Page 1
Page # 11
APPLICATION OF DERIVATIVES
1. Equation of tangent and normal
Tangent at (x
1
, y
1
) is given by (y – y
1
) = f ?(x
1
) (x – x
1
) ; when, f ?(x
1
) is real.
And normal at (x
1
, y
1
) is (y – y
1
) = –
) x ( f
1
1
?
(x – x
1
), when f ?(x
1
) is nonzero
real.
2. Tangent from an external point
Given a point P(a, b) which does not lie on the curve y = f(x), then the
equation of possible tangents to the curve y = f(x), passing through (a, b)
can be found by solving for the point of contact Q.
f ?(h) =
a h
b ) h ( f
?
?
And equation of tangent is y – b =
h a
f(h) b
?
?
(x – a)
3. Length of tangent, normal, subtangent, subnormal
(i) PT =
2
m
1
| k | 1 ? = Length of Tangent
p p( (h h, ,k k) )
N N M M T T
(ii) PN =
2
| k | 1 ? m = Length of Normal
(iii) TM =
m
k
= Length of subtangent
(iv) MN = |km| = Length of subnormal.
Page 2
Page # 11
APPLICATION OF DERIVATIVES
1. Equation of tangent and normal
Tangent at (x
1
, y
1
) is given by (y – y
1
) = f ?(x
1
) (x – x
1
) ; when, f ?(x
1
) is real.
And normal at (x
1
, y
1
) is (y – y
1
) = –
) x ( f
1
1
?
(x – x
1
), when f ?(x
1
) is nonzero
real.
2. Tangent from an external point
Given a point P(a, b) which does not lie on the curve y = f(x), then the
equation of possible tangents to the curve y = f(x), passing through (a, b)
can be found by solving for the point of contact Q.
f ?(h) =
a h
b ) h ( f
?
?
And equation of tangent is y – b =
h a
f(h) b
?
?
(x – a)
3. Length of tangent, normal, subtangent, subnormal
(i) PT =
2
m
1
| k | 1 ? = Length of Tangent
p p( (h h, ,k k) )
N N M M T T
(ii) PN =
2
| k | 1 ? m = Length of Normal
(iii) TM =
m
k
= Length of subtangent
(iv) MN = |km| = Length of subnormal.
Page # 12
4. Angle between the curves
Angle between two intersecting curves is defined as the acute angle between
their tangents (or normals) at the point of intersection of two curves (as shown
in figure).
tan ? =
2 1
2 1
m m 1
m m
?
?
5. Shortest distance between two curves
Shortest distance between two non-intersecting differentiable curves is always
along their common normal.
(Wherever defined)
6. Rolle’s Theorem :
If a function f defined on [a, b] is
(i) continuous on [a, b]
(ii) derivable on (a, b) and
(iii) f(a) = f(b),
then there exists at least one real number c between a and b (a < c < b) such
that f ?(c) = 0
7. Lagrange’s Mean Value Theorem (LMVT) :
If a function f defined on [a, b] is
(i) continuous on [a, b] and (ii) derivable on (a, b)
then there exists at least one real numbers between a and b (a < c < b) such
that
b a
f( )b f( )a
?
?
= f ?(c)
Page 3
Page # 11
APPLICATION OF DERIVATIVES
1. Equation of tangent and normal
Tangent at (x
1
, y
1
) is given by (y – y
1
) = f ?(x
1
) (x – x
1
) ; when, f ?(x
1
) is real.
And normal at (x
1
, y
1
) is (y – y
1
) = –
) x ( f
1
1
?
(x – x
1
), when f ?(x
1
) is nonzero
real.
2. Tangent from an external point
Given a point P(a, b) which does not lie on the curve y = f(x), then the
equation of possible tangents to the curve y = f(x), passing through (a, b)
can be found by solving for the point of contact Q.
f ?(h) =
a h
b ) h ( f
?
?
And equation of tangent is y – b =
h a
f(h) b
?
?
(x – a)
3. Length of tangent, normal, subtangent, subnormal
(i) PT =
2
m
1
| k | 1 ? = Length of Tangent
p p( (h h, ,k k) )
N N M M T T
(ii) PN =
2
| k | 1 ? m = Length of Normal
(iii) TM =
m
k
= Length of subtangent
(iv) MN = |km| = Length of subnormal.
Page # 12
4. Angle between the curves
Angle between two intersecting curves is defined as the acute angle between
their tangents (or normals) at the point of intersection of two curves (as shown
in figure).
tan ? =
2 1
2 1
m m 1
m m
?
?
5. Shortest distance between two curves
Shortest distance between two non-intersecting differentiable curves is always
along their common normal.
(Wherever defined)
6. Rolle’s Theorem :
If a function f defined on [a, b] is
(i) continuous on [a, b]
(ii) derivable on (a, b) and
(iii) f(a) = f(b),
then there exists at least one real number c between a and b (a < c < b) such
that f ?(c) = 0
7. Lagrange’s Mean Value Theorem (LMVT) :
If a function f defined on [a, b] is
(i) continuous on [a, b] and (ii) derivable on (a, b)
then there exists at least one real numbers between a and b (a < c < b) such
that
b a
f( )b f( )a
?
?
= f ?(c)
Page # 13
8. Useful Formulae of Mensuration to Remember :
1. Volume of a cuboid = ?bh.
2. Surface area of cuboid = 2( ?b + bh + h ?).
3. Volume of cube = a
3
4. Surface area of cube = 6a
2
5. Volume of a cone =
3
1
? ?r
2
h.
6. Curved surface area of cone = ?r ? ( ? = slant height)
7. Curved surface area of a cylinder = 2 ?rh.
8. Total surface area of a cylinder = 2 ?rh + 2 ?r
2
.
9. Volume of a sphere =
3
4
?r
3
.
10. Surface area of a sphere = 4 ?r
2
.
11. Area of a circular sector =
2
1
r
2
?, when ? is in radians.
12. Volume of a prism = (area of the base) × (height).
13. Lateral surface area of a prism = (perimeter of the base) × (height).
14. Total surface area of a prism = (lateral surface area) + 2 (area of
the base)
(Note that lateral surfaces of a prism are all rectangle).
15. Volume of a pyramid =
3
1
(area of the base) × (height).
16. Curved surface area of a pyramid =
2
1
(perimeter of the base) ×
(slant height).
(Note that slant surfaces of a pyramid are triangles).
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