Category: Maths Notes PPT

All About Calculus Application 

Maths is a subject of practical information. Concepts of Mathematics are used widely in practical terms to understand its nature. As it is used widely, hence requires deep learning of the concept. To understand it even better, an app to learn calculus is introduced which guides students further conveniently and easily.

Calculus is a branch of Mathematics which deals in the study of rates of change. Before calculus, all Maths was static. Earlier it could only calculate the objects that were still. Seeing this, it is practically not possible as the universe is never constant. All objects from stars in space to cells in the human body are never at rest. These all are constantly moving. Calculus helps in determining how particles, stars, and matter move and change in real life.

Calculus is not only confined to Mathematics, but it is also used widely in other fields as well

Some of the known fields where calculus concepts are used are:

1. Physics

2. Engineering

3. Economics

4. Statistics

5. Medicine

Calculus is used in other disparate areas as well including:

1. Space travel

2. Determining how medications interact with the human body

3. How to build safer structures

Using calculus, scientists, astronomers, mathematicians, and chemists could chart the orbit of planets and stars and also the path of electrons and protons at the atomic level.

Calculus apps like provide the best solution to learn calculus. Here Calculus formulas like integral formula, limits, and derivative formula are explained in an easy manner. Calculus help app provides assistance to understand calculus in a fun and exciting way and also ensures that the context is easy to understand. 

Types of Calculus

Basic calculus is the study of differentiation and integration. In basic calculus, there are two branches of calculus namely –

1. Differential calculus

2. Integral calculus

Differential calculus studies the rate of change of quantity and also examines the rate of change of slopes and curves. This part is concerned with continuous change and its application. Here are many topics to shield on in differential calculus. These are as follows:

1. Limits: It is a degree of closeness to any value or the approaching term.

2. Derivatives: It is an instantaneous rate of change of quantity with respect to the other.

3. Continuity and differentiability: Any function is always continuous if it is differentiable at any point.

4. Chain rule: To find the derivative of the composition of a function, the chain rule is applied.

5. Quotient rule: To find the differentiation of a function, the quotient rule is used.

Integral calculus finds the quantity where the rate of change is known. It basically focuses on concepts as slopes of tangent lines and velocities and concerns with space under the curve. It resembles the reverse of differentiation. It is the study of integrals and their property. Below listed are other topics covered in integral calculus:

1. Integration: It is simply defined as the reciprocal of differentiation.

2. Definite Integral: In a definite integral, the upper limit and lower limit of the independent variable of a function is specified.

3. Indefinite integral: Here, it is not confined in a specified boundary and hence the integration value is always accompanied by a constant value.

Application of Calculus 

Calculus is not only confined to Mathematics subjects, but it has many practical applications in the outside world. Here are some of the concepts that use calculus including:

1. Motion

2. Heat

3. Electricity

4. Harmonics

5. Astronomy

6. Acoustics

7. Photography

8. AI

9. Robotics

10. Video Games

11. Movies

12. Predict birth and death rate

13. Study of gravity

14. Planetary motion

15. Bridge engineering

Talking about economics, Calculus is even used in economics to determine the price elasticity of demand. Calculus allows us to determine points on changing supply and demand curves.

Advanced Calculus

Advanced calculus includes topics such as advanced series and power series and it covers the basic calculus topics such as differentiation, derivatives, and so on. The important areas to be covered are vector spaces, matrices, linear transformation. It helps us to understand the knowledge on few concepts such as:

1. Vector fields

2. Multilinear algebra

3. Continuous differentiability

4. Integration of forms

5. Quadratic forms

6. Tangent space

7. Normal space via gradients

8. Critical point analysis 

In this article, students will be able to learn about the topic of the centroid of a trapezoid. We will also look at the centroid of the trapezoid formula. But before we learn how to find the centroid of a trapezoid, students need to focus on the basics and start from the beginning.

The first thing that one needs to learn is the definition of a trapezoid. A trapezoid can be defined as a quadrilateral in which there are two parallel sides. A trapezoid is also known as a trapezium. So, if you see trapezium written in some other book, then don’t be confused. It means the same thing as a trapezoid.

A trapezoid can also be defined as a four-sided figure that is closed. It also covers some areas and has its perimeter. We will learn the formula for both area and perimeter of a trapezoid at a later point in this article.

It should be noted that a trapezoid is a two-dimensional figure and not a three-dimensional figure. The sides that are parallel to one another are known as the bases of the trapezoid. On the other hand, the sides that are not parallel to each other are known as lateral sides or legs. The distance between the two parallel sides is also known as the altitude.

