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## Kerala Plus Two Maths Notes Chapter 6 Application of Derivatives

Introduction

In this chapter we analyses the physical and geometrical applications of derivatives in real life such as to determine rate of change, to find tangents and normal to a curve, to find turning points, intervals in which the curve is increasing and decreasing, to find approximate value of certain quantities.

I. Rate of Change

\(\frac{d y}{d x}\), we mean the rate of change of y with respect to x. If s is the displacement function in terms of time t and v the velocity at that time. Then, \(\frac{d s}{d t}\) = velocity, Acceleration = \(\frac{d v}{d t}=\frac{d^{2} s}{d t^{2}}\)

II. Tangents and Normals

If a tangent line to the curve y = f(x) makes an angle θ with the positive direction of the x-axis, then f'(x) = slope of the tangent = tanθ.

Equation of tangent to the curve y = f (x) at the point (x_{1}, y_{1}): y – y_{1} = f'(x_{1})(x – x_{1})

Equation of normal to the curve y = f (x) at the point (x_{1}, y_{1}): y – y_{1} = –\(\frac{1}{f^{\prime}\left(x_{1}\right)}\)(x – x_{1})

III. Increasing and decreasing functions

Nature of a function on a given interval;

Strictly increasing on [a, b]: f'(x) > 0, x ∈ (a, b) Increasing on [a, b]: f'(x) ≥ 0, x ∈ (a, b)

Strictly decreasing on [a, b]: f'(x) < 0, x ∈ (a, b) Decreasing on[a, b]: f'(x) ≤ 0, x ∈ (a, b).

1. Between two consecutive points at which f'(x) = 0 the function has only one nature either it is increasing or decreasing, not both.

IV. Approximation

Consider a function y = f(x). Let ∆x denote a small increment in x and ∆y be the corresponding increment in y. Then, ∆y can be approximated by dy, where dy = \(\frac{d y}{d x}\) × ∆x.

V. Maxima and Minima

A function y = f(x) is said to have a local maximum at x = a, if f(a) is the maximum value obtained by the function in the neighbourhood of x = a.

A function y = f(x) is said to have a local minimum at x = a, if f(a) is the minimum value obtained by the function in the neighbourhood of x = a.

Point on the curve at which f'(x) = 0 is called stationary point or turning point. The following are methods to find the local maximum and local minimum at points where f'(x) = 0.

First Derivative Test:

- If f'(c) = 0 and f'(x)changes its sign from positive to negative from left to right of x = c, then the point is a local maximum point.
- If f'(c) = 0 and f'(x) changes its sign from negative to positive from left to right of x = c, then the point is a local minimum point.
- If f'(c) = 0 and if there is no change of sign for f'(x) from left to right of x = c, then the point is a inflexion point.

Second Derivative Test:

- If f'(c) = 0 and f”(c) < 0 , then x = c is a local maximum point.
- If f'(c) = 0 and f”(c) > 0, then x = c is a local minimum point.
- If f'(c) = 0 and f”(c) = 0, then the test fails and go to first derivate test for checking maxima and minima.

Absolute Maxima and Minima:

Let f(x) be a function defined on [a, b] and if , f'(x) = 0 ⇒ x = x_{1}, x_{2}, x_{3},……etc, then

- Absolute maximum value of

= max{f(a), f(x_{1}), f(x_{2}),…….f(b)} - Absolute minimum value of

= min{f(a), f.(x_{1}), f(x_{2}),…..f(b)}