Application of Differential Calculus
Date
A-Level Pure Mathematics
Chapter 5 Application of Differential Calculus
Exercise 5A (L’Hospital’s Rule)
Name : ________________
:
x  sin x
x3
1.
Evaluate lim
2.
Evaluate the following limits:
x 0
(a)
3.
1  cos x
x 0
x2
lim
sin x 2
x 0
x
(c)
lim
sec x  1
x 0
x2
(c)
lim
(b) lim
sin nx
x  0 sin x
Evaluate the following limits:
(a)
lim x sin x
x 0
(b) lim
1
sin 2 x
x0
x2
Application of Differential Calculus
4.
Evaluate the following limits:
(a)
1  sin x
 cos x
x
1 
1
(b) lim  

x 0 x
tan x 

lim
2
5.
x 0
x  tan x
x3
Evaluate the following limits:
(a)
Ans:
lim
(c)
tan x  x
lim 2
x 0 x sin x
1.
1
6
(b) lim (e  x)
x
1
x
(c)
x 0
2.
1
, 0, n
2
3.
1,
1
, 1 4.
2
2
2

lim  cos 
x  
x

0, 0, 
1
3
5.
x
1
, e2 , 1
3
Application of Differential Calculus
Date
1
A-Level Pure Mathematics
Chapter 5 Application of Differential Calculus
Exercise 5B(L’Hospital’s Rule)
Name : ________________
:
Evaluate the following limits:
1
(a)
2.
lim [ln( x  e)] x
x 0
1
(b)
lim (sin x) ln x
x 0
Evaluate:
(a)
lim
x 1
1 x
1 5 x
(b)
lim (sec x  tan x)
x

2
[HKAL 1994]
3
Application of Differential Calculus
3.
Evaluate:
 3e  2 

(b) lim 
x 0
5


e  1  sin x
x2
x
x
(a)
lim
x 0
1
x
[HKAL 1998]
Ans:
1.
1
e
e , e
2.
5
, 0
2
3
3.
1
, e5
2
4
Application of Differential Calculus
Date
A-Level Pure Mathematics
Chapter 5 Application of Differential Calculus
Exercise 5C (Monotonic Functions)
Name : ________________
:

x3
is strictly increasing for 0  x  .
2
3
1.
Show that the function f ( x)  tan x  x 
2.
Show that the function f ( x) 
3.
Determine the interval for which the function f ( x) 
4.
Determine the interval in [0,2 ] for which the function f ( x)  x  2 sin x is decreasing.
Ans:
3.
[1 , 1]
4.
[
1
 2 x  3 is strictly increasing for x  1.
x
2 4
, ].
3 3
5
2x
is increasing.
1 x2
Application of Differential Calculus
Date
:
A-Level Pure Mathematics
Chapter 5 Application of Differential Calculus
Exercise 5D(Monotonic Functions)
Name : ________________
x
 tan 1 x  x for x  0 .
1 x2
1.
Prove
2.
Prove that 0  ln( 1  x) 
2x
x3

for x  0
2  x 12
6
Application of Differential Calculus
3.
ln x
.
x
(a) Show that f (x) is strictly increasing on the interval (0, e) .
Let f ( x ) 
(b) Hence, show that if 0  a  b  e ,
ab  ba
4.
Let f ( x)  ln( 1  x)  x .
By finding the greatest value of f (x) , prove that ln( 1  x)  x .
7
Application of Differential Calculus
5.
(a) Show that for 0  x  1,
(i) ln( 1  x)  x
(ii)  ln( 1  x)  x
(b) Let n be a positive integer greater than 1 . Deduce from (a) that
ln( n  1)  ln n 
1
 ln n  ln( n  1)
n
Hence show that
ln(
3n  1 1
1
1
3n
) 

 ln(
)
n
n n 1
3n
n 1
(c) Use the above results to evaluate the limit
1
1
1
lim  
  
n  n
n 1
3n 

(Ans: ln 3 )
8
Application of Differential Calculus
Date
1.
A-Level Pure Mathematics
Chapter 5 Application of Differential Calculus
Exercise 5E (Maxima and Minima)
Name : ________________
:
Find the maximum or minimum points of y 
x
x
x
2x
2(1  x 2 )
and
.
y
'

