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## Calculus

Calculus is known to be the branch of mathematics, which deals in the study rate of change and its application in solving the equations. Differential calculus and integral calculus are the two major branches of calculus. Differential calculus formulas deal with the rates of change and slopes of curves.

Integral Calculus deals mainly with the accumulation of quantities and the areas under and between curves. In calculus, we use the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. These fundamentals are used by both differential and integral calculus.

In other words, we can say that in differential calculus, an area splits up into small parts to calculate the rate of change. Whereas in integral calculus small parts are joined to calculates the area or volume and it is the method of reasoning or calculation.

For calculating very small quantities, we use Calculus. Initially, the first method of doing the calculation with very small quantities was by infinitesimals. For example, an infinitesimal number could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3, etc.

In other words, its value is less than any positive real number. According to this point of view, we can say that calculus is a collection of techniques for manipulating infinitesimals. In calculus formulas, the symbols dx and dy were taken to be infinitesimal, and the derivative formula dy/dx was simply their ratio.

**What are the Limits?**

Suppose we have a function f(x). The value, a function attains, as the variable x approaches a particular value let’s say suppose a that is., x → a is called its limit. Here, ‘a’ is some pre-assigned value. It is denoted as

lim x→a f(x) = l

- The expected value of the function f(x) shown by the points to the left of a point ‘a’ is the left-hand limit of the function at that point. It is denoted as limx→a− f(x).
- The point to the right of the point that is ‘a’ which generally shows the value of the function is the right-hand limit of the function at that point. It can be denoted as limx→a + f(x).

Limits of functions at a point are the common and coincidence value of the left and right-hand limits.

(Image to be added soon)

The value of a limit of a function f(x) at a point a that is, f(a) may vary from the value of f(x) at the point ‘a’.

**Math Limit Formula List**

**Differential Calculus & Differential Calculus Formulas**

The study of the definition, properties, and applications of the derivative of a function is known as Differential calculus. Differentiation is the process of finding the derivative. Here are some calculus differentiation formulas by which we can find a derivative of a function.

**Differential Calculus Formulas**

**Integration Formulas**

The branch of calculus where we study about integrals, accumulation of quantities, and the areas under and between curves and their properties is known as Integral Calculus. Let’s discuss some integration formulas by which we can find integral of a function. Here’s the Integration Formulas List

## Integral Calculus Formulas

The basic use of integration is to add the slices and make it into a whole thing. In other words, integration is the process of continuous addition and the variable “C” represents the constant of integration. But often, integration formulas are used to find the central points, areas and volumes for the most important things. Also, it helps to find the area under the curve of a function. There are certain important integral calculus formulas helps to get the solutions. These integral calculus formulas help to minimize the time taken to solve the problem. The list of integral calculus formulas is given below:

**Questions to Be Solved**

**Question 1. Solve the Following Definite Integral**

### Differentiation Formulas

In the first section of this chapter we saw the definition of the derivative and we computed a couple of derivatives using the definition. As we saw in those examples there was a fair amount of work involved in computing the limits and the functions that we worked with were not terribly complicated.

For more complex functions using the definition of the derivative would be an almost impossible task. Luckily for us we won’t have to use the definition terribly often. We will have to use it on occasion, however we have a large collection of formulas and properties that we can use to simplify our life considerably and will allow us to avoid using the definition whenever possible.

We will introduce most of these formulas over the course of the next several sections. We will start in this section with some of the basic properties and formulas. We will give the properties and formulas in this section in both “prime” notation and “fraction” notation.

Note that we have not included formulas for the derivative of products or quotients of two functions here. The derivative of a product or quotient of two functions is not the product or quotient of the derivatives of the individual pieces. We will take a look at these in the next section.

Next, let’s take a quick look at a couple of basic “computation” formulas that will allow us to actually compute some derivatives.

This formula is sometimes called the **power rule**. All we are doing here is bringing the original exponent down in front and multiplying and then subtracting one from the original exponent.

Note as well that in order to use this formula nn must be a number, it can’t be a variable. Also note that the base, the xx, must be a variable, it can’t be a number. It will be tempting in some later sections to misuse the Power Rule when we run in some functions where the exponent isn’t a number and/or the base isn’t a variable.

See the Proof of Various Derivative Formulas section of the Extras chapter to see the proof of this formula. There are actually three different proofs in this section. The first two restrict the formula to nn being an integer because at this point that is all that we can do at this point. The third proof is for the general rule but does suppose that you’ve read most of this chapter.

