Limits and Derivatives

Edited By Team Careers360 | Updated on May 07, 2022 12:41 PM IST

Introduction:
Limits and derivatives are incredibly important topics in mathematics that have applications not only in math but also in other fields such as physics. The whole ideas of limits and derivatives, as well as their characteristics and formulae, are presented in this article. This subject is well-explained in the 11th-grade curriculum. We'll go through the key formulae, characteristics, and problems to help you fully grasp the notion of limits and derivatives. Sir Isaac Newton established the essential rules of differential calculus based on his conceptions of rate and change, whereas integral calculus emerged as a result of the reverse process. Differentiation and calculus are the foundations for advanced mathematics, modern physics, and a variety of other modern sciences and engineering disciplines. For CBSE students, limits and derivatives class 11 serves as an introduction to calculus. A limit is a value that a function approaches as an input and creates some value in mathematics. Limits are used to define integrals, derivatives, and continuity in calculus and mathematical analysis.

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List of topics according to NCERT and JEE Main/NEET syllabus:

Limits
Direct method of solving limit
Derivatives
First principle to differentiate
Algebra of derivatives of functions
Standard simplifications
Sandwich theorem
Trigonometric function

Important concepts and Laws:

Let's pretend we have a function f(x). The value that a function achieves when the variable x approaches a specific value, say a, is referred to as its limit. 'a' denotes a value that has already been allocated. Lim x → a f(x) = 1 is the symbol for it.

The function's predicted value displayed by the points to the left of a point 'a' represents the function's left-hand limit at that point. It is abbreviated as Lim x → a⁻ f(x)

The right-hand limit of the function at that point is the points to the right of a point 'a' that indicates the value of the function. Lim x → a⁺ f(x)is the symbol for it (x).

The common and coincidence value of the left and right-hand limits are the limits of functions at a location. Assume that f(x) is a function that exists inside its domain. Let's imagine this function is used to calculate the rate of change of velocity, or acceleration, of a vehicle as it travels from one location to another. Because the function is reliant on both the vehicle's speed and direction, the rate of change of velocity is obviously instantaneous. The bounds of the function at that time are required if we are to determine the vehicle's instantaneous acceleration. A derivative of a function is a function that represents the rate of change of another function. To put it another way, a derivative is used to define a function's rate of change. Calculating the slope of a line is the most typical example. To determine the slope of any point on a line, we draw a tangent to it and calculate the tan of the angle it makes with the base, as we know. A function's derivative at any point 'a' in its domain is given by:

Lim h→0 [f(a+h) – f(a)]h

NCERT Notes Subject Wise Link:

Importance of Limits and Derivatives class 11

Calculus is the study of the rate of change and quantity continuity. About half of calculus is made up of "Limits and Derivatives." The most essential thing is to pay attention in class and comprehend the material thoroughly. All of your questions should be sent to your professors as soon as possible. For a better understanding, you may also view videos. The more you practice this chapter, the better you will get at it. It is essential to practice example papers and question papers. Learning how ideas in mathematics are used in real-life situations aids in understanding the chapter and, as a result, in performing well on tests. Limits and derivatives are useful not just in advanced mathematics, but also in other fields such as physics and engineering. They may be used to determine electric and magnetic fields, find a quantity's rate of change, and examine temperature fluctuations, among other things.

NCERT Solutions Subject wise link:

NCERT Exemplar Solutions Subject wise link:

Frequently Asked Question (FAQs)

1. What is the relationship between integrals and limit derivatives?

The idea of the limit is used to connect derivative and integral in such a way that they are both defined by it. Furthermore, they are the inverse operations of one another. The fundamental theorem of calculus is named after this property. Furthermore, both are essential to modern science.

2. In differential calculus, what does limit mean?

The term "limit" refers to the value that a sequence or function approaches when the input approaches a certain value.

3. Derivative of sin x is?

Derivative of sin x is cos x

4. Is it possible to call the derivative a slope?

Yes, a slope may be defined as a derivative. Because the derivative assesses the steepness of a graph's steepness belonging to a function at a given point on the graph, this is the case.

5. What is the difference between a derivative and an integral?

The area under the curve of the mathematical function f(x) whose graphing takes place as a function of x is referred to as the integral of a function. The slope of the curve of the mathematical function f(x) whose graphing takes place as a function of x, on the other hand, is referred to as the derivative of a function.