Chapter 13

Limits and Derivatives

Although in a primitive form, Calculus is covered in Class 11 CBSE and is an essential subject in the curriculum. The topics include function derivatives and integrals. We also cover Chapter 13 Class 11 Maths NCERT Solutions here.

Limits and Derivatives are incredibly important topics in mathematics that have applications in math and other subjects, such as physics. In this article, you will find most concepts pertaining to NCERT Solutions For Class 11 Maths Limits And Derivatives, as well as their properties and formulae. 

Class 11 Maths Limits and Derivatives is an important chapter that requires thorough preparation. We have tailored our Class 11 Maths NCERT Solutions Limits And Derivatives to meet the students’ needs. In this chapter, many aspects of Limits and Derivatives are explored. The Chapter 13 Class 11 Maths NCERT Solutions format follows a step-by-step procedure that aids in comprehending the topics.

All of the core principles are elaborated in Class 11 Maths Chapter 13 NCERTSolutions. These solutions will help you in understanding every problem of your NCERT Textbook.

Topics covered in this Chapter: (Content Table)

Sr. No.

Topic

1.

Introduction

2.

Intuitive Idea of Derivatives

3.

Limits

4.

Limits of Trigonometric Functions

5.

Derivatives

6.

Solved Examples for CBSE Class 11 Maths Chapter 13

7.

FAQs

 

Introduction

Sir Isaac Newton formulated the essential laws of differential calculus centered on his conceptions of rate and change, while integral calculus emerged due to the reverse process. Differentiation and calculus are the foundations for advanced mathematics, contemporary physics, and other modern engineering and science disciplines. For CBSE students, NCERT Solutions Class 11 Maths Limits And Derivatives serves as an introduction to calculus.

Limits and derivatives account for over half of calculus subjects and are like an oasis in a desert of difficulty. This chapter summarizes the calculus, derivative, limits, and algebra of limits branches of mathematics. Various standard functions' derivatives are also discussed.

      • Calculus is used to determine an object's velocity and trajectory.

      • In electrical engineering, calculus is used to determine the length across two stations that are kilometres apart.

      • In operations research, calculus is utilized to increase effectiveness, productivity, and profit.

Let's have a look at the topics below to obtain a better understanding of these topics.

 

This section explains how to calculate the derivative of a function using a real-life example of distance covered at various time intervals.

The derivative refers to the pace at which one quantity changes in relation to another. If you run a business (for example, selling sweets), derivatives can assist you in figuring out how much to sell. Gather the data on your company's performance over the previous few months, and the trend can be assessed by plotting a curve. The instantaneous slope (dy/dx) of such a graph is defined as ‘derivative.’

The slope indicates the marginal cost of generating each extra unit at that time (cost change/unit produced change). If this cost exceeds the selling price of 10 dollars, a loss is incurred at this moment.

For instance,

If the cost exceeds the sales for 0-300 units, it's a loss.

If sales exceed costs for 400-600 units, a profit is made.

 

Derivatives and Limits Class 11 includes a variety of topics that serve as the foundation for Advanced Calculus. Here is a list of possible topics:

Defining Limits 

Limits in Calculus are assigning values to specific functions in areas where no earlier values have been defined. A limit is a value that a function approaches as input and provides a certain value, as described in mathematics. Integrals, derivatives, and continuity are all defined by limits in mathematical analysis and calculus.

The limit of a function is expressed as:

0/0 Form's Limits

It denotes an 'indeterminant,' where the solution is 0/0 after considering the limit. Infinity or 1/0 is an example of an undefined number.

xⁿ Formula's Limits

The concept is related to the derivatives power rule.

Derivatives by other Trigonometric Formula

There are derivatives of four other forms of trigonometric functions.

 

There are six trigonometric functions. The limit of each function approaching a point can be estimated based on the function's continuity when its domain and range are taken into account.

Sine Function

The function f(x) = sin(x) is a continuous function throughout its whole domain, including all real numbers as its domain.

Cosine Function

The function f(x) = cos(x) is a continuous function throughout its whole domain, with all real numbers as its domain.

Tangent Function

The function f(x) = tan(x) gets defined for all real numbers except for those where cos(x) is equivalent to 0, that is, for all integers n, the values /2 + n. As a result, it has a domain that includes all real numbers except /2 + n, n € Z.

This function has a range of  (-∞, +c).

Cosec

The function f(x) = cosec(x) is defined for all real numbers except for those where sin(x) equals 0, that is, for all integers n. As a result, it has a domain that includes all real numbers except n, n € Z.

This function's range is (-∞,-1] U [1,+∞).

Secant Function

The function f(x) = sec(x) gets defined for all real numbers besides those where cos(x) is equal to 0, for all integers n, the values /2 + n. As a result, it has a domain that includes all real numbers excluding /2 + n, n € Z.

The range of this function is (-∞, -1] U [1, +∞).

Cot Function

The function f(x) = cot(x) is defined including all real numbers besides those where tan(x) equals 0, that is, for all integers n. As a result, it has a domain that includes all real numbers excluding n, n € Z.

The range of this function is (-∞, +∞).

 

The instantaneous rate of change of one quantity with respect to another is referred to as a derivative. It aids in determining the nature of an amount at any given time. The derivative of a function is expressed using the formula below:

Given the equation above, the derivative of the function f is f'(x). All derivative formulas for trigonometric functions, hyperbolic functions, inverse functions, and other functions are as follows:

Properties of Derivatives

The following are some of the most important properties of derivatives:

 

Example 1

Example 2

Example 3

    • There are numerous practice sums available for students to complete. The more you practice, the better prepared you will be for the exam.
    • The answer to a number of questions is provided in the solution. It aids students in comprehending the various types of questions that may appear in the examination.
    • The solutions are formatted in accordance with the CBSE syllabus. In the examination, several questions are taken directly or indirectly from the NCERT text. Students can effectively tackle such questions by studying from the Class 11 Maths NCERT Solutions Limits And Derivatives.

 

How many exercises are there in Limits and Derivatives class 11?

There are two exercises and one Miscellaneous Exercise in Class 11 Maths Chapter 13 NCERT Solutions.

What is the significance of Limits?

Limits should be studied since they offer the framework for understanding other concepts in Calculus. As a result, limits are fundamental in mathematics, and it is beneficial to learn about their laws and applications.

What are the topics included in Chapter 13 of NCERT Solutions for Class 11 Maths?

Students must comprehend that the subjects covered in Class 11 Maths NCERT  Solutions Chapter 13 provide a solid basis for differentiation and integration. This chapter covers the following topics:

1. An overview

2. Derivatives: An Intuitive Concept

3. Limits

4. Trigonometric Functions' Limits

5. Derivatives

 

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