5565-7-96PE
AID: 1112 | 30/09/2012
The LHospitals rule states that
If f x
and g x
for some x
Then
f x
f' x
lim
lim
x a g x
x a g' x
a
Consider the limit
x
3
lim 1
x 0
x
For x 0 , the limit tends to
0
.
Consider a function
x
3
f x
1
x
Taking logarithmic on

5565-7-119PE
AID: 1112 | 30/09/2012
It can be easily observed from the figure that the sum of the three angles including ,the
angles subtended by both the buildings on the point at a distance x from the tall building
is 180 .
By applying trigonometry in t

5565-7-108PE
AID: 1112 | 30/09/2012
As we can see from the graph, the value of the function goes to negative infinity as we go
near to 0.So, the function has no absolute minimum.
This can be also verified by our knowledge from calculus.
For a function to

5565-7-108PE
AID: 1112 | 30/09/2012
(a)
We can observe from the graph that the maximum value of the function is about 0.73 and
it is for x 7.3
The minimum value of the function tends to negative infinity.
We can verify this by using calculus.
Here
ln x
f

5565-7-106PE
AID: 1112 | 30/09/2012
For the rectangle, the length is the x coordinate of the upper right hand vertex and the
breadth is the y coordinate of the upper right hand vertex.
So area of the rectangle is
Area xy
ln x
Here y
x2
So, area is given b

5565-7-105PE
AID: 1112 | 30/09/2012
For the rectangle, the length is the x coordinate of the upper right hand vertex and the
breadth is the y coordinate of the upper right hand vertex.
So area of the rectangle is
Area xy
2
Here y e x
So, area is given by

5565-7-101PE
AID: 1112 | 30/09/2012
Area under a graph is calculated by integrating the function depicting the graph with
suitable limits.
2ln x
Now, we have to find the area between the curve y
and the x-axis from x 1 to
x
x e
Area is given by the integr

5565-7-99PE
AID: 1112 | 30/09/2012
Consider the function
y x ln 2 x x
We know that a function has its absolute maximum and minimum in a given interval
either at the end points of the interval or at the points in the interval where the first
derivative of

5565-7-87PE
AID: 1112 | 30/09/2012
The LHospitals rule states that
If f x 0 and g x 0 for some x
Then
f x
f' x
lim
lim
x a g x
x a g' x
a
Now, consider the following limit
2sin x 1
lim x
x 0 e
1
For x 0 , the numerator and the denominator are zero. So, we

5565-7-120PE
AID: 1112 | 30/09/2012
Radius of the core is r 1cm .
Thickness of the insulation is h.
r
x
h
Speed of transmission signal is
1
v x 2 ln
x
2
x ln x
Substituting the value of x
1 1
v h
ln
h2 h
ln h
h2
For speed to be maximum, its first derivati