Chapter 6
APPLICATION OF DERIVATIVES
6.1 Overview
6.1.1 Rate of change of quantities
d
For the function y = f (x), (f (x)) represents the rate of change of y with respect to x.
dx
ds
Thus if ‘s’ represents the distance and ‘t’ the time, then represents the rate of
dt
change of distance with respect to time.
6.1.2 Tangents and normals
A line touching a curve y = f (x) at a point (x1, y1) is called the tangent to the curve at
dy
that point and its equation is given y − y1 = ( x , y ) ( x – x1 ) .
dx 1 1
The normal to the curve is the line perpendicular to the tangent at the point of contact,
and its equation is given as:
–1
y – y1 = ( x − x1 )
dy
(x , y )
dx 1 1
The angle of intersection between two curves is the angle between the tangents to the
curves at the point of intersection.
6.1.3 Approximation
f ( x + ∆x) – f ( x)
Since f ′(x) = lim , we can say that f ′(x) is approximately equal
∆x →0 ∆x
f ( x + ∆x) – f ( x)
to
∆x
⇒ approximate value of f (x + ∆ x) = f (x) + ∆x .f ′ (x).
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6.1.4 Increasing/decreasing functions
A continuous function in an interval (a, b) is :
(i) strictly increasing if for all x1, x2 ∈ (a, b), x1< x2 ⇒ f (x1) < f (x2) or for all
x ∈ (a, b), f ′ (x) > 0
(ii) strictly decreasing if for all x1, x2 ∈ (a, b), x1 < x2 ⇒ f (x1) > f (x2) or for all
x ∈ (a, b), f ′(x) < 0
6.1.5 Theorem : Let f be a continuous function on [a, b] and differentiable in (a, b) then
(i) f is increasing in [a, b] if f ′ (x) > 0 for each x ∈ (a, b)
(ii) f is decreasing in [a, b] if f ′ (x) < 0 for each x ∈ (a, b)
(iii) f is a constant function in [a, b] if f ′ (x) = 0 for each x ∈ (a, b).
6.1.6 Maxima and minima
Local Maximum/Local Minimum for a real valued function f
A point c in the interior of the domain of f, is called
(i) local maxima, if there exists an h > 0 , such that f (c) > f (x), for all x in
(c – h, c + h).
The value f (c) is called the local maximum value of f .
(ii) local minima if there exists an h > 0 such that f (c) < f (x), for all x in
(c – h, c + h).
The value f (c) is called the local minimum value of f.
A function f defined over [a, b] is said to have maximum (or absolute maximum) at
x = c, c ∈ [a, b], if f (x) ≤ f (c) for all x ∈ [a, b].
Similarly, a function f (x) defined over [a, b] is said to have a minimum [or absolute
minimum] at x = d, if f (x) ≥ f (d) for all x ∈ [a, b].
6.1.7 Critical point of f : A point c in the domain of a function f at which either
f ′ (c) = 0 or f is not differentiable is called a critical point of f.
Working rule for finding points of local maxima or local minima:
(a) First derivative test:
(i) If f ′ (x) changes sign from positive to negative as x increases through
c, then c is a point of local maxima, and f (c) is local maximum value.
, APPLICATION OF DERIVATIVES 119
(ii) If f ′ (x) changes sign from negative to positive as x increases through
c, then c is a point of local minima, and f (c) is local minimum value.
(iii) If f ′ (x) does not change sign as x increases through c, then c is
neither a point of local minima nor a point of local maxima. Such a
point is called a point of inflection.
(b) Second Derivative test: Let f be a function defined on an interval I and
c ∈ I. Let f be twice differentiable at c. Then
(i) x = c is a point of local maxima if f ′(c) = 0 and f ″(c) < 0. In this case
f (c) is then the local maximum value.
(ii) x = c is a point of local minima if f ′ (c) = 0 and f ″(c) > 0. In this case
f (c) is the local minimum value.
(iii) The test fails if f ′(c) = 0 and f ″ (c) = 0. In this case, we go back to
first derivative test.
6.1.8 Working rule for finding absolute maxima and or absolute minima :
Step 1 : Find all the critical points of f in the given interval.
Step 2 : At all these points and at the end points of the interval, calculate the
values of f.
Step 3 : Identify the maximum and minimum values of f out of the values
calculated in step 2. The maximum value will be the absolute maximum
value of f and the minimum value will be the absolute minimum
value of f.
6.2 Solved Examples
Short Answer Type (S.A.)
