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CHAPTER 6 : APPLICATION OF DERIVATIVES
Application of Derivatives, is an important topic for students who are pursuing higher studies in mathematics. The chapter deals with the concept of derivatives and its applications in solving different mathematical problems. It comprises of a total of 3 exercises with 24 questions in total. Students can access the NCERT solutions for class 12 maths chapter 6 from the official website of NCERT or through various educational websites.
The chapter starts with a brief introduction of the derivative concept and its importance in calculus. It further explains the different methods of finding derivatives of various functions, such as algebraic, trigonometric, logarithmic, and exponential functions. The chapter also covers the concept of second-order derivatives, which are used in analyzing the shape and behavior of a curve.
The first exercise of the chapter deals with finding the derivative of a given function using different methods. It comprises of a total of 9 questions, which include functions such as polynomial, exponential, logarithmic, and trigonometric functions. The questions are designed to help students understand the different techniques used to find the derivative of a function.
The second exercise of the chapter deals with the applications of derivatives in solving real-life problems. It comprises of a total of 10 questions, which include problems related to optimization, rate of change, and approximations. The questions are designed to help students understand the practical applications of derivatives in various fields such as economics, physics, and engineering.
The third and final exercise of the chapter deals with the concept of second-order derivatives and their applications. It comprises of a total of 5 questions, which include problems related to the analysis of curvature and convexity of a curve. The questions are designed to help students understand the importance of second-order derivatives in analyzing the behavior of a curve.
The NCERT solutions for class 12 maths chapter 6 are designed to help students understand the concepts of derivatives and their applications in a systematic manner. The solutions provide step-by-step explanations of the questions and help students to understand the underlying concepts.
One of the important applications of derivatives is in the field of optimization. Optimization problems involve finding the maximum or minimum value of a function. The process involves finding the derivative of the function and equating it to zero. The critical points obtained are then analyzed to find the maximum or minimum value of the function.
FOR EXAMPLE,
consider a rectangular field with a fixed perimeter of 100 meters. We need to find the dimensions of the field such that the area is maximum. Let the length and breadth of the field be l and b, respectively.
We know that the perimeter of the field is 2(l+b)=100. Rearranging the terms, we get l+b=50. The area of the field is given by A=lb.
To find the maximum value of A, we need to find the derivative of A with respect to l and equate it to zero.
dA/dl=b
To find the value of b, we substitute the value of l+b=50 in the equation A=lb.
A=l(50-l)
Taking the derivative of A with respect to l, we get
dA/dl=50-2l
Equating dA/dl to zero, we get
50-2l=0
l=25
Substituting the value of l in the equation l+b=50, we get
b=25
Therefore, the dimensions of the field are 25 meters by 25 meters, and the area is 625 square meters.
Another application of derivatives is in finding the rate of change of a function. For example, consider a circle with radius r.
CHAPTER 6 NOTES PDF DOWNLOAD
NCERT SOLUTIONS FOR CLASS 12TH MATHS :
CHAPTER 6 : APPLICATION OF DERIVATIVES
NCERT Solutions for Class 12 Maths Chapter 6 is an important study material for students preparing for board exams and other competitive exams. The chapter deals with the application of derivatives in solving real-life problems. Derivatives are an important concept in calculus, and their applications are found in various fields like physics, economics, engineering, etc. In this article, we will discuss the NCERT Solutions for Class 12 Maths Chapter 6 in detail.
SECTION 6.1 : INTRODUCTION
The first section of Chapter 6 gives an introduction to the concept of derivatives and their applications. The concept of derivatives is essential in finding the rate of change of a function with respect to its independent variable. The chapter discusses the concept of the derivative of a function, the meaning of the derivative, and the notation used for the derivative. It also explains the concept of the derivative of a composite function and the chain rule.
SECTION 6.2 : RATE OF CHANGE OF QUANTITIES
The second section of Chapter 6 explains how derivatives can be used to find the rate of change of quantities. It discusses the concept of average rate of change and instantaneous rate of change. The chapter also explains how the concept of derivatives can be used to find the velocity and acceleration of a moving object.
SECTION 6.3 : INCREASING AND DECREASING FUNCTIONS
The third section of Chapter 6 discusses increasing and decreasing functions. The chapter explains how the sign of the derivative can be used to determine whether a function is increasing or decreasing. It also discusses the first derivative test and the second derivative test for finding the maximum and minimum values of a function.
SECTION 6.4 : TANGENTS AND NORMALS
The fourth section of Chapter 6 discusses tangents and normals. The chapter explains how the concept of derivatives can be used to find the equations of tangent and normal to a curve at a given point. It also discusses the angle between two curves and the condition for perpendicularity of two curves.
SECTION 6.5 : APPROXIMATIONS
The fifth section of Chapter 6 discusses approximations. The chapter explains how derivatives can be used to find the approximate value of a function near a given point. It also discusses the concept of differentials and their applications in finding approximate values.
SECTION 6.6 : MAXIMA AND MINIMA
The sixth section of Chapter 6 discusses the concept of maxima and minima. The chapter explains how the concept of derivatives can be used to find the maximum and minimum values of a function. It also discusses the necessary and sufficient conditions for a function to have a maximum or minimum value.
SECTION 6.7 : SUMMARY
The last section of Chapter 6 gives a summary of the concepts discussed in the chapter. It provides a brief overview of the applications of derivatives in finding the rate of change of quantities, increasing and decreasing functions, tangents and normals, approximations, and maxima and minima.
CONCLUSION
NCERT Solutions for Class 12 Maths Chapter 6 is an important study material for students preparing for board exams and other competitive exams. The chapter deals with the application of derivatives in solving real-life problems. Derivatives are an important concept in calculus, and their applications are found in various fields like physics, economics, engineering, etc. The chapter discusses the concept of the derivative of a function, the meaning of the derivative, and the notation used for the derivative. It also explains the concept of the derivative of a composite function and the chain rule. The chapter explains how derivatives can be used to find the rate of change of quantities, increasing and decreasing functions, tangents and normals, approximations, and maxima and minima. Students are advised to practice the problems given in the NCERT.
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