Some readers might find it interesting to learn that there is also a disagreement over the exact definition of a trapezoid. There are different schools of mathematics that take up different definitions.

According to one of those schools of mathematics, a trapezoid can only have one pair of parallel sides. Another school of mathematics dictates that a trapezoid can have more than one pair of parallel sides.

This means that if we consider the first school of thought to be true, then a parallelogram is not a trapezoid. But according to the second school of thought, a parallelogram is a trapezoid. There are also different types of trapezoids. And those different types of trapezoids are:

A right trapezoid contains a pair of right angles. We have also attached an image of a right trapezoid below.

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In an isosceles trapezoid, the non-parallel sides of the legs of the trapezoid are equal in length. An image depicting an isosceles trapezoid is attached below.

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A scalene trapezoid is a figure in which neither the sides nor the angles of the trapezium are equal. For your better understanding, an image of a scalene trapezoid is attached below.

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The Formula for Area and Perimeter of a Trapezoid

Now, let’s look at the formula for calculating the area and perimeter of a trapezoid. According to experts, the area of a trapezoid can be calculated by taking the average of the two bases and multiplying the answer with the value for the altitude. This means that the formula for the area of a trapezoid can also be depicted by:

Area = ½(a + b) x h

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Moving on to the formula for the perimeter of a trapezoid, it can be described as the simple sum of all the sides. This means that if a trapezoid has four sides like a, b, c, and d, then the formula for the perimeter of a trapezoid can be represented by:

Perimeter = a + b + c + d.

The Properties of a Trapezoid

There are various important properties of a trapezoid. We have discussed those properties in the list that is mentioned below.

• The diagonals and base angles of an isosceles trapezoid are equal in length.

• If a median is drawn on a trapezoid, then the median will be parallel to the bases. And the length will also be the average of the length of the bases.

• The intersection point of the diagonals is collinear to the midpoints of the two opposite sides.

• If there is a trapezoid that has sides, including a, b, c, and d, and diagonals p and q, then the following equation stands true.

p² + q² = c² + d² + 2ab

In the next section, we will look at the centroid of a trapezoid formula.

The Formula for Centroid of a Trapezoid

In this section, we will look at the trapezoid centroid and the centroid formula for the trapezoid. As you must already know, a trapezoid is a quadrilateral that has two sides parallel. The centroid, as the name indicates, lies at the centre of a trapezoid. This means that for any trapezoid that has parallel sides a and b, the trapezoid centroid formula is:

X = {b + 2a / 3 (a + b)} x h

In this formula, h is the height of the trapezoid. Also, a and b are the lengths of the parallel sides.

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The terms clockwise and counterclockwise are used to describe the direction in which a rotating object turns. This demonstration will help you understand how this directly affects pressure on each side of a propeller or fan blade as it spins.  This demo is to help you understand the effects of clockwise and counterclockwise rotating propellers on pressure. To do this, there are two things that will be demonstrated: the effect of a fan blade rotating clockwise vs. counterclockwise when placed in front of an opening, and the difference in airflow through two fans rotating in opposite directions. Take a fan blade and place it flat side down on top of the opening to your hand while you are holding your hand up to the opening. Spin it by using your other hand to spin the blade clockwise or counterclockwise. You will notice that when spun clockwise, the blade will create a high-pressure zone around your hand. The opposite happens when you spin the blade counterclockwise: there is a low-pressure zone around your hand. This is because the spinning of a blade creates lift, and in turn causes airflow which pushes against surfaces on one side with high pressure and pulls on the surfaces of an adjacent object with low pressure. This is what causes lift, as well as the force that will cause a spinning propeller to push or pull an aircraft.

Clockwise and Counterclockwise in Detail

The rotation in a two-dimensional plane about a fixed point can be of two types- clockwise and counterclockwise. These two terms are used for indicating the direction.

Clockwise

When we move something in a clockwise direction, it is moving in the same direction as the hands on the clock.

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Example of clockwise: Most screws, lock,s and taps are closed by turning in the clockwise direction.

Counterclockwise

When something moves in the opposite direction to the way in which the hands of a clock move round in known as counterclockwise.  The counterclockwise direction is also known as the anticlockwise direction.

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Examples of Counterclockwise: Just opposite of clockwise when we open screw, lock, and taps we have to move in the anticlockwise direction.

Let’s have a look at some examples of clockwise and counterclockwise For measuring angle we use clockwise and counterclockwise directions.

When we measure the positive angle, we should move in the clockwise direction, as shown in the below diagram.

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When we measure the negative angle, we should move in the counterclockwise direction. Shown in the below diagram.