Example 1 For the curve y = 5x – 2x3, if x increases at the rate of 2 units/sec, then
how fast is the slope of curve changing when x = 3?
dy
Solution Slope of curve = = 5 – 6x2
dx
d dy dx
⇒ = –12x.
dt dx dt
APPLICATION OF DERIVATIVES
6.1 Overview
6.1.1 Rate of change of quantities
d
For the function y = f (x), (f (x)) represents the rate of change of y with respect to x.
dx
ds
Thus if ‘s’ represents the distance and ‘t’ the time, then represents the rate of
dt
change of distance with respect to time.
6.1.2 Tangents and normals
A line touching a curve y = f (x) at a point (x1, y1) is called the tangent to the curve at
dy
that point and its equation is given y − y1 = ( x , y ) ( x – x1 ) .
dx 1 1
The normal to the curve is the line perpendicular to the tangent at the point of contact,
and its equation is given as:
–1
y – y1 = ( x − x1 )
dy
(x , y )
dx 1 1
The angle of intersection between two curves is the angle between the tangents to the
curves at the point of intersection.
6.1.3 Approximation
f ( x + ∆x) – f ( x)
Since f ′(x) = lim , we can say that f ′(x) is approximately equal
∆x →0 ∆x
f ( x + ∆x) – f ( x)
to
∆x
⇒ approximate value of f (x + ∆ x) = f (x) + ∆x .f ′ (x).
,118 MATHEMATICS
6.1.4 Increasing/decreasing functions
A continuous function in an interval (a, b) is :
(i) strictly increasing if for all x1, x2 ∈ (a, b), x1< x2 ⇒ f (x1) < f (x2) or for all
x ∈ (a, b), f ′ (x) > 0
(ii) strictly decreasing if for all x1, x2 ∈ (a, b), x1 < x2 ⇒ f (x1) > f (x2) or for all
x ∈ (a, b), f ′(x) < 0
6.1.5 Theorem : Let f be a continuous function on [a, b] and differentiable in (a, b) then
(i) f is increasing in [a, b] if f ′ (x) > 0 for each x ∈ (a, b)
(ii) f is decreasing in [a, b] if f ′ (x) < 0 for each x ∈ (a, b)
(iii) f is a constant function in [a, b] if f ′ (x) = 0 for each x ∈ (a, b).
6.1.6 Maxima and minima
Local Maximum/Local Minimum for a real valued function f
A point c in the interior of the domain of f, is called
(i) local maxima, if there exists an h > 0 , such that f (c) > f (x), for all x in
(c – h, c + h).
The value f (c) is called the local maximum value of f .
(ii) local minima if there exists an h > 0 such that f (c) < f (x), for all x in
(c – h, c + h).
The value f (c) is called the local minimum value of f.
A function f defined over [a, b] is said to have maximum (or absolute maximum) at
x = c, c ∈ [a, b], if f (x) ≤ f (c) for all x ∈ [a, b].
Similarly, a function f (x) defined over [a, b] is said to have a minimum [or absolute
minimum] at x = d, if f (x) ≥ f (d) for all x ∈ [a, b].
6.1.7 Critical point of f : A point c in the domain of a function f at which either
f ′ (c) = 0 or f is not differentiable is called a critical point of f.
Working rule for finding points of local maxima or local minima:
(a) First derivative test:
(i) If f ′ (x) changes sign from positive to negative as x increases through
c, then c is a point of local maxima, and f (c) is local maximum value.
, APPLICATION OF DERIVATIVES 119
(ii) If f ′ (x) changes sign from negative to positive as x increases through
c, then c is a point of local minima, and f (c) is local minimum value.
(iii) If f ′ (x) does not change sign as x increases through c, then c is
neither a point of local minima nor a point of local maxima. Such a
point is called a point of inflection.
(b) Second Derivative test: Let f be a function defined on an interval I and
c ∈ I. Let f be twice differentiable at c. Then
(i) x = c is a point of local maxima if f ′(c) = 0 and f ″(c) < 0. In this case
f (c) is then the local maximum value.
(ii) x = c is a point of local minima if f ′ (c) = 0 and f ″(c) > 0. In this case
f (c) is the local minimum value.
(iii) The test fails if f ′(c) = 0 and f ″ (c) = 0. In this case, we go back to
first derivative test.
6.1.8 Working rule for finding absolute maxima and or absolute minima :
Step 1 : Find all the critical points of f in the given interval.
Step 2 : At all these points and at the end points of the interval, calculate the
values of f.
Step 3 : Identify the maximum and minimum values of f out of the values
calculated in step 2. The maximum value will be the absolute maximum
value of f and the minimum value will be the absolute minimum
value of f.
6.2 Solved Examples
Short Answer Type (S.A.)
Example 1 For the curve y = 5x – 2x3, if x increases at the rate of 2 units/sec, then
how fast is the slope of curve changing when x = 3?
dy
Solution Slope of curve = = 5 – 6x2
dx
d dy dx
⇒ = –12x.
dt dx dt
