, x  1, y  1
(b)
x  1, x  1, y  0
(d)
x  0, y  x  1
(e)
y  x
1
Ans:
x
x 1
(b)
(c)
2
1
yx
xa
xa
2
(c)
x  a, y  x  a
3
2
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Applications of Differential Calculus
Advanced Level Pure Mathematics
x
e e
.
e x  e x
x
Example
Find the asymptotes to the curve y  x 
Example
Find the horizontal and vertical asymptotes to the curve y 
(Ans: y  x, y   x )
x2 1
.
x 2  5x  6
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Page 20
Applications of Differential Calculus
Let F( x) 
Theorem
Advanced Level Pure Mathematics
f ( x)
, where f ( x ) and g( x) are polynomials in x and the degree of f ( x )
g( x )
exceeds that of g( x) by one; let F ( x ) be written in the form F( x )  ax  b 
R( x)
, where
g( x )
a and b are constants, and R( x) is a polynomial of degree less than that of g( x) . (This
can be done by long division.) Then the line y  ax  b is an oblique asymptote to the curve
y  F( x ).
2 x 3  x 2  3x  1
Find the oblique asymptotes to the curve y 
.
x2 1
Example
x
2 x 3  x 2  3x  1
= 2x 1  2
y
2
x 1
x 1
Hence the oblique asymptote is y  2 x  1 .
Example
Find the oblique asymptotes to the curve
(a)
Ans:
(a)
y
y  x3
x 2  6x  3
x3
(b)
(b)
y
2 x 3  x 2  3x  1
x2  2
y  2x  1
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Page 21
Applications of Differential Calculus
5.7
Advanced Level Pure Mathematics
Curve Sketching
The following information is useful for sketching the graph of y  f (x)
(1)
The domain of f (x) , i.e. the range of values of x within which y is well-defined.
(2)
Determine whether f (x) is periodic, odd or even, so that the graph may be symmetric about the
(3)
(4)
coordinate axes or about the origin.
Turning points and monotonicity of f (x) .
Inflexional points and convexity of f (x) .
(5)
(6)
Asymptotes including horizontal, vertical and oblique ones (if any).
Some special points on the graph, such as intercepts.
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Page 22
Applications of Differential Calculus
Advanced Level Pure Mathematics
HKAL 01 Paper II
2
1
Let f ( x)  x 3 (6  x) 3 .
(a)
(i)
(ii)
Find f ' ( x) for x  0,6 .
Show that f ' (0) and f ' (6) do not exist.
(iii)
Show that f ' ' ( x) 
8
4
3
x (6  x )
(b)
(c)
(d)
(e)
5
3
for x  0,6
Determine the values of x for each of the following cases:
(i) f ' ( x)  0 ,
(ii) f ' ( x)  0 ,
(iii) f ' ' ( x)  0
(iv) f ' ' ( x)  0 .
Find all relative extreme points and points of inflexion of f (x) .
Find all asymptotes of the graph of f (x) .
Sketch the graph of f (x) .
(4 marks)
(3 marks)
(3 marks)
(2 marks)
(3 marks)
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Applications of Differential Calculus
Advanced Level Pure Mathematics
HKAL 98 Paper II
1
2
Let f ( x)  x 3 ( x  1) 3 .
(a)
(i)
Find f ' ( x) for x  1,0 .
(ii)
Show that f ' ' ( x) 
2
5
3
9 x ( x  1)
4
3
for x  1,0
(2 marks)
(b)
Determine with reasons whether f ' ( 1) and f ' (0) exist or not.
(2 marks)
(c)
Determine the values of x for each of the following cases:
(i) f ' ( x)  0 ,
(ii) f ' ( x)  0 ,
(iii) f ' ' ( x)  0
(iv) f ' ' ( x)  0 .
Find all relative extrema and points of inflexion of f (x) .
Find all asymptotes of the graph of f (x) .
Sketch the graph of f (x) .
(3 marks)
(3 marks)
(d)
(e)
(f)
(2 marks)
(3 marks)
Prepared by K. F. Ngai(2003)
Page 24
Applications of Differential Calculus
Advanced Level Pure Mathematics
HKAL 97 Paper II
Let f ( x) 
(a)
2
3
x
x 1
(i)
(ii)
( x  1)
Find f ' ( x) for x  1,0 .
Does f ' (0) exist? Explain.
Show that f ' ' ( x) 
2(2 x 2  8 x  1)
4
3
9 x ( x  1)
(b)
(c)
(d)
(e)
for x  1,0
(4 marks)
3
Determine the values of x for each of the following cases:
(i) f ' ( x)  0 ,
(ii) f ' ( x)  0 ,
(iii) f ' ' ( x)  0
(iv) f ' ' ( x)  0 .
Find all relative extreme points and points of inflexion of f (x) .
Find all asymptotes of the graph of f (x) .
Sketch the graph of f (x) .
(4 marks)
(3 marks)
(1 marks)
(3 marks)
Prepared by K. F. Ngai(2003)
Page 25
Applications of Differential Calculus
Advanced Level Pure Mathematics
HKAL 99 Paper II
1
Let f ( x)  xe x for x  0 .
(a)
Find lim f ( x ) and show that f (x)   as x  0  .
(3 marks)
(b)
Find f ' ( x) and f ' ' ( x) for x  0
(2 marks)
(c)
Determine the values of x for each of the following cases:
(i) f ' ( x)  0 ,
(ii) f ' ' ( x)  0
Find all relative extrema of f (x) .
Find all asymptotes of the graph of f (x) .
Sketch the graph of f (x) .
(3 marks)
(2 marks)
(3 marks)
(2 marks)
(d)
(e)
(f)
x 0
Prepared by K. F. Ngai(2003)
Page 26
Applications of Differential Calculus
Advanced Level Pure Mathematics
HKAL 95 Paper II
Let f ( x) 
(a)
(b)
(d)
(e)
x
(1  x) 2
, where x  1.
(i)
Find f ' ( x) and f ' ' ( x) for x  0
(ii)
Find f ' ( x) and f ' ' ( x) for x  0
(iii)Show that f ' (0) does not exist.
(4 marks)
Determine the values of x for each of the following cases:
(i) f ' ( x)  0 ,
(ii) f ' ( x)  0 ,
(iii) f ' ' ( x)  0
(iv) f ' ' ( x)  0 .
Find all relative extreme point(s) and point(s) of inflexion of f (x) .
Find the asymptote(s) and sketch the graph of f (x) .
(4 marks)
(3 marks)
(4 marks)
Prepared by K. F. Ngai(2003)
Page 27
Applications of Differential Calculus
Advanced Level Pure Mathematics
HKAL 96 Paper II
( x  1) 3
.
( x  1) 2
Find f ' ( x) and f ' ' ( x) for x  1
(2 marks)
(c)
(d)
(e)
Determine the values of x for each of the following cases:
(i) f ' ( x)  0 ,
(ii) f ' ( x)  0 ,
(iii) f ' ' ( x)  0
(iv) f ' ' ( x)  0 .
Find all relative extreme point(s) and point(s) of inflexion of f (x) .
Find the asymptote(s) of f (x) .
Sketch the graph of f (x) .
(3 marks)
(2 marks)
(2 marks)
(2 marks)
(f)
Let g ( x)  f ( x) .Does g ' (1) exist ?
Let f ( x) 
(a)
(b)
Find the asymptote(s) and sketch of g (x ) .
(4 marks)
Prepared by K. F. Ngai(2003)
Page 28