NCERT Class 12-Mathematics: Chapter –6 Application of Derivatives Part 1

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6.1 Overview

6.1.1 Rate of change of quantities

For the function represents the rate of change of with respect to .

Thus if ‘’ represents the distance and ‘’ the time, then, represents the rate of change of distance with respect to time.

6.1.2 Tangents and normal

A line touching a curve at a point is called the tangent to the curve at that point and its equation is given

The normal to the curve is the line perpendicular to the tangent at the point of contact, and its equation is given as:

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

Since , we can say that is approximately equal to

value of

6.1.4 Increasing/decreasing functions

A continuous function in an interval is :

(i) Strictly increasing if for all or for all

(ii) Strictly decreasing if for all or for all

6.1.5 Theorem:

Let be a continuous function on and differentiable in then

(i) is increasing in if for each

(ii) is decreasing in if for each

(iii) is a constant function in if for each .

6.1.6 Maxima and minima

Local Maximum/Local Minimum for a real valued function

A point in the interior of the domain of , is called

(i) Local maxima, if there exists an , such that , for all in .

The value is called the local maximum value of.

(ii) Local minima if there exists an such that , for all in .

The value is called the local minimum value of .

A function defined over is said to have maximum (or absolute maximum) at for all .

Similarly, function defined over is said to have a minimum [or absolute minimum] at , if for all .

6.1.7 Critical point of: A point in the domain of a function at which either or f is not differentiable is called a critical point of.

Working rule for finding points of local maxima or local minima:

(a) First derivative test:

(i) If changes sign from positive to negative as x increases through, then is a point of local maxima, and is local maximum value.

(ii) If changes sign from negative to positive as x increases through, then is a point of local minima, and is local minimum value.

(iii) If does not change sign as increases through, then 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 be a function defined on an interval I and . Let be twice differentiable at . Then

(i) is a point of local maxima if f and . In this case is then the local maximum value.

(ii) 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 and . 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 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 out of the values calculated in step 2. The maximum value will be the absolute maximum value of and the minimum value will be the absolute minimum value of.

6.2 Solved Examples

Short Answer Type (S.A.)

Question 1:

For the curve , if x increases at the rate of units/sec, then how fast is the slope of curve changing when ?

Answer:

Slope of curve

us, slope of curve is decreasing at the rate of units/sec when x is increasing at the rate of units/sec.

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