NCERT Solutions Class 12 Mathematics Chapter 6 – Application of Derivatives
Mathematics is a subject requiring a strong conceptual understanding of its applications. Hence, students are advised to study the subject from good resources to gain in-depth knowledge about Mathematics chapters with examples, solutions, theorems as well as exercises and to score well in exams.
The Class 12 Mathematics Chapter 6 ‘Application of derivatives’ is a part of the Calculus section of Mathematics. It covers the various applications of derivatives and their role while doing calculations. The vital topics covered in Chapter 6 Class 12 Mathematics include:
- Introduction
- Rate of Change of Quantities
- Increasing and Decreasing Functions
- Tangents and Normals
- Approximations
- Maxima and Minima
- Maximum and Minimum Values of a Function in a Closed Interval
The chapter’s important formulas are listed in the NCERT Solutions Class 12 Mathematics Chapter 6. Also, one can find easy ways to carry out calculations after referring to them thoroughly and consistently.
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Key Topics Covered in NCERT Solutions Class 12 Mathematics Chapter 6
NCERT Solutions Class 12 Mathematics Chapter 6 is about derivatives and their applications.
This chapter is a part of Calculus and requires the study of differential calculus. One should have a good command of it to excel in this chapter.
If you have a hold on calculus, this chapter will help students to learn to carry out calculations easily. As a result, they will be able to approach problems more logically. The entire chapter is covered thoroughly in the NCERT Solutions Class 12 Mathematics Chapter 6 and is available on the Extramarks website.
After completing Chapter 6 Mathematics Class 12, students will learn to analyse the problems with a better approach and be able to solve them easily.
Introduction
A derivative is the rate of change or the amount on which a particular function changes at one given point.
In the above-given figure, the function is represented in black colour, and a tangent is represented in red colour.
Rate of Change of Quantities
The derivative ds/dt is used to show the rate of change of distance s to the time t.
Assume a particular quantity y varies with another quantity x which satisfies y = f(x), here dy/dx or f’(x) shows the rate of change of y w.r.t. x and [dy/dx]x = x0 or f’(x0) shows the rate of change of y w.r.t. x at x = x0.
Now, assume the two variables x and y are varying w.r.t. another variable t i.e. x = f(t) and y = g(t)
Thus, by chain rule
dy/dx = (dy/dt)/(dx/dt) here dx/dt ≠ 0
So, the rate of change of y to x could be calculated using the rate of change of y at a given time and that of x to t.
Increasing and Decreasing Functions
The function’s derivative might determine if it is increasing or decreasing at any intervals in its domain.
Assume I will be an open interval contained in the domain of the real-valued function f. Then f is said to be
(i) increasing on I when x1 < x2 in I => f(x1) ≤ f(x2) for all x1, x2 Є I.
(ii) strictly increasing on I if x1 < x2 in I => f(x1) < f(x2) for all x1, x2 Є I.
(iii) decreasing on I if x1 < x2 in I => f(x1) ≥ f(x2) for all x1, x2 Є I.
(iv) strictly decreasing on I if x1 < x2 in I => f(x1) > f(x2) for all x1, x2 Є I.
However, some functions are neither increasing nor decreasing.
The graphical representations of all these functions are given below:
function f will be said to be increasing at x0 when there exists an interval I = (x0 – h, x0 + h), h > 0 such that for x1, x2 ∈ I,
x1 < x2 in I => f (x1) ≤ f (x2)
With the help of a theorem listed in the NCERT Solutions Class 12 Mathematics Chapter 6, you can test for increasing and decreasing functions for a given interval.
Tangents and Normals
For the given equation of a straight line that is passing through a given point (x0, y0) having a finite slope, m is represented as y – y0 = m(x – x0)
Assume the given curve y = f(x), and the tangent of the curve at that point (x0, y0) will be
[dy/dx](x0, y0) or f’(x0)
Now, the equation of the tangent of the curve y = f(x) at (x0, y0) will be
y – y0 = f’(x0)(x – x0)
The slope of the normal to curve y = f(x) at (x0, y0) is given by -1/ f’(x0) here f’(x0) ≠ 0
For the equation of normal to the curve, y = f(x) at (x0, y0) will be
Y – y0 = (-1/f’(x0))(x – x0)
(y – y0)* f’(x0) +(x – x0) = 0
Particular cases
(i) If the slope of the tangent line is zero, then tan θ = 0, i.e. θ = 0, which means a tangent line is parallel to the x-axis. For this case, the equation of the given tangent at the point (x0, y0) will be y = y0.
(ii) If θ -> π/2 then tan θ → ∞, that means the tangent line will be perpendicular to the x-axis, that is parallel to the y-axis. For this case, the equation of the tangent at (x0, y0) will be x = x0.
