NCERT Solutions for Class 12 Maths Chapter 6 - Application of Derivatives
Download the CBSE NCERT Solutions for Class 12 Maths Chapter-6 PDF from the official website of Vedantu. These solutions are prepared by subject matter experts who give lucid explanations for each of the topics discussed in the Application of Derivatives Class 12. The solution providers have a lot of experience in the education domain and they understand how to design a solution that caters to the understanding level of students of a particular class. In AOD Class 12 Solutions, even complex solutions are made easy by these experienced teachers by breaking them up into smaller chunks and giving you tips to remember major formulas and functions. You will also receive timely help from our experts and teachers if you face any doubts while going through the Application of Derivatives Class 12 NCERT Solutions.
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NCERT Solutions for Class 12 Maths Chapter 6 - Application of Derivatives
NCERT Solutions for Class 12 Maths – Free PDF Download
In today’s age, when students have so many assignments and tests to take, it becomes burdensome to handle them all. So, finding quick and quality solutions online would provide you with tremendous support in progressing with your studies. Finding NCERT Solutions for Class 12 Maths in the PDF format on the official website of Vedantu would give you all the help that is needed in tackling all the problems of Chapter 6 Maths Class 12.
Chapter 6 – APPLICATION OF DERIVATIVES
NCERT Solutions for Class 12 Maths Chapter 6
In Ch 6 Maths Class 12, the introduction part will be a recap of the last chapter where you got acquainted with concepts of derivative of a composite function, implicit functions, logarithmic functions, inverse trigonometric functions and exponential functions.
In this chapter, you will learn how the application of derivatives works in different disciplines like social science, engineering, science, etc. This chapter will go through how to apply the application of derivatives to find out the rate of change of quantities.
Other Key Concepts Discussed in This Chapter Are
Equations of a tangent and normal to a curve at a point.
Find out turning points on the graph which will help you determine the points where maximum and minimum values of a function occur.
Find intervals where a function increases and decreases.
Approximations and errors.
The chapter will also give you ample practice with a variety of exercises of different kinds like short and long answers, objective types questions, etc. These activities would further strengthen all the important topics learned in this chapter and help you tremendously in your preparation for your board’s exams.
6.2 Rate of Change of Quantities
In NCERT Solutions for Class 12 Chapter 6 Applications of Derivatives, you would brush up all the learnings from the previous class. You would be reminded that to represent the rate of change of quantity with respect to another, we use the notation dy/dx, this means that y changes when x is changed.
You would further expand this concept to find out if changes in x and y are dependent on a 3rd variable (say z) i.e. x= f(z) and y=g(z), Chain rule can be applied as shown below:
dy/dx = (dy/dz)/(dx/dz), given dx/dz <> =0
Exercise 6.1 Solutions: 18 Questions (10 Long Questions, 6 Short Questions, 2 MCQs)
6.3 Increasing and Decreasing Functions
In this section of AOD Class 12 NCERT solutions, you would understand what is meant by increasing and decreasing function and how to determine whether a function is increasing or decreasing in a given range by using differentiation. You would define that a function is increasing in a range if the first derivative is positive in that range. Similarly, you would learn that a function is decreasing in a range if the derivative is negative in that range.
You will learn that in the case of trigonometric functions, one has to determine the quadrant in which they lie in order to know if they are increasing or decreasing. So, in this case, you would understand that for increasing value of x, y is increasing then it is an increasing function, and if y decreases with an increase in the value of x, then it is a decreasing function. For example, sin x increases in the first quadrant while cos x is a decreasing function in the same quadrant.
You Should Write Down The Above Laws in The Formula As Depicted Below
Increasing Functions –> x1 < x2 in interval i implies f(x1) < f(x2) where x1 and x2 є i
Decreasing Functions -> x1 < x2 in interval i implies f(x1) > f(x2) where x1 and x2 є i
Exercise 6.2 Solutions: 19 Questions (10 Long questions, 7 Short questions, 2 MCQ)
6.4 Tangents and Normals
In this part of NCERT Solutions for Class 12 Maths Application of Derivatives, you would build your knowledge of the equation of straight lines learned in previous chapters. You would rekindle that the equation of a straight line on a curve y = f(x) with a finite slope s, passing through a given point (a, b) is given by:
y - b = s (x – a)
You would learn that a tangent to a point (a,b) on a curve is given by:
[dy/dx](a,b) and is denoted by f’(a) hence the equation of a tangent to the curve is given by : y – b = f’(a) (x – a).
You would also get to know that normal is perpendicular to the tangent whose equation is given by y – b = (-1/ f’(a)) (x – a)
With Application of Derivatives Class 12 Solutions, you would get a good idea about what methods to adopt to calculate the expressions utilizing the slope formula.
