Category: Maths Notes PPT

What is Triangle Inequality?

We have studied the equality of sides and angles between two triangles or in a triangle. Sometimes we may come across unequal objects or figures.

In our instances of comparisons, we take into consideration every part of the object. Similarly, when we compare various parts of a triangle, we compare every part individually. So let us study what is the triangle inequality. For instance, take the line segment shown in the figure:

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From the above figure, we can see that line segment AB is smaller than the line segment PQ. The difference is clear, yet to know the difference and confirm our findings we measure both of them. After measuring both, we come to the conclusion that they are unequal. Likewise, see the figure of angles below:

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The angles AOB and POQ are unequal. We can clearly see that ∠ POQ is greater than ∠AOB. These unequal figures have a close relationship between unequal sides and unequal angles of a triangle. 

The term triangle inequality means unequal in their measures. 

Let us consider any triangle of length AB, BC, and AC of three sides of a triangle.  Then the triangle inequality definition or triangle inequality theorem states that

The sum of any two sides of a triangle is greater than or equal to the third side of a triangle.

i.e.,

AB + BC ≥ AC

Now let us understand the relation between the unequal sides and unequal angles of a triangle with the help of the triangle inequality theorems.

Triangle Inequality Theorem

Theorem 1: If two sides of a triangle are unequal, the longer side has a greater angle opposite to it.

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Triangle inequality theorem-proof:

Given: in [triangle ABC] in which AC > AB

To prove :∠ ABC > ∠ ACB

Construction: Take a point D on AC such that AB = AD, Join BD.

Proof:

In [triangle ABC], we have 

AB = AD…by construction

∠ ABD =  ∠ ADB….(1)……angle opposite to equal side are equal

Now, consider BCD, we find that  ∠ADB is the exterior angle of BCD and an exterior angle is always greater than the interior opposite angle.

therefore,

∠ADB >  ∠DCB

∠ ADB =  ∠ACB…(2)

From 1 and 2 we get

∠ABD =  ∠ ACB…..(3)

But  ∠ABD is a part of  ∠ABC 

So,  ∠ABC =  ∠ABD…..(4)

From 3 and 4 we get 

∠ABC > ∠ ACB……..thus proved

Theorem 2: (Converse of Theorem 1) in a triangle the greater angle has the longer side opposite to it.

Triangle inequality theorem proof

Given:  In [triangle ABC] ∠ ABC > ∠ ACB

To prove: AC > AB

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Proof:

 In  [triangle ABC], we have the following three possibilities

1. AC =AB

2. AC < AB 

3.  AC > AB

Case 1 : When AC = AB

AC = AB

∠ ABC = ∠ ACB…angles equal to opposite sides are equal

This is contradiction

Since given that ∠ ABC > ∠ ACB

AC [neq] AB

Case 2: when AC < AB

AC < AB

∠ ACB  > ∠ ABC……..[Longest side has the greater angle opposite to it]

This also contradicts the given hypothesis.

Thus we are left with the only one possibility, AC > AB, 

Hence AC > AB

Theorem 3: The sum of any two sides of a triangle is greater than the third side of a triangle.

Given: in [triangle ABC]

To prove: AB + AC > BC,

AB + BC > AC and 

BC + AC > AB

Construction: produce side BA to D such that AD = AC. join CD

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Proof: 

in [triangle ACD], we have 

AC = AD …by construction

∠ ADC = ∠ ACD…..angle opposite to equal sides are equal

∠ ACD = ∠ ADC 

∠ BCA + ∠ ACD > ∠ ADC ……[ ∠ BCA + ∠ ACD > ∠ ACD]

∠ BCD > ∠ ADC

∠ BCD > ∠ BDC………………..[because ∠ ADC = ∠ BDC]

BD > BC …………………………[ side opposite to greater angle is larger]

BA + AD > BC

BA + AC > BC……………………[ since AC = AD by construction]

AB + AC > BC

Thus, AB + AC > BC 

Similarly, we can prove  AB + BC > AC and BC + AC > AB

Triangle Inequality Formula

In [triangle ABC]

AB + AC > BC,

AB + BC > AC and 

BC + AC > AB

Solved Examples

Example 1 : if side lengths 8cm, 5cm and 7cm are given can it form a triangle?

Solution: If 8cm, 5cm and 7cm are the sides of the triangle, then they should satisfy the inequality theorem.AB + AC > BC,

AB + BC > AC and 

BC + AC > AB

Let AB = 8cm, BC = 5cm and AC = 7cm

Then

AB + BC > AC

8 + 5 >  7

⇒ 13 > 7

BC + AC > AB

5 + 7 > 8 

⇒ 12 > 8

AB + AC > BC,

8 + 7 > 5 

⇒ 15 > 5 

All the three conditions are satisfied, therefore a triangle can be formed with sides 8cm, 5cm and 7cm.

Quiz Time

1. if side lengths 18cm, 5cm and 4cm are given can it form a triangle?

2. if side lengths 7cm, 15cm and 9cm are given can it form a triangle?

Trigonometric Ratios in Trigonometry are derived from the three sides of a right-angled triangle: the hypotenuse, the base (adjacent) and the perpendicular (opposite). These trigonometric formulas and trigonometric identities are used widely in all sciences related to geometry, mechanics and many others. Trigonometric Ratios help us to find missing angles and missing sides of a triangle. To be more specific, they are used in right-angled triangles, the triangles with one angle equal to 90°. These Trigonometric Ratios help us to find the values of trigonometric standard angles. 

