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NCERT Exemplar Class 12 Maths Chapter 6 Solutions: Application of Derivatives

NCERT Exemplar Class 12 Maths Solutions Chapter 6 is a comprehensive list of questions on the topic of applications of derivatives. This chapter will teach you how to apply the concepts of the application of derivatives in many spheres like engineering, science, etc. This chapter of NCERT Exemplar Class 12 Maths is built on the previous chapter which is on differentiability and continuity. You will get in-depth knowledge of the rate of change of quantities, approximations, maxima, minima, increasing and decreasing functions, equations of tangents and normal to a curve, and the min and max values of a function in the closed interval.

This chapter has a total of 64 exemplar problems divided into 4 sections, which has a thorough coverage of this topic. The different sections are arranged based on the type of answers required. Instasolv aims not only at providing NCERT Exemplar solutions to the problems but also to strengthen the concepts and equip you with the skill to analyze the problem with clarity and ease. With the help of Instasolv solutions, you will get the flow of solving a problem and will be able to solve any complex problem related to derivatives independently.

Important Topics for NCERT Maths Exemplar Solutions Class 12 Chapter 6

  • The rate at which a quantity changes: The change in the quantity of one variable with respect to another quantity is called its rate of change. Hence if a quantity “a” changes with respect to “b” i.e. a = f(b) then:

da/db denotes the rate of change of a in relation with b.

E.g.: area of a square changes with its side changes. 

  • Increasing and decreasing function: One can determine if a function is increasing or decreasing within any intervals in its domain by its derivative. Let there be an interval “int’, such that f(x) > 0 at every point of “int”, then the function is said to be increasing in that interval. Similarly, the law applies for decreasing function if f(x) < 0 at every point in int. One can generalize this by considering f as a continuous function on [p,q] and differentiable in (p,q)
  • if f’ (x) > 0 for each x ∈ (p,q) then f is said to be increasing in [p,q]
  • if f’ (x) < 0 for each x ∈ (p,q) then f is said to be decreasing in [p,q]
  • if f’ (x) = 0 for each x ∈ (p,q) then f is said to be constant in [p,q]
  • Tangents and normals: Let there be a curve c = f(a) and a line touches the curve at P (p1,p2) then that line is called a tangent to the curve at that point then:
    • The slope of the tangent sl = (dc/da)p
    • Equation of the line passing through the curve at P (p1,p2) is; c – p1 = sl ( a – p2)
    • Equation of a tangent to the curve at P(p1,p2) is; c – p2 = (dc/da)p (a-p2)
    • Equation of normal to the tangent at the point of contact P(p1,p2) is; c – p2 = (-1 / (dc/da)p) * (a-p2)
  • Approximation – If one variable changes minimally with respect to another variable then we get the approximate value by applying differentiation.

f(a+Δa)=f(a)+f′(a)Δa

Here Δa = small change in the value of a

  • Maxima and minima – Maxima and minima of a function, also called extrema, are its largest and smallest value respectively within a given range or the entire domain. 
  • Local maxima – For a function f(x), its local maximum within an interval is greater than or equal to any other value in that range. So if “a” is the maximum point then: f(a) >= f(x), for any x in the interval
  • Local minima – For a function f(x), its local minimum within an interval is less than or equal to any other value in that range. So if “b” is the minimum point then: f(b) <= f(x), for any x in the interval
  • The critical point of a function f – The point in the domain of a function where either the function is not differentiable or its derivative is 0, is called the critical point of that function. f’(x) = 0

Discussion of Exercises of NCERT Maths Exemplar Class 12 Solutions Chapter 6

  1. The first section of NCERT Exemplar Class 12 Maths problems for Chapter 6 has 24 questions which are short answer types and based on the concept of approximation, rate of change of quantity, increasing/decreasing functions, and tangents and normals.
  2. The second section has 10 questions which are long answer types and based on the concept of maxima and minima, and tangents and normals.
  3. The third section has 25 questions which are objective type and they brush up on all the concepts discussed in this chapter. This section has questions on the rate of change of quantity, tangents and normals, increasing/decreasing functions, and minima/maxima.
  4. The fourth section has 5 questions which are filled in the blank types.  The topics in this section are based on tangents and normals and minima/maxima of a function.
  5. The NCERT exemplar questions are designed in collaboration with DESM (Department of Education in Science and Mathematics) to get the best quality problems to you which will test you in varying levels of difficulty on any given topic. The variety in NCERT exemplar questions is apt for you from the point of view of boards as well as competitive exams.

Why Use NCERT Science Exemplar Problems with Solutions Class 12 Chapter 6 by Instasolv?

Instasolv has given NCERT Exemplar solutions to all the problems in this chapter in clear steps which clarifies all your doubts and will let you solve any kind of problem. Instasolv is free of cost but gives priceless and effective tools to you using which you can get excellent results in their CBSE exams and other exams like IIT JEE. It helps you to do better time management while in a stressful exam environment. To score well in the exam, you should only look up to NCERT Solutions and NCERT Exemplar Class 6 Science Solutions as they are self-sufficient and cover the entire syllabus of CBSE.