1 x2
(1  x 2 ) 2
x
x
f ' ( x)
f (x)
2.
3.
Maximum point
=
Minimum point
=
Find the maximum or minimum points of y  x(3 x  1) .
2
3
Find the maximum or minimum points of y  3( x  4)  2 .
9
x
Application of Differential Calculus
(ln x) 2
.
x
4.
Find the maximum or minimum points of y 
5.
Find the maximum or minimum points of y  x ( x  4)
Ans:
1.
Max. pt ( 1,1 ) Min. pt (  1,1 )
3.
Min. pt ( 4,2 )
5.
Min. pt ( 0, 0 )
4.
2.
3 33
2)
Min. pt ( ,
4 8
Max. pt ( e 2 ,4e 2 ) Min. pt ( 1,0 )
10
Application of Differential Calculus
Date
1.
A-Level Pure Mathematics
Chapter 5 Application of Differential Calculus
Exercise 5F (points of inflexion)
Name : ________________
:
Find the points of inflexion of the curve y  3x 2  x 3 and y ' '  6  6 x .
x
x
x
x
y''
y

(
,
)
is point of inflexion.
5
3
2.
Find the points of inflexion of the curve y  x  x .
3.
Find the points of inflexion of the curve y  1  x 2 .
11
Application of Differential Calculus
4.
Find the points of inflexion of the curve y  ln x , ( x  0 ).
5.
Find the points of inflexion of the curve y  e  x .
2
1
Ans:
1.
(1,2)
2.
(0,0)
3.
no point of inflexion
12
4.
(1,0)
5.
1 
( , e 2 )
2
Application of Differential Calculus
Date
:
A-Level Pure Mathematics
Chapter 5 Application of Differential Calculus
Exercise 5G(asymptotes)
Name : ________________
1.
Find the asymptotes to the curve y 
1
.
x 9
2.
Find the asymptotes to the curve y 
2x 2
.
1 x2
3.
Find the asymptotes to the curve y 
x3  x2  1
.
x 2  3x  4
4.
Find the asymptotes to the curve y 
9x 2  4
.
x2
Ans:
1.
4.
x  3, x  3, y  0
2.
2
y2
3.
x  2, y  3, y  3 .
13
x  4, x  1, y  x  2
Application of Differential Calculus
Date
:
A-Level Pure Mathematics
Chapter 5 Application of Differential Calculus
Exercise 5H(Curve Sketching)
Name : ________________
2
 2
 x ( x  1) 3
Let f ( x)  
2
 x 2 ( x  1) 3

1.
if
x0
if
x0
(a)
(b)
For x  0 and  1 , find f ' ( x) and f ' ' ( x)
Show that f ' (0)  0 but both f ' ' (0) and f ' ( 1) do not exist.
(c)
Show that the graph of y  f (x) has extreme points at x  1 and 
inflexional points at x  0 and
(d)
3
and has
4
 15  3 5
.
20
Sketch the graph of y  f (x) .
Vision Ex 5.10(12)
14
Application of Differential Calculus
2.
Let f ( x)  3 x 3  x 2  x  1 .
(a) Show that f ' (1) and f ' ( 1) do not exist.
(b) Find f ' ( x) for x  1 and  1 .
(c) Find the range of values of x such that
(i) f ' ( x)  0 ,
(ii) f ' ( x)  0
(d) Find the maximum, minimum and inflexional points of the graph of y  f (x) .
(e) Find the asymptote(s) of the graph of y  f (x) .
(f) Sketch the graph of f (x) .
Vision Ex 5.10(14)
15
Application of Differential Calculus
Date
A-Level Pure Mathematics
Chapter 5 Application of Differential Calculus
Exercise 5I(Curve Sketching)
Name : ________________
:
Let f ( x) 
1.
(a)
(b)
(c)
x
2
3
x 1
2
( xR)
(i) Evaluate f ' ( x) for x  0 . Prove that f ' (0) does not exist.
(ii) Determine those values of x for which f ' ( x)  0 and those values of x for which
f ' ( x)  0 .
(iii) Find the relative extreme points of f (x) .
(i) Evaluate f ' ' ( x) for x  0 . Hence determine the points of inflexion of f (x) .
(ii) Find the asymptote of the graph of f (x) .
Using the above results, sketch the graph of f (x) .
HKAL 94 Paper II
16
Application of Differential Calculus
2.
8
( x  1)
x 1
(a) Find f ' ( x) and f ' ' ( x) for x  1 .
Let f ( x)  x 2 
(b) Determine the range of 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 .
(c) Find the relative extreme point(s) and point(s) of inflexion of f (x) .
(d) Find the asymptote(s) of f (x) .
(e) Sketch the graph of f (x) .
(f)
Let g ( x)  f ( x )
( x  1)
(i) Is g (x ) differentiable at x  0 ? Why?
(ii) Sketch the graph of g (x ) .
HKAL 02 Paper II
17