These are the only properties and formulas that we’ll give in this section. Let’s compute some derivatives using these properties.

In this case we have the sum and difference of four terms and so we will differentiate each of the terms using the first property from above and then put them back together with the proper sign. Also, for each term with a multiplicative constant remember that all we need to do is “factor” the constant out (using the second property) and then do the derivative.

Notice that in the third term the exponent was a one and so upon subtracting 1 from the original exponent we get a new exponent of zero. Now recall that x^{0}=1. Don’t forget to do any basic arithmetic that needs to be done such as any multiplication and/or division in the coefficients.

Note that we left the 3 in the denominator and only moved the variable up to the numerator. Remember that the only thing that gets an exponent is the term that is immediately to the left of the exponent. If we’d wanted the three to come up as well we’d have written,

so be careful with this! It’s a very common mistake to bring the 3 up into the numerator as well at this stage.

Now that we’ve gotten the function rewritten into a proper form that allows us to use the Power Rule we can differentiate the function. Here is the derivative for this part.

All of the terms in this function have roots in them. In order to use the power rule we need to first convert all the roots to fractional exponents. Again, remember that the Power Rule requires us to have a variable to a number and that it must be in the numerator of the term. Here is the function written in “proper” form.

In the last two terms we combined the exponents. You should always do this with this kind of term. In a later section we will learn of a technique that would allow us to differentiate this term without combining exponents, however it will take significantly more work to do. Also, don’t forget to move the term in the denominator of the third term up to the numerator. We can now differentiate the function.

Make sure that you can deal with fractional exponents. You will see a lot of them in this class.

In all of the previous examples the exponents have been nice integers or fractions. That is usually what we’ll see in this class. However, the exponent only needs to be a number so don’t get excited about problems like this one. They work exactly the same.

The answer is a little messy and we won’t reduce the exponents down to decimals. However, this problem is not terribly difficult it just looks that way initially.

There is a general rule about derivatives in this class that you will need to get into the habit of using. When you see radicals you should always first convert the radical to a fractional exponent and then simplify exponents as much as possible. Following this rule will save you a lot of grief in the future.

Back when we first put down the properties we noted that we hadn’t included a property for products and quotients. That doesn’t mean that we can’t differentiate any product or quotient at this point. There are some that we can do.

In this function we can’t just differentiate the first term, differentiate the second term and then multiply the two back together. That just won’t work. We will discuss this in detail in the next section so if you’re not sure you believe that hold on for a bit and we’ll be looking at that soon as well as showing you an example of why it won’t work.

It is still possible to do this derivative however. All that we need to do is convert the radical to fractional exponents (as we should anyway) and then multiply this through the parenthesis.

As with the first part we can’t just differentiate the numerator and the denominator and the put it back together as a fraction. Again, if you’re not sure you believe this hold on until the next section and we’ll take a more detailed look at this.

We can simplify this rational expression however as follows.

So, as we saw in this example there are a few products and quotients that we can differentiate. If we can first do some simplification the functions will sometimes simplify into a form that can be differentiated using the properties and formulas in this section.

Before moving on to the next section let’s work a couple of examples to remind us once again of some of the interpretations of the derivative.

** Example 5** The position of an object at any time tt (in hours) is given by,

Determine when the object is moving to the right and when the object is moving to the left.

*Solution*

The only way that we’ll know for sure which direction the object is moving is to have the velocity in hand. Recall that if the velocity is positive the object is moving off to the right and if the velocity is negative then the object is moving to the left.

We need the derivative in order to get the velocity of the object. The derivative, and hence the velocity, is,

s′(t)=6t^{2}−42t+60=6(t^{2}−7t+10)=6(t−2)(t−5)

The reason for factoring the derivative will be apparent shortly.

Now, we need to determine where the derivative is positive and where the derivative is negative. There are several ways to do this. The method that we tend to prefer is the following.

Since polynomials are continuous we know from the Intermediate Value Theorem that if the polynomial ever changes sign then it must have first gone through zero. So, if we knew where the derivative was zero we would know the only points where the derivative *might* change sign.

We can see from the factored form of the derivative that the derivative will be zero at t=2 and t=5. Let’s graph these points on a number line.

Now, we can see that these two points divide the number line into three distinct regions. In each of these regions we **know** that the derivative will be the same sign. Recall the derivative can only change sign at the two points that are used to divide the number line up into the regions.