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What is Clockwise?

Clockwise is the motion that proceeds in a similar direction as a clock’s hands move. The clockwise movement usually starts from the top going around to the right, then to the bottom, to the left, and then back to the top position.

What is Counterclockwise?

Counterclockwise is the opposite sense of clockwise rotation. The movement in the counterclockwise direction, starts from the top, heads to the right, goes down, then follows to the right side, and ends up at the top position.

Abbreviations Used for Clockwise and Counterclockwise:

• Clockwise is usually abbreviated as CW.

• Counterclockwise abbreviated as CCW. As we know, Counterclockwise is also known as anticlockwise so abbreviated as ACW.

Difference Between Clockwise and Counterclockwise:

The major difference between the two is based on rotations.

• In clockwise rotation, the motion starts from the top position, heads to the right side, goes to the bottom position, heads to left, and finally goes up to the top position.

• Whereas on the other hand counterclockwise starts from the top position, goes to the left side, proceeds to the bottom position, then heads to the right side, and finally reaches the top position.

Solved Question:

1. Kabir is facing north. He turns 45° in the clockwise direction and then another 90in the same direction and then again 180° in the clockwise direction. Which direction he is facing now?

Sol: Here the starting point is north as Kabir is facing north. Draw a diagram from the given question to visualize in all directions.

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Let the starting point be O and A be the point in the north direction. So, from A point, he moves 45° in a clockwise direction that is OB. Then turn 90° in the same direction that is OC. Finally, he turns 180° in the same direction at point D. From the above diagram, we can say that Kabir is facing the northwest direction and the distance is OD.

Multiples are used in whole mathematics. The extensive use of multiples is because of getting commons. It is used in calculation to simplify a fraction. The fundamental principle of arithmetic is also telling us about the multiples only.

Multiples

Multiples are numbers which we’ll get after multiplying a number with counting numbers. Let say we have a number n and we know that counting numbers are 1, 2, 3, 4, 5, 6…. Then, the product of n and 1, 2, 3, 4, 5, 6.. Is called multiples and this scenario is called multiplication.

Example: Let say we have a number 5. For getting it’s multiples we need to multiply it by counting numbers that are multiplication with 1, 2, 3, and so on. We’ll get

5 x 1 = 5

5 x 2 = 10

5 x 3 = 15

5 x 4 = 20

5 x 5 = 25

5 x 6 = 30

Hence, we’ll get multiples of 5 as 5, 10, 15, 20, 25 and so on.

Common Multiples

Whenever we want to talk about the common immediately, more than one quantity comes in our mind. Hence, we’ll start with 2 numbers whose multiples we have. Let say we have numbers as 5 and 6. First of all we’ll find there multiples.

Multiples of 5 will be 5, 10 ,15, 20, 25, 30, 35, 40, and so on.

Multiples of 6 will be 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 and so on.

Now to get common multiples we just need to find common numbers in both the multiples. Observe that in both the multiples set 30 is common. So, we’ll say 30 is the first common multiples or Least common multiple of 5 and 6. Least common multiple is commonly called LCM.

Now the obvious question comes in our mind, is there any multiple which is not least. The answer is yes. There are many common multiples after first or least. For example observe that for 5 and 6 first common multiples is 30 then second will be 60 third will be 90 and so on.

Representation of Common Multiples

Grid Method

Common multiples are the numbers which come common in both the multiples table of given numbers. If we take number 6 and 7 and we have to find the common multiples of them one way of representing them is following:

We have marked the circle for all the multiples of 6 and cross for all the multiples of 7 till 100 numbers. Now what will be our common multiples?

Common multiples will simply be those numbers that are crossed as well as circled. That’s how they’ll come in both the multiple tables.

Hence, 42, 84 and so on will be our common multiples. Also, the least common multiple will be 42.

Vann Diagram

Another way to represent common multiples is the Vann diagram. Suppose we have to find the common multiples of 3 and 4. Then we’ll close all the multiples of 3 and 4 in a circle separately. Then we’ll find commons in them. Kindly refer the following:

Solved Examples

1. Write the common multiples of 7 and 9.

Ans: First of all we’ll find the multiple of 7 which are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70. Now, multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 90. Now the Common multiples of 7 and 9 will be the number that occurs in both the tables. Which are 63, 126, 189, and so on.

2. A and B are two numbers such that B is a multiple of A. Which of the following is the value of LCM (A,B)?

Ans: We have given two numbers A and B. For finding the least common multiple first we need to find their common multiples. Then we can find least out of them. We have also given that B is a multiple of A means when we’ll be writing multiples of A then B will occur in its table. Also, all the common multiples will be the multiples of B. Hence the LCM(A, B) will be B.

