Application of Derivatives Class 12th Notes - Free NCERT Class 12 Maths Chapter 6 Notes - Download PDF

Introduction: Class 12 Math chapter 6 notes are regarding the Application of Derivatives. Application of derivatives Class 12 notes is about finding the derivatives of the functions. This Class 12 Maths chapter 6 notes contains the following topics: finding the derivatives of the equations, rate of change of quantities, Increasing and decreasing functions, Tangents and normal, Approximations, Maxima and minima, and many more. NCERT Class 12 Math chapter 6 notes also contain theorems and their proofs that are to be remembered for the implementation in problems. NCERT Class 12 Math Chapter 6 contains a detailed explanation of topics, theorems, examples, exercises. By going through the document students can cover all the topics that are in NCERT Notes for Class 12 Math chapter 6 textbook. This also contains examples, exercises, a few interesting points, and most importantly contains FAQ’s that are frequently asked questions by students which can clarify many other students with the same doubt. Every concept that is in CBSE Class 12 Maths chapter 6 notes is explained here in a simple and understanding way that can reach students easily. All these concepts can be downloaded from Class 12 Maths chapter 6 notes pdf download. Class 12 application of derivatives notes, application of derivatives Class 12 notes pdf download. Notes also contain a few of previous year’s question papers.

Also, students can refer,

• NCERT Solutions for Class 12 Maths Chapter 6 Application of Derivatives
• NCERT Exemplar Class 12 Maths Chapter 6 Solutions Application of Derivatives

NCERT Class 12 Chapter 6 Notes

Functions were discussed in chapter 5, now in this chapter, we are going to discuss the applications of those derivatives like a) determining the rate of change, b) finding the equations of tangent and normal to a curve at a point, c) to finding turning points on the graph of a function. We also will be finding increasing and decreasing functions.

Rate of change:

According to CBSE Class 12 chapter 6 notes the rate of change is:

DERIVATIVE:

DEFINITION: The rate of change of distance (S) with respect to time (t) is called the rate of change.

Mathematical Representation:

Similarly, when one quantity (x) differs with another quantity (y), satisfying y=f(x) then or f’(x) represents a rate of change with respect to y at x=𝑥0.

Mathematical Representation:

SO, the rate of change of y with x can be calculated using the rate of change of y and x with respect to t.

Increasing and Decreasing Functions:

DEFINITION: Strictly increasing:

function f:X→R is defined on an X⊂R is increasing if f(x)≤f(y) where x<y. If there is inequality that is strict, i.e., f(x)<f(y) where x<y, then f is strictly increasing. Strictly decreasing:

function f:X→R is defined on an X⊂R is decreasing if f(x)≥f(y) where x<y. If there is inequality that is strict, i.e., f(x)>f(y) where x<y, then f is strictly decreasing.

Mathematical equations:

a) increasing if x1 < x2 ⇒ f (x1) < f (x2) for all x1, x2 ∈ (interval)

b) decreasing if x1 < x2 ⇒f(x1) < f(x2) for all x1, x2 ∈ (interval)

c) if f(x) = c for all x ∈ intervals, where c is a constant.

d) decreasing if x1 < x2 ⇒ f (x1) ≥ f (x2) for all x1, x2 ∈ intervals.

e) strictly decreasing if x1 < x2 ⇒ f(x1) > f(x2) for all x1, x2 ∈ interval THEOREM:

Let f which denotes a function be a continuous function on [a, b] and is differentiable on the interval (a, b). Then

(a) f is said to be increasing in interval [a, b] if f ′(x) > 0 where x ∈ (a, b)

(b) f is said to be decreasing in interval [a, b] if f ′(x) < 0 where x ∈ (a, b)

(c) f is a said to be a constant function interval in [a, b] if f ′(x) = 0 where x ∈ (a, b)

Proof: Let x1, x2 ∈ to interval [a, b] where x1 < x2. Then, by Mean Value Theorem that is learned earlier

f(x2) > f(x1)

So x1<x2 in interval (a, b).

Hence, f is said to be an increasing function in [a, b].

Similarly, it can be done to decrease functions as well. By simply interchanging the signs.

Graph for the Functions:

Tangents and Normals:

We all know that a line passing through points (𝑥0, 𝑦0) having slope m is given by:

y - 𝑦0 = 𝑚(𝑥 − 𝑥0)

The slope of the tangent of the curve y = f(x) at point (𝑥0, 𝑦0) is at (𝑥0, 𝑦0).