Therefore, all that we need to do is to check the derivative at a test point in each region and the derivative in that region will have the same sign as the test point. Here is the number line with the test points and results shown.

Here are the intervals in which the derivative is positive and negative.

We included negative tt’s here because we could even though they may not make much sense for this problem. Once we know this we also can answer the question. The object is moving to the right and left in the following intervals.

**Example 1: Let f(y) = y ^{2} and g(y) = e^{y}. Use the chain rule in calculus to calculate h′(y) where h(y) = f(g(y)).**

**Solution:**

Given, f(y) = y^{2} and g(y) = e^{y}. First derivative above functions are f'(y) = 2y and g'(y) = e^{y}

To find: h′(y)

Now, h(y) = f(g(y))

h'(y) = f'(g(y))g'(y)

h'(y) = f'(e^{y})e^{y}

By substituting the values.

h'(y) = 2e^{y} × e^{y}

or h'(y) = 2e^{2y}

**Example 2: y is a function of x, and the function definition is given as: y = f(x) = 1/(1 + x^{2}). Find the output values of the function for x = 0, x = −1, and x = √2.**

**Solution:**

We have:

f(0) = 1/[1 + (0)^{2}]^{ }= 1/1 = 1

f(−1) = 1/[1 + (−1)^{2}] = 1/2

f(√2) = 1/[1 + (√2)^{2}] = 1/3

**Example 3: Find the slope of the curve y = 2x ^{2} + 3x + 1, at the point (-1, 0)**

**Solution:**

The given equation of the curve is y = 2x^{2} + 3x + 1.

The aim is to find the slope of the curve and hence we need to differentiate the equation.

(d/dx).y = d/dx. (2x^{2} + 3x + 1)

dy/dx = 4x + 3

Slope at at point (-1, 0) is m = 4(-1) + 3 = -1**Answer:** Hence the slope of the curve is -1

**Example 4: Differentiate x tan x.**

**Solution:**

d/dx. x tanx = x.( d/dx) tan x + tan x.(d/dx) x

Using calculus formula,

d/dx. x tan x = x.sec^{2}x + tan x.1

= x sec^{2}x + tan x**Answer:** (d/dx) x tan x = x sec^{2}x + tan x

**What is Meant by Calculus?**

Calculus is the branch of mathematics, which deals in the study rate of change and its application in solving the equations. Differential calculus and integral calculus are the two major branches of calculus. Differential Calculus deals with the rates of change and slopes of curves.

Integral Calculus deals mainly with the accumulation of quantities and the areas under and between curves. In calculus, we use the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. These fundamentals are used by both differential and integral calculus.

In other words, we can say that in differential calculus, an area splits up into small parts to calculate the rate of change. Whereas in integral calculus** **small parts are joined to calculates the area or volume and it is the method of reasoning or calculation.

For calculating very small quantities, we use Calculus. Initially, the first method of doing the calculation with very small quantities was by infinitesimals. For example, an infinitesimal number could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3, etc.

In other words, its value is less than any positive real number. According to this point of view, we can say that calculus is a collection of techniques for manipulating infinitesimals. In calculus the symbols dxdxanddydy were taken to be infinitesimal,

## What is Calculus?

Calculus focuses on some important topics covered in math such as differentiation, integration, limits, functions, and so on. Calculus, a branch of mathematics, deals with the study of the rate of change, was developed by Newton and Leibniz.

**Calculus Definition:** Calculus in Mathematics is generally used in mathematical models to obtain optimal solutions and thus helps in understanding the changes between the values related by a function. Calculus is broadly classified into two different sections:

- Differential Calculus
- Integral Calculus

Both differential and integral calculus serve as a foundation for the higher branch of Mathematics known as “Analysis”, dealing with the impact of a slight change in dependent variable, as it leads to zero, on the function.

## Calculus Topics

Based on the complexity of the concepts covered under calculus, we classify the topics under different categories as listed below,

- Precalculus
- Calculus 1
- Calculus 2

### Precalculus

Precalculus in mathematics is a course that includes trigonometry and algebra designed to prepare students for the study of calculus. In precalculus, we focus on the study of advanced mathematical concepts including functions and quantitative reasoning. Some important topics covered under precalculus are,

- Functions
- Inverse Functions
- Complex Numbers
- Rational Function

### Calculus 1

Calculus 1 covered the topics mainly focusing on differential calculus and the related concepts like limits and continuity. Some of the topics covered under calculus 1 are,

- Limits
- Derivatives
- Applications of Derivatives
- Integrals

### Calculus 2

Calculus 2 focuses on the mathematical study of change first introduced during the curriculum of Calculus 1. Some of the important topics under Calculus 2 are,

- Differential Equations
- Sequence and Series
- Application of Integrals
- Trapezoidal Rule

## Functions

Functions in calculus represent the relationship between two variables, which are the independent variable and the dependent variable. Let’s consider the following diagram.