Did You Know?

1. There are infinite common multiples possible for any two finite real numbers. We just have to write multiples of LCM.

2. The smallest common multiple is known as LCM. Which refers to least common multiple. The representation is LCM(a, b) which represents the least common multiple of a and b.

In the subject of mathematics or math, composite numbers are defined as the type of numbers that have more than two factors. These numbers are different from prime numbers as prime numbers only have two factors. These factors are one and the number itself. Composite numbers are also simply known as composites.

Readers need to remember that all numbers that are not prime numbers can be categorized as being composite numbers. These numbers can also be divided by more than two numbers. For example, the number six is composite because it can be divided by 1, 2, 3, and 6.

Composite numbers can also be defined as the integers that can be generated by multiplying the two smallest positive integers. These numbers should also contain at least one divisor other than the number one. These numbers also have more than two composite factors. For example, any even number that is greater than 2 is a composite number.

Students are often asked to mention the difference between prime numbers and composite numbers. This is why it is advised that students should understand the concept of prime and composite numbers properly.

Also, another common question that students have on this topic is whether zero falls under the category of prime or composite numbers. The answer to this question is that zero does not fall under either of these categories of prime or composite numbers.

Keep in mind everything that we have discussed till now. Can you mention all the major composite numbers up to 200? If you can’t, then go through the section that is mentioned below.

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132, 133, 134, 135, 136, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 150, 152, 154, 155, 156, 157, 158, 159, 160, 162, 164, 165, 166, 168, 170, 172, 174, 175, 176, 177, 178, 180, 182,184, 185, 186, 187, 188, 189, 190, 192, 194, 195, 196, 198, and 200.

One should also look at lists mentioning composite numbers from 1 to 1000 to become more familiar with this topic.

 

How Can You Determine Composite Numbers?

Now, you must understand the meaning of consecutive composite numbers. You should also have a ready list prepared of all the composite numbers from 1 to 100. This is why it is now time to learn the process of finding out the values of composite numbers.

The process that you can follow to determine composite numbers is mentioned below.

• Begin by finding out all the factors of a positive integer.

• If a number only has two factors, which are one and the number itself, then classify it as a prime number.

• If a number has more than two factors, then categorize it as a composite number.

For example, let’s assume that you have to find whether 14 is a prime or composite number. You must first begin with the understanding of what prime and composite numbers are. After that, you should find out the factors of 14.

We know that, 14 / 1 = 14

14 / 2 = 7

14 / 7 = 2

14 / 14 = 1

From this, it can be concluded that 14 is a composite number as it has more than one factor. Also, students should be familiar with all the composite numbers less than 20 without consulting a book.

The Different Types of Composite Numbers

In this section, we will look at the different types of composite numbers. According to experts, there are mainly two divisions of composite numbers. These divisions are:

Odd composite numbers are all the composite numbers that are odd integers. For example, 9, 15, 21, 25, and 27.

On the other hand, even composite numbers are all the composite numbers that are also even integers. For example, 4, 6, 8, and 10.

What are The Properties of Composite Numbers?

Composite numbers refer to the positive integers that are numbers that are formed by multiplying two smaller positive integers. The major properties of these composite numbers are that these numbers are evenly divisible by both smaller composite and prime numbers, that is, no special preference is given to the prime numbers or the composite numbers for becoming a factor of any number. The second property of composite numbers is that all composite numbers are made up of a combination of two or more prime numbers.

How are Composite Numbers Different From Prime Numbers?

The major difference between a prime number and composite number is that the prime numbers are only divisible by 1 and themselves, that is, such numbers have only two factors whereas composite numbers have more than two factors, that is, it is divisible by numbers other than 1 and itself also. Prime numbers can only be written in the form of a product of two numbers only while composite numbers can be written in the form of the product of more than two numbers. For example, 5 is a prime number as it has no factor other than 1 and 5 whereas 4 is a composite number because other than 1 and 4, 2 is also a factor of 4.

Learn About the Smallest Composite Number

Four is the smallest composite number. This is because, in the case of 4, there are more than two factors. The divisors of 4 are 1, 2, and 4. Also, you might want to note that the smallest prime number is 2, the smallest odd composite number is 9, and the two-digit smallest composite number is 12.

Prime Factorization of Commonly Used Composite Numbers

It is recommended that students should be familiar with composite numbers between 1 to 100 and their prime factorization. This is because students can be asked to answer questions related to that from the prime and composite number chapter. This is why we have created a list of prime factorization of composite numbers till 50. That list is mentioned below.