Exercise 6.3 Solutions: 27 Questions (25 Short Questions, 2 MCQ)
This part of Ch 6 Class 12 Maths NCERT Solutions will introduce you to the concept of using differentiation in order to find approximate values of certain quantities. You would get to know how approximations are useful when an exact numerical number is not easy to obtain. You would understand how to derive the approximate value of a quantity that changes in small measures by a change in another variable. So if y = f(x) and dx is a small increment in x which makes an increment of dy in the value of y, you would learn that dy = f (x + dx) – f(x) or dy = f’(x) dx.
The small change is generally denoted by Δ and approximation by ≈. So you will learn how to use these symbols to express the above equation as dy = (dy/dx) Δx.
From the learnings in this chapter, you will be able to figure out that the derivative of the dependent variable (y) is not equal to the increase in the variable while the derivative of the independent variable (x) is equal to the increase in the variable.
Exercise 6.4 Solutions: 9 Questions (7 Short Questions, 2 MCQ)
6.6 Maxima and Minima
This unit of NCERT Solutions for Class 12 Maths Chapter 6 describes the minimum and maximum values of a function in the form of derivatives. You will learn about turning points on the graph of a function where the graph reaches its highest point and the lowest point locally in a domain. You would understand how these points can be used to sketch the graph of a function.
You will also be able to find the absolute maximum and absolute minimum of a function which are useful in solving many applied problems. We would learn various definition around maxima and minima which are used for solving problems related to it like:
In a range r, function f is said to have maximum value if there exists a point p in r where f(p) >= f(x) for all x є r. The value f(p) defines the maximum value of f in r, and the point p is termed as the point of the maximum value in r.
In a range r, function f is said to have minimum value if there exists a point p in r where f(p) <= f(x) for all x є r. The value f(p) defines the maximum value of f in r, and the point p is termed as the point of the minimum value in r.
In a range r, function f is said to have extreme value if there exists a point p in r where f(p) is either a maximum value or a minimum value of f in r and the point p is termed as the extreme point in r.
You would also learn how local maxima and local minima are represented and many theorems like the first derivative test, second derivative test around this concept.
Exercise 6.5 Solutions: 29 Questions (15 Long Questions, 11 Short Questions, 3 MCQ)
Key Features of NCERT Solutions for Class 12 Maths Chapter 6
If you want to ace your exams and score well, having solutions to Maths problems becomes essential. Maths is an important subject for boards and also for many competitive exams. Hence if you get the solutions readily available by a stellar team of experts at Vedantu, you have the edge over others. The key benefits of Maths NCERT Solutions Class 12 Chapter 6 are:
The solutions are well-crafted by subject matter experts who have given a proper explanation for every step of the problems.
Using these solutions, you can get better clarity of the topic and be able to solve even difficult problems with proper time management.
Many tips and tricks of remembering important formulas will be shared.
You can always seek assistance from the teachers online in case you get doubts while going through the solutions.
FAQs (Frequently Asked Questions)
1. Give me an overview of the topics/ subtopics of Class 12 Maths Chapter 6?
A. There are a total of six topics/ sub-topics including a Miscellaneous Q&A at the end. Take a look at the list given below.
6.1 - Introduction
6.2 - Rate of Change of Quantities
6.3 - Increasing and Decreasing Functions
6.4 - Tangents and Normals
6.5 - Approximations
6.6 - Maxima and Minima
2. Why should I refer to Vedantu’s NCERT Solutions for Class 12 Maths Chapter 6?
A. There are various reasons why you should always opt for the NCERT Solutions for Maths Class 12 Chapter 6 created by Vedantu.
CBSE highly recommends the NCERT books to every class 12 student as these NCERT books proved to be the best guide with detailed study material including the important topics. Those appearing for the board examinations should opt for the NCERT Solutions for class 12 maths chapter 6 while solving the questions from the textbook.
Our NCERT solutions for Class 12 maths chapter 6 play a significant role for you as you can get to know the answers to all the questions at one single place.
NCERT solutions for Class 12 maths Chapter 6 are prepared by our subject matter expert and faculties to help you in board exam preparations. These solutions help you to solve the problems easily and in a concise manner, so that you score good marks in exams.
NCERT solutions provide a detailed and step-by-step explanation of each answer to the questions asked in the exercises of the Class 12 Maths textbook Chapter 6.
3. What will I learn from Chapter 6 of Class 12 Maths NCERT textbook?
A. In NCERT solutions for class 12 maths chapter 6, you will learn the application of derivatives, finding rate of change, show increasing/decreasing in whole domain, in intervals, find intervals of increasing/decreasing, Rolle’s theorem, Lagrange’s Mean Value theorem, finding slope of tangent/normal, point when tangent is parallel/ perpendicular, when point and curve is known, when slope and curve are known, approximate value of numbers, function, minimum and maximum values from graph, local maxima and minima, absolute minima/maxima.
4. How many questions are there in each exercise of this chapter?
A. There are various types of questions in each exercise of this chapter. We have provided a list below containing the number and types of questions asked in the exercises of Chapter 6 of Class 12 Maths.
Exercise 6.1: 18 Questions (10 Long answer type, 6 Short answer type, 2 MCQ)