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

The application of derivatives is a key concept in calculus used to analyze the behavior of functions in real-life and mathematical problems. Derivatives help us understand the variation of a function with respect to the variable. For example rate of change of the position of a particle is velocity, and this rate is known as the derivative of position with respect to time. In the applications where optimization needs to be done, derivative plays an important role in finding the points of maximum and minimum value. Class 12 Maths Chapter 6 notes contain the topics such as finding the derivatives of the equations, rate of change of quantities, increasing and decreasing functions, Tangents and Normals, Maxima and minima, etc. In NCERT Class 12 Math Chapter 6 notes, there are related theorems and their proofs, which are very important from an examination point of view.

NCERT notes Class 12 Maths Chapter 6 offer well-explained and structured content to help the students grasp the concepts of application of derivatives easily. These notes of Class 12 Maths Chapter 6 are developed by the Subject Matter Experts as per the latest CBSE syllabus, ensuring that students won't miss any concept. NCERT solutions for class 12 maths and NCERT solutions for other subjects and classes can be downloaded from NCERT Solutions.

Application of Derivatives:

Rate of change:

For the function $y=f(x),\frac{d}{dx}(f(x))$ represents the rate of change of $y$ with respect to $x$. Thus, if ' $s$ ' represents the distance and ' $t$ ' the time, then $\frac{ds}{dt}$ represents the rate of change of distance with respect to time.

DERIVATIVE:

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

Mathematical Representation: $\frac{ds}{dt}$

Mathematical Representation: $\frac{dy}{dx}=\frac{dy}{dt}\frac{dt}{dx}$ if $\frac{dx}{dt}\ne 0$

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 the interval [a, b] if f ′(x) > 0 where x ∈ (a, b)

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

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

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

$\begin{array}{rl}& f^{\prime }(c)=\frac{f(x2)-f(x1)}{(x2-x1)}\Rightarrow f(x2)-f(x1)=f^{\prime }(c)(x2-x1)\\ & \Rightarrow f(x2)-f(x1)>0\end{array}$

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:

A line touching a curve $y=f(x)$ at a point $(x_1,y_1)$ is called the tangent to the curve at that point and its equation is given $y-y_1={\left(\frac{dy}{dx}\right)}_{(x_1,y_1)}(x-x_1)$. The normal to the curve is the line perpendicular to the tangent at the point of contact, and its equation is given as:

$y-y_1=\frac{-1}{{\left(\frac{dy}{dx}\right)}_{(x_1,y_1)}}(x-x_1)$

The angle of intersection between two curves is the angle between the tangents to the curves at the point of intersection.

APPROXIMATIONS:

Since $f^{\prime }(x)=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}$, we can say that $f^{\prime }(x)$ is approximately equal to $\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}$ $\Rightarrow$ approximate value of $f(x+\mathrm{\Delta }x)=f(x)+\mathrm{\Delta }x\cdot f^{\prime }(x)$.

Increasing/decreasing functions

A continuous function in an interval $(a,b)$ is :
(i) strictly increasing if for all $x_1,x_2\in (a,b),x_1<x_2\Rightarrow f\left(x_1\right)<f\left(x_2\right)$ or for all $x\in (a,b),f^{\prime }(x)>0$
(ii) strictly decreasing if for all $x_1,x_2\in (a,b),x_1<x_2\Rightarrow f\left(x_1\right)>f\left(x_2\right)$ or for all $x\in (a,b),f^{\prime }(x)<0$

Theorem : Let $f$ be a continuous function on $[a,b]$ and differentiable in $(a,b)$ then
(i) $f$ is increasing in $[a,b]$ if $f^{\prime }(x)>0$ for each $x\in (a,b)$
(ii) $f$ is decreasing in $[a,b]$ if $f^{\prime }(x)<0$ for each $x\in (a,b)$
(iii) $f$ is a constant function in $[a,b]$ if $f^{\prime }(x)=0$ for each $x\in (a,b)$.

Maxima and Minima :

Local Maximum/Local Minimum for a real-valued function $f$
A point $c$ in the interior of the domain of $f$ is called
(i) local maximum, if there exists an $h>0$, such that $f(c)>f(x)$, for all $x$ in $(c-h,c+h)$.

The value $f(c)$ is called the local maximum value of $f$.
(ii) local minima if there exists an $h>0$ such that $f(c)<f(x)$, for all $x$ in $(c-h,c+h)$.

The value $f(c)$ is called the local minimum value of $f$.
A function $f$ defined over $[a,b]$ is said to have maximum (or absolute maximum) at $x=c,c\in [a,b]$, if $f(x)\le f(c)$ for all $x\in [a,b]$.

Similarly, a function $f(x)$ defined over $[a,b]$ is said to have a minimum [or absolute minimum $]$ at $x=d$ if $f(x)\ge f(d)$ for all $x\in [a,b]$.
Critical point of $f$: A point $c$ in the domain of a function $f$ at which either $f^{\prime }(c)=0$ or $f$ is not differentiable is called a critical point of $f$.

First Derivative Test:

f is a function in the interval I, and f be continuous at 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 the right of c, then c is called a 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 the left of c and f’(x)>0 to the right of c, then c is called a 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 is a function in the interval I, then f is 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 the 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 the interval I. f is a differentiable function in the 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 the 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 helpful for students to score maximum marks in their 12 board exams. In the Application of Derivatives Class 12 chapter 6 notes, we have discussed many topics like finding the rate of change, maximum and minimum values, tangent and normal, local maximum and minimum values, and absolute maximum and minimum values, along with their conditions and rules of solving the problems. NCERT Class 12 Mathematics chapter 6 is also very useful to cover the 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 the previous year’s questions and the NCERT textbook PDF. The next part contains the 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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• NCERT problems are very important in order to perform well in exams. Students must try to solve all the NCERT problems, including miscellaneous exercises, and if needed, refer to the NCERT Solutions for Class 12 Maths Chapter 6 Application of Derivatives
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