How to Recognize if a Number is Prime or Not?

The rules with which you can recognize if a number is prime or not. These rules are:

• If a number is greater than 2 and is a multiple of 2, then, in that case, the number is not a prime number because it will have at least three factors, 1, 2 and the number itself.

• If a number is greater than 3 and a multiple of 3 as well, then that number can also not be a prime number, because it has at least three factors 1, 3 and the number itself.

• If a particular number is a multiple of 4, then it will also be a multiple of 2, therefore, such numbers can also not be considered as prime numbers.

• If any number is greater than 5 and is also a multiple of 5, then it will not be a prime number. In other words, if a number ends with 0 or 5, then it will not be a prime number because in this case also, the number is divisible by at least 1, 5 and the number itself.

• Similarly, if a number is a multiple of 6, then it will also be a multiple of 2 and 3, therefore such numbers can automatically rule out from the list of prime numbers as it will have at least five factors, that is, 1, 2, 3, 6 and the number itself.

Fun Facts About Composite Numbers

Did you know that prime numbers can only be divided by 1 and the number itself? On the other hand, composite numbers have more than two factors. Prime numbers can also be just written as products of two numbers. But composite numbers can be written as the products of more than two numbers. For example, 5 is a prime number, while 4 is a composite number.

How to Draw Congruent Figures? 

You can take a sheet of paper and draw two similar figures on it. Cut the figures and place them on one another. Both the statistics will be congruent when you put one picture on top of the other image.

In Geometry, when we say that one (A) figure is equal to figure (B), we write down as figure A ≅ to figure B. Below, you will learn more about congruent shapes, congruent line segments, and corresponding shapes and angles.

How to find The Congruence of two Figures?

To know what are congruent figures you have to keep two plane figures in front of you, and you have to find if they are identical with each other or not. Use the following method to discover if the two figures are congruent figures or not:

• Take a tracing paper and trace the outline of figure A.

• Cut out the shape of figure A and place it over the figure B.

• You can set the paper over figure B or flip it and then put it over figure B.

• If the two figures completely cover each other, they are congruent to each other. You can symbolically represent it, as figure A is ≅ to figure B.

Congruent Lines Segments

Like two figures can be congruent, so can two lines and lines segments. Two line segments are congruent if they have equal or the same length.

If you want to find out if two figures are corresponding to each other or not, you can use the same method as given above. Take two line segments and place them on top of each other. So, two line segments are congruent if they are equal or the same.

What are Congruent Angles?

Besides, line segments, angles can be congruent to each other too. If angle A and angle B are 36 degrees each, they are congruent. However, if the corners are different, the angles are not congruent with each other.

When two angles have the same measure in degree, they are congruent to each other. The angles may differ in position and orientation on the plane. You can use the method given above to find out the congruency of the angles.

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Here are two triangles with congruent angles. 

If you have two figures, ∠ABC and ∠QPR, with the same angle, which you can assume as 45 degrees, when you use the method given above, you will see that the ∠ABC ≅ ∠QPR means they are congruent to each other.

How to prove if two Angles are Congruent to each other?

There is another way to check for the congruence of two angles. Follow the below-given method to prove that two angles are congruent:

• When two angles and the side between them are equal or similar in both the triangles, it is an ASA congruence.

• When two angles and a side that is not between them is identical or identical in both the triangles, it’s an AAS congruence.

• If three sides of a triangle are the same or equal to each other, it is SSS congruence. 

• When two hands and one angle between them makes the two triangles congruent, it’s the SAS congruence.

 

Congruent Angles Solved Examples

Here are a few congruent angles solved examples that will help you understand how to find the congruence of two angles.

 Example 1:

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Here are two figures which you need to prove as congruent.

The angles G and S of the diagram above are 42 degrees, which means that they are equal. When two angles of a figure are identical or similar, it means that they are congruent.

 Example 2:

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Here are two figures which you have to prove as congruent. 

The angles of the two figures are 155 degrees each, so It means that R and Q are equal to each other. If two aspects are similar to each other, it means that they are congruent to each other.

Fun Facts About Congruent Shapes

Here are some fun facts about the corresponding figures:

• Squares that have equal length on all four sides are congruent shapes.

• An equilateral triangle which has equal distances on all the three sides are congruent.

• Slices of pizza are a perfect example of a congruent triangle and congruent angles.

• When you combine two congruent triangles, they form vertical angles at a corner.

• Most buildings in the United States are congruent triangles.