There is an INPUT, a black box, and an OUTPUT. For example, suppose we want to make a pizza. We would need the following basic ingredients.

- Pizza Base
- Pizza Sauce
- Cheese
- Seasoning

The above real-life example can be represented in the form of a function as explained below,

The taste of our pizza depends on the quality of the ingredients. Let’s take another example

Suppose that: y = x^{2}

Value of x | Value of y |

1 | 1 |

2 | 4 |

9 | 81 |

11 | 121 |

Using the diagram given above, we get:

We can see that the value of y **depends** on the value of x. We can conclude that

- INPUT is independent of the OUTPUT
- OUTPUT depends on the INPUT
- Black Box is responsible for the transformation of the INPUT to the OUTPUT

In calculus,

- INPUT is an independent variable
- OUTPUT is a dependent variable
- Black Box is a function

## Applications of Calculus

Calculus is a very important branch, a mathematical model that helps in:

- Analyzing a system to find an optimal solution to predict the future of any given condition for a function.
- Concepts of calculus play a major role in real life, either it is related to solve the area of complicated shapes, evaluating survey data, the safety of vehicles, business planning, credit card payment records, or finding the changing conditions of a system that affect us, etc.
- Calculus is a language of economists, biologists, architects, medical experts, statisticians. For example, Architects and engineers use different concepts of calculus in determining the size and shape of the construction structures.
- Calculus is used in modeling concepts like birth and death rates, radioactive decay, reaction rates, heat and light, motion, electricity, etc.

## FAQs (Frequently Asked Questions)

**1. What is the Calculus Formulas?**

Calculus formulas basically describes the rate of change of a function for the given input value using the derivative of a function/differentiation formula. The process of finding the derivative of any given function is known as differentiation.

**2. What is the Basic Calculus?**

In basic calculus, we learn rules and differentiation formula, which is the method by which we calculate the derivative of a function, and integration, which is the process by which we calculate the antiderivative of a function.

**3. What Are the 4 Concepts of Calculus?**

Outline of calculus. Calculus is known to be a branch of mathematics that is focused on limits, functions, derivatives, integrals, and mainly infinite series. This subject does constitute a major part of contemporary mathematics education.

**4. What is the Study of Calculus?**

Calculus is the study of how things change. It provides a framework for modeling systems in which there are changes and a way to deduce the predictions of such models.

**5. Who is the Father of Calculus?**

Calculus, is the branch of Mathematics that is known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series. Gottfried Wilhelm Leibniz and Isaac Newton were the ones who independently developed the theory of indefinitesimal calculus in the later 17th century.

**What is Differential Calculus?**

Differential calculus focuses on solving the problem of finding the rate of change of a function with respect to the other variables. The derivative of a function, y with respect to variable x, is represented by dy/dx or f’(x).

**What do you Use Calculus for in Real Life?**

Concepts of calculus play a major role in real life, either it is related to solve the area of complicated shapes, evaluating survey data, the safety of vehicles, business planning, credit card payment records, or finding the changing conditions of a system that affect us, etc. Calculus is a language of economists, biologists, architects, medical experts, statisticians. For example, architects and engineers use different concepts of calculus in determining the size and shape of construction structures.

**How to Find the Maxima and Minima of a Function in Calculus?**

To find the optimal solution, derivatives are used to calculate the maxima and minima values of a function. Maxima and minima are the highest and lowest points of a function respectively, which could be determined by finding the derivative of the function.

**What are the Four Main Concepts of Calculus?**

The four main concepts covered under calculus are as given below,

- Limits
- Differential Calculus
- Integral Calculus
- Multivariable Calculus

**What is Integral Calculus?**

Integral calculus is the study of integrals and the properties associated with them. It is helpful in calculating f from f’ (i.e. from its derivative). If a function, say f is differentiable in any given interval, then f’ is defined in that interval. It enables us to calculate the area under a curve for any function.

## Formulas and Theorems for Reference

## Integral Calculus Formula Sheet

## Calculus Summary Formulas

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