Approximations
An approximation is anything which is similar but not the same as something else.
Assume f : D => R, D Ì R, will be a given function and assume y = f (x). Assume ∆x denotes a small increment in terms of x.
Now, consider the increment in y corresponding to the increment in x will be ∆y = f (x + ∆x) – f (x).
(i) The differential of x is represented as dx = ∆x.
(ii) The differential of y is represented as dy = f’(x) dx or dy = (dy/dx) * ∆x
Maxima and Minima
In the section, you will learn the different methods of calculating a function’s maximum and minimum values in a given domain. You will also learn about the absolute maximum and minimum of a function used to find the solution to many applied problems.
You will clearly understand this topic through the definitions and theorems included in the NCERT Solutions Class 12 Mathematics Chapter 6 available on the Extramarks official website.
Maximum and Minimum Values of a Function in a Closed Interval
You will learn about the two theorems to find out the absolute maximum and minimum values of a function upon a closed interval I.
Theorem: Let f be a continuous function on an interval I = [a, b]. Then f has the absolute maximum value, as well as f attains it at least once in I. Also, f has the absolute minimum value and attains it at least once in I.
Theorem: Let f be a differentiable function on a closed interval I and let c be any interior point of I. Then
(i) f′(c) = 0 when f attains its absolute maximum value at c.
(ii) f′(c) = 0 when f attains its absolute minimum value at c.
NCERT Solutions Class 12 Mathematics Chapter 6 Exercise & Solutions
Find all the NCERT solutions to the exercises covered in the chapter in the NCERT Solutions Class 12 Mathematics Chapter 6. It is prepared by the subject experts while adhering to the NCERT book and following the CBSE guidelines and curriculum.
All the vital topics and key concepts are covered in a point-wise manner. NCERT Solutions Class 12 Mathematics Chapter 6 is written end-to-end, highlighting everything in detail by the faculty experts. . Therefore, it is trusted by students and teachers across the private and government schools.
You can avail of the NCERT Solutions Class 12 Mathematics Chapter 6 from the Extramarks website.
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Click on the below links to view exercise-specific questions and solutions for NCERT Solutions Class 12 Mathematics Chapter 6:
- Class 12 Mathematics Chapter 6: Questions and Answers
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NCERT Solutions Class 12 Mathematics Chapter 6 Exercise & Answer Solutions
NCERT Solutions Class 12 Mathematics Chapter 6 Exercise and Answer Solutions are available for students to refer for free on the Extramarks website. The material offers step-by-step solutions that help students understand how to solve problems relating to the chapter. The solution also helps students solve complex questions in a simplified manner. Students may refer to NCERT Solutions Class 12 Mathematics Chapter 6 to strengthen their basics.
Students may refer to the links below to download exercise-specific questions and solutions for NCERT Solutions Class 12 Mathematics Chapter 6 Application of Derivatives:
NCERT Exemplar Class 12 Mathematics
NCERT Exemplar Class 12 Mathematics book is an excellent source for students preparing for JEE Mains, NEET, MHT-CET etc. It has questions according to the topics and concepts covered in the NCERT textbook. This aids students in solving all types of questions confidently.
It has different questions with varying levels of difficulty, which helps students to improve their performance in various tests and exams and definitely builds their confidence level in the process. The questions are selected from different sources for students to prepare and improve their performance. It is also a complete source of information for CBSE students preparing for their 12th standard examinations. The students think logically about a problem after referring to the NCERT Solutions Class 12 Mathematics Chapter 6 and NCERT Exemplar.
The NCERT solutions Class 12 Mathematics Chapter 6 is prepared after analysing CBSE past years’ question papers. It contains extra questions from the NCERT Class 12 Mathematics textbook. Students can refer to NCERT Exemplar for Class 12 Mathematics for more practice and face their examinations with courage. They can assure themselves that nothing remains untouched in the chapter and will score very well in all their examinations. To get good grades in exams students must refer to multiple study resources, practice a lot of questions and stick to a study schedule and follow it religiously to come out with flying colours.
Key Features for NCERT Solutions Class 12 Mathematics Chapter 6
Regular practice is quite necessary for the students to excel. Hence, NCERT Solutions Class 12 Mathematics Chapter 6 helps students develop the habit of regular practice and to clarify their doubts then and there, take regular tests to assess their performance and get proper feedback to step up their learning. . The key features are as follows:
- You can find all the topics covered as well as in text to end text exercises from the chapter covered in the NCERT Solutions Class 12 Mathematics Chapter 6.
- It helps students analyse the easy and difficult topics in the chapter.
- After completing the NCERT Solutions Class 12 Mathematics Chapter 6, students will become experts in solving derivatives problems.