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# Calculus

## What is Calculus?

Last updated date: 18th Mar 2023
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Many students find class 12 Calculus a hard subject. Due to the unfamiliar and challenging concepts they are learning, many students find it difficult. The study of rates of change is calculus. There are two major branches of calculus. The first is differential calculus, which determines a quantity's rate of change. The second is integral calculus. It finds the quantity where the rate of change is known. Calculus explains how values change for any function, both on a small and large scale. Applications of calculus can be found in science, economics, and engineering. It can address a wide range of issues that algebra alone cannot.

## What is Differential Calculus?

This is related to the problem of determining how quickly a function changes with other variables. Differentiation is the process of identifying the derivatives. The function's derivatives are generally expressed by $\dfrac{\text{d}y}{\text{d}x}$ or $f\prime(x$). The function is therefore the derivative of y to the variable x.

## What is Integral Calculus?

This type of calculus is the exact opposite (reciprocal) of differentiation. Differentiation can be explained as breaking a large object into numerous smaller ones. Correspondingly, we can define integration as the assembling of numerous minor components to create a larger whole.

$\int f(x)dx$ which means Integral of f with respect to x.

## Limit and Continuity

The core aspects of many calculus concepts are limits. Limits were developed to solve the mathematical problem of division by zero. Instead, we use limits to think about what happens when we divide by a very small number that is nearly equal to, but not exactly, zero.

Limit: In general, we define the limit for any function f(x), if f(x) approaches a value L as x approaches a, we define the limit of f(x) as x approaches c as L. It is written as:

$\lim_{x \to c} f(x) = L$

1. Left-Hand limit of f(x) at c: $\lim_{x \to c-} f(x)$ is the expected value of f at x = c given the values of f(x) near x to the left of c.

2. Right-Hand limit of f(x) at c: $\lim_{x \to c+} f(x)$ is the expected value of f(x) at x = c given the values of f(x) near x to the right of c.

3. $\lim_{x \to c} f(x)$ means the right-hand and left-hand limits coincide and have the common value as the limit of f(x) at x = c.

Continuity: The possibility to trace a function's graph with a pencil without taking the pencil off the paper is a feature of many functions. These are referred to as continuous functions. A function is continuous at a particular point if its graph does not break at that point. The following three conditions must all be fulfilled for a function to be considered continuous at a specific point.

1. The function f(a) is defined

2. For the function f(x); $\lim_{x \to a} f(x)$ exists

3. For the function f(x); $\lim_{x \to a+} f(x)=\lim_{x \to a-} f(x)=f(a)$

## Derivative

The derivative is the most effective component in differential calculus. To display the rate of change, use the derivative. It is useful to illustrate how much the function is altering at each point. The derivative is referred to as a slope.

Slope of a Tangent

On a graph, a derivative is defined as the slope of the tangent, which either meets the curve at a certain point or yields a derivative at that point.

### Solved Examples

Example 1: Evaluate $\lim_{x \to -2} 3x^2+5x-2$

Solution: Substitute x as -2; we get

$\lim_{x \to -2} 3x^2+5x-2$

$=3(-2)^2+5(-2)-2$

$=3(4)+10-2$

$=12-10-2$

$=0$

Example 2: Check the contiuity for the function f(x)=2x+3 at x=1.

Solution: It is given that the function f(x)=2x+3 is defined at x=1 which means

f(1)=2(1)+3=5

Now limit of the function at x=1,

$\lim_{x \to 1} f(x) = \lim_{x \to 1} f(2x+3) = 2(1)+3=5$

$\lim_{x \to 1} f(x)=5=f(1)$

Therefore, f(x)=2x+3 is continuous at x=1.

### Practise Problem

1. Find the derivative of $g(x) = \sin x^2$.

1. $-\sin x^2$

2. $2x \cos x^2$

3. $-2x \cos x^2$

4. $-2x \sin x^2$

Ans: B.

2. Find derivative of $\tan (x+7)$.

1. $\sec^2 (x+7)$

2. $7 \sec^2 (x+7)$

3. $7x \sec^2 (x+7)$

4. $\sec^2x$

Ans: A.

## Conclusion

This article covers the study of rates of change called calculus. Differential calculus and integral calculus are the two main subfields of calculus. Calculus discusses the small- and large-scale value changes for any function. The concepts of limit, continuity, derivatives, differentiation, and integration are also covered in this article. You can better comprehend calculus by using the examples that have been solved.

Competitive Exams after 12th Science

## FAQs on Calculus

1. What is the meaning of Calculus?

The calculus chapter is a part of mathematics that is related to the properties of derivatives and integrals of quantities; for example velocity, area, acceleration, volume, etc., by processes initially dependent on the summation of infinitesimal differences. Calculus is helpful in the determination of the changes between the values that are related to the functions. We study differentiation and integration under the concept of basic calculus. Both types of calculus are based on the idea of limits and functions.

2. What are the applications of Calculus?

Calculus is a model of mathematics which is helpful in analyzing a system to find an optimal solution to predict the future. The basic calculus concepts play an important role whether it is related to solving the area of complex functions or shapes, the safety of vehicles, evaluating survey data for business planning, records of payment that is done by credit card and determining the changing conditions of a system affect us, etc. in our normal life routine. We can say that it is a language of physicians, economists, biologists, architects, medical experts, statisticians. For example- Engineers and architects use the concept of calculus to determine the shape and size of the curves to design or construct buildings,  tunnels, bridges and roads etc.

3. What are some real-world applications of calculus?

Calculus is used in a wide variety of fields, including physics, engineering, medicine, economics, biology, engineering, space exploration, statistics, pharmacology, and many others.

1. Calculus is used in economics to determine the price elasticity of supply and demand.

2. Calculus is used by astronomers to analyse the various motions of planets and meteorites. Based on the laws of planetary motion, it is used to determine how quickly a moving body's position changes over time.

3. In many applications, including calculating the mass moment of inertia, the centre of gravity, and many others, physicists use integration calculus.

4. Calculus is used to estimate the rate at which compounds undergo chemical reactions.

4. How to calculate areas using integral calculus?

Integral calculus divides a shape into numerous small boxes and adds the areas of each box to determine the area of the shape. This provides a rough idea of the area. As the boxes get smaller and smaller, there are more and more of them, and their combined surface area gets very close to the shape's total area. Calculus is based on the idea that if we had an infinite number of these boxes, each one infinitely narrow, we would have the exact area of the shape.

5. What is the relationship between continuity and differentiability?

If a function can be drawn continuously without raising the pencil, it is said to have continuity. The slope of a function's graph at any point in its domain is defined as its differentiability. Differentiability is only a possibility if the given function is continuous at a given point. For the function to be differentiable at that point, continuity must first be verified. At a point x = a, if the function y = f(x) is not continuous, it cannot be differentiated.