, XII- MATHS Application of Derivatives- Results
I. Rate of change of quantities.
𝑑
For the function y = f (x), 𝑑𝑥 (𝑓(𝑥)) represents the rate of change of y with respect to x.
𝑑𝑠
Thus if ‘s’ represents the distance and ‘t’ the time, then 𝑑𝑡 represents the rate of change of
distance with respect to time.
𝒅𝒚
(𝒅𝒙) Represents the rate of change of y w.r.t. x at x = a.
𝒙=𝒂
𝒅𝒚
is positive if y increases as x increases
𝒅𝒙
and is negative if y decreases as x increases.
If C(x) is cost of producing x items,
𝐶(𝑥) 𝑑
then average cost is 𝑥 and Marginal cost is 𝑑𝑥 C(x)
If R(x) is revenue of selling x items,
𝑅(𝑥) 𝑑
then average revenue is 𝑥 and Marginal revenue is 𝑑𝑥 R(x)
II. Increasing and Decreasing Function
Let f be a real-valued continuous function and let I be any interval in the domain of f.
Then f is said to be
a) Strictly increasing on I, if for all 𝑥1 , 𝑥2 ∈ I, 𝒙𝟏 < 𝒙𝟐 ⇒ 𝒇(𝒙𝟏 ) < 𝒇(𝒙𝟐 )
b) Increasing on I, if for all 𝒙𝟏 < 𝒙𝟐 ⇒ 𝒇(𝒙𝟏 ) ≤ 𝒇(𝒙𝟐 )
c) Strictly decreasing in I, if for all 𝑥1 , 𝑥2 ∈ I, we have 𝒙𝟏 < 𝒙𝟐 ⇒ 𝒇(𝒙𝟏 ) > 𝒇(𝒙𝟐 )
d) Decreasing on I, if for all 𝒙𝟏 < 𝒙𝟐 ⇒ 𝒇(𝒙𝟏 ) ≥ 𝒇(𝒙𝟐 )
Test for Increasing and Decreasing Function.
Let f be a continuous function on [a, b] and differentiable in (a, b) then
(i) f is strictly increasing in [a, b] if f’ (x) > 0 for each x ∈(a, b)
(ii) f is increasing in [a, b] if 𝒇’(𝒙) ≥ 𝟎 for each x ∈(a, b)
(iii) f is strictly decreasing in [a, b] if f’ (x) < 0 for each x ∈(a, b)
(iv) f is decreasing in [a, b] if 𝒇’(𝒙) ≤ 𝟎 for each x ∈(a, b)
(v) f is a constant function in [a, b] if f’ (x) = 0 for each x ∈(a, b).
STEPS
(i) Find f’(x) and factorise completely, if it is a polynomial function.
(ii) Solve f’(x) = 0 and get the critical points and split the real numbers to intervals.
(iii) Find the sign of each term and f’(x) in each of the interval obtained.
(iv) Conclusion should be taken as per the sign of f’(x), +ve : increasing and – ve :
decreasing
(v) In case of trigonometric functions. Check the sign within the given domain only
as per the quadrants.
© 0509312916 XII Mathematics
I. Rate of change of quantities.
𝑑
For the function y = f (x), 𝑑𝑥 (𝑓(𝑥)) represents the rate of change of y with respect to x.
𝑑𝑠
Thus if ‘s’ represents the distance and ‘t’ the time, then 𝑑𝑡 represents the rate of change of
distance with respect to time.
𝒅𝒚
(𝒅𝒙) Represents the rate of change of y w.r.t. x at x = a.
𝒙=𝒂
𝒅𝒚
is positive if y increases as x increases
𝒅𝒙
and is negative if y decreases as x increases.
If C(x) is cost of producing x items,
𝐶(𝑥) 𝑑
then average cost is 𝑥 and Marginal cost is 𝑑𝑥 C(x)
If R(x) is revenue of selling x items,
𝑅(𝑥) 𝑑
then average revenue is 𝑥 and Marginal revenue is 𝑑𝑥 R(x)
II. Increasing and Decreasing Function
Let f be a real-valued continuous function and let I be any interval in the domain of f.
Then f is said to be
a) Strictly increasing on I, if for all 𝑥1 , 𝑥2 ∈ I, 𝒙𝟏 < 𝒙𝟐 ⇒ 𝒇(𝒙𝟏 ) < 𝒇(𝒙𝟐 )
b) Increasing on I, if for all 𝒙𝟏 < 𝒙𝟐 ⇒ 𝒇(𝒙𝟏 ) ≤ 𝒇(𝒙𝟐 )
c) Strictly decreasing in I, if for all 𝑥1 , 𝑥2 ∈ I, we have 𝒙𝟏 < 𝒙𝟐 ⇒ 𝒇(𝒙𝟏 ) > 𝒇(𝒙𝟐 )
d) Decreasing on I, if for all 𝒙𝟏 < 𝒙𝟐 ⇒ 𝒇(𝒙𝟏 ) ≥ 𝒇(𝒙𝟐 )
Test for Increasing and Decreasing Function.
Let f be a continuous function on [a, b] and differentiable in (a, b) then
(i) f is strictly increasing in [a, b] if f’ (x) > 0 for each x ∈(a, b)
(ii) f is increasing in [a, b] if 𝒇’(𝒙) ≥ 𝟎 for each x ∈(a, b)
(iii) f is strictly decreasing in [a, b] if f’ (x) < 0 for each x ∈(a, b)
(iv) f is decreasing in [a, b] if 𝒇’(𝒙) ≤ 𝟎 for each x ∈(a, b)
(v) f is a constant function in [a, b] if f’ (x) = 0 for each x ∈(a, b).
STEPS
(i) Find f’(x) and factorise completely, if it is a polynomial function.
(ii) Solve f’(x) = 0 and get the critical points and split the real numbers to intervals.
(iii) Find the sign of each term and f’(x) in each of the interval obtained.
(iv) Conclusion should be taken as per the sign of f’(x), +ve : increasing and – ve :
decreasing
(v) In case of trigonometric functions. Check the sign within the given domain only
as per the quadrants.
© 0509312916 XII Mathematics
