 # 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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• Q1 Marks:1

(e, ).

##### Explanation: • Q2

The function f(x) = cos x is

Marks:1

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:1

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:1

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)= x2+2x+3 increases in the interval

Marks:1

(–1, )

##### Explanation:

We have f(x)= x2+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, ).