Equation of tangent: y –𝑦0 = f ‘(𝑥0 )(x – 𝑥0) NOTE:

a) If a tangent makes a line with X-axis in a positive direction, then the slope of the tangent is tan 𝜃.

b) If the slope is 0°, that means the lines are parallel to X-axis, then tan 𝜃 = 0;

Then the tangent equation is: y=𝑦0

c) If the slope is 90°, that means the lines are parallel to the y-axis, then tan 𝜃 = infinity;

Then the tangent equation is x = 𝑥0

NORMAL: Normal is perpendicular to the tangent. So, the slope of normal to the -1/f’(x).

Equation of normal is: (y –𝑦0) f ‘(𝑥0 )+(x – 𝑥0)=0;

APPROXIMATIONS:

A function f: D → R, D is a subset of R (Real numbers), be function and y = f (x).

Then the small increment ∆𝑥 𝑏𝑒 𝑎 𝑑𝑒𝑛𝑜𝑡𝑒𝑑 𝑏𝑦 ∆𝑦 𝑖𝑠 𝑔𝑖𝑣𝑒𝑛 𝑎𝑠

dy = f (x + ∆x) – f (x).

This is called approximation.

a) here we have

b)

c)

Maxima and Minima :

If f be a function on an interval I. Then

(a) f will have a maximum value, if there is a point c in the interval in a way that f (c) > f(x), in the interval. Then f(c) is called the maximum value of the function f and that point where the maximum value is found is said to be a point of maximum for function f.

(b) b) f will have a minimum value, if there is a point c in the interval in a way that f (c) < f(x), in the interval. Then f(c) is called the minimum value of the function f and that point where the minimum value is found is said to be a point of minimum for function f.

(c) f(c) is said to be extreme if it is either a maximum or a minimum value. Point c is said to have an extreme value at that extreme point.

First Derivative Test:

f be a function in the interval I, f be continuous along with point c in I, then

a) When f’(x) changes sign from positive to negative when x increases passing through c, that is f’(x)>0 to left of c and f’(x)<0 to right of c, then c is called point of maxima.

b) When f’(x) changes sign from negative to positive when x increases passing through c, that is f’(x)<0 to left of c and f’(x)>0 to right of c, then c is called point of minima.

c) When f’(x) does not change when x increases, then c is neither a point of local maxima nor a point of local minima. It is called the point of inflection.

Second Derivative Test:

F be a function in the interval I, then f is the twice derivable (f’’(x)) at c, then

a) x=c, a point of local maxima only when f’(c) =0 and f’’(c) < 0 then f(C) is called local maximum of the function f.

b) x=c, a point of local minima only when f’(c) =0 and f’’(c) > 0 then f(C) is called a local minimum of the function f.

c) It doesn’t work if both f’(c)=0 and f’’(c)=0;

Absolute Maximum and Minimum Values:

f is continuous in interval I= [a, b]. f has both absolute

maximum and minimum values and f has at least one value in the interval I;

Theorem: f be a continuous function in interval I. f is a differentiable function in Interval I. c be any point

a) f’(c) =0 if f gets its absolute maximum value at point c.

b) f’(c) =0 if f gets its absolute minimum value at point c.

We have a few working rules to find these values:

Step 1: Finding the critical points

Find f’(x) =0 or not differentiable;

Step 2: Take the extreme points in the interval;

Step 3: calculate all the values of f found above in step 1,2;

Step 4: Find maximum and minimum values that are found in step 3. Among those maximum values, the greatest value of f will be the absolute maximum, and the least value will be the absolute minimum value of f.

Significance of NCERT Class 12 Maths Chapter 6 Notes:

NCERT Class 12 Maths chapter 6 notes will be very much helpful for students to score maximum marks in their 12 board exams. In Application of Derivatives Class 12 chapter 6 notes, we have discussed many topics like finding rate of change, maximum and minimum values, tangent and normal, local maximum and minimum values, absolute maximum and maximum values along with their conditions and rules of solving the problems. NCERT Class 12 Mathematics chapter 6 is also very useful to cover major topics of the Class 12 CBSE Mathematics Syllabus.

The CBSE Class 12 Maths chapter 6 will help to understand the theorems, examples along solutions in detail. This pdf also contains previous year’s questions and NCERT TextBook pdf. The next part contains FAQ’s most frequently asked questions along with topic-wise explanations. By referring to the document you can get a complete idea of all the topics of Class 12 chapter 6 Application of Derivatives pdf download.

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