Application of Derivatives  I
We can use derivatives to measure rate of change of quantities, to find increasing and decreasing functions, to find tangents and normals etc.
• If a quantity y varies with another quantity x, satisfying some rule y = f(x), then dy/dx represents the rate of change of y with respect to x.
• If two variables x and y are varying with respect to another variable t, i.e., if y = f(t) and x = g(t), then dy/dx = (dy/dt)/(dx/dt), provided that dx/dt is not equal to 0.
• A function f is said to be increasing function on an interval (a, b) if x1 < x2 in (a, b)
f(x1) ≤ f(x2) for all x1, x2∈ (a, b).
• A function f is said to be decreasing function on an interval (a, b) if x1 < x2 in (a, b)
f(x1) ≥ f(x2) for all x1, x2∈ (a, b).
• If f(x) be a function continuous on [a, b] and differentiable on (a, b), then
f(x) is strictly increasing in (a, b) if f´(x) > 0 for all x in (a, b).
f(x) is strictly decreasing in (a, b) if f´(x) < 0 for all x in (a, b).
• The equation of tangent at (x1, y1) to a curve y = f(x) is given by
y  y1 = (x  x1) f' (x1).
• The equation of normal at (x1, y1) to a curve y = f(x) is given by
y  y1 =  (x  x1)/ f' (x1).
Keywords: Derivative, Rate of Change, increasing functions, decreasing functions, tangents and normals
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Q1Marks:1
Answer:
(e, ).
Explanation:

Q2
The function f(x) = cos x is
Marks:1Answer:
strictly decreasing on (0, ).
Explanation:
Given f(x) = cos x f'(x) = sin x < 0 x (0, ) (Becasue sin x > 0 x (0, ))f'(x) < 0 x (0, )
f(x) is strictly decreasing on (0, ).

Q3
On R, the function f(x) = 7x – 3 is
Marks:1Answer:
strictly increasing.
Explanation:
Given f(x) = 7x  3, x R f'(x) = 7 > 0, x R
f is strictly increasing on R.

Q4
The function f(x) = 2 – 3x is
Marks:1Answer:
strictly decreasing.
Explanation:
Given f(x) = 2 – 3x, x R.f’(x) = – 3 < 0
f is strictly decreasing on R.

Q5
The function f(x)= x^{2}+2x+3 increases in the interval
Marks:1Answer:
(–1, )
Explanation:
We have f(x)= x^{2}+2x+3
f'(x)= 2x+2
For turning point f'(x)=0 x= 1
f'(x)> 0 if x (–1, )
f (x) is increasing in the interval (–1, ).