According to the trigonometric ratio in Math, there are three basic or primary Trigonometric Ratios also known as trigonometric identities.Here they are:

What is Sin, Cos and Tan?

1. Sine: Sine of an angle is defined as the ratio of the side opposite the angle (to the hypotenuse (longest side) in the triangle.

2. Cosine: The cosine of an angle is defined as the ratio of the side which is adjacent to the angle to the hypotenuse (longest side) in the triangle.

3. Tangent: Tangent of an angle is defined as the ratio of the side which is opposite the angle to the adjacent in the triangle. 

The Different Values of Sin, Cos and Tan concerning Radians have been Listed Down in the Table Given Below 

Tricks to Remember the above Values

Step 1: Divide the numbers 0, 1, 2, 3 and 4 by 4,

Step 2: Take the positive square roots.

Step 3: These numbers give the values of sin 0°, sin 30°, sin 45°, sin 60° and sin 90° respectively.

Step 4: Write down the values of sin 0°, sin 30°, sin 45°, sin 60°, and sin 90° in reverse order and now you will get the values of cos, tan, cosec, sec, and cot ratios respectively.

Here’s a little description of how we got the values. Let’s take an acute angle θ. The values of sin θ and cos θ lie between 0 and 1. The sin of the standard angles 0°, 30°, 45°, 60°, and 90° are the positive square roots of 0/4,1/4, 2/4,3/4 and 4/4 respectively. The sine value of the standard angle table 0°, 30°, 45°, 60°,90°, 180°, and 360°:

Derivation of Values of Trigonometric Standard Angles

We will deal with the Trigonometric Ratios of 30 degrees and 60 degrees first. 

To do so, take an equilateral triangle ABC whose side is 2 units.

Then AB=BC=CA=2.

Let AD be the altitude of this aforementioned triangle we have taken.

This implies that BD=1.

From the Pythagoras theorem, we then get AD = √3.

From ΔABD, we will use the definitions of the various trigonometric quantities to find their standard values. Thus, 

• Sin 60 = √3/2 which implies that Cosec 60 = 2/ √3

• Cos 60 = 1/2 which implies that Sec 60 = 2

• Tan 60 = √3 which implies that Cot 60 = 1/√3

Again from ΔABD, we will determine the values again. 

• Sin 30 = ½ which implies that Cosec 30 = 2

• Cos 30 = √3/2 which implies that Sec 30 = 2/√3

• Tan 30 = 1/√3 which implies that Cot 30 = √3.

Let us now derive the values of 45 degrees of these trigonometric quantities. 

This time, we will take an isosceles right-angled triangle ABC with a right angle at B and AB=BC =1 unit.

Using the definitions of Trigonometric Ratios, we see that:

• Sin 45 = 1/√2 which implies that Cosec 45 = √2

• Cos 45 = 1/√2 which implies that Sec 45 = √2

• Tan 45 = 1 which implies that Cot 45 = 1.

Here are Values of the Trigonometry Standard Angles 

Similarly, we will find the cosine values of the values of other Trigonometric Ratios of standard angles are respectively the positive square roots of 4/4, 3/4, 2/4, 1/4, 0/4.

The cos value of the standard angles 0°, 30°, 45°, 60°,90°, 180° and 360°:

Here are Values of the Trigonometry Standard Angles 

Now, we know the sin and cos values of other Trigonometric Ratios of standard angles can easily be found.

The tangent value of the standard angles 0°, 30°, 45°, 60°,90, 180° and 360°:

Here are Values of the Trigonometry Standard Angles

Here are the cosecant values of the standard angles 0°, 30°, 45°, 60°, 90°, 180° and 360°:

Here is the Standard Angles Table 

Here are the secant values of the standard angles 0°, 30°, 45°, 60°, 90°, 180° and 360°:

Here is the Standard Angles Table 

Here, the cotangent values of the standard angles 0°, 30°, 45°, 60°,90°, 180° and 360° are listed below:

Here is the Standard Angles Table

The Trigonometric Ratios of standard angles are listed below 0°, 30°, 45°, 60° and 90°.The values of Trigonometric Ratios of standard angles are very helpful and important to solve the trigonometric problems. Therefore, it is necessary for you to remember the value of the Trigonometric Ratios of standard angles. Here’s the Trigonometric Ratios of the standard angles table.

Table Showing the Value of Each Ratio with Respect to Different Angles

( Trigonometric Ratios of Standard Angles Table)

Few Applications of Trigonometry

• Trigonometry is used in cartography, which is the creation of maps. 

• It has its applications in satellite systems. 

• It is used in the aviation industry. 

• The functions of Trigonometry are used to describe sound and light waves.

Questions to be Solved

1. Calculate cos(A) from the triangle given below.

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Solution) We know the formula of 

[cos (A) = frac{Adjacent}{Hypotenuse}]

In the given question, 

Adjacent = 12

Hypotenuse = 13

Then, cos (A) =12/13

2. Evaluate the value of Sin 90 + Cos 90.

Solution) As we know that the value of Sin 90 = 1

And the value of Cos 90 = 0

Substituting the values of Sin 90 and Cos 90 ,

Therefore, Sin 90 + Cos 90 = 1 + 0

= 1

A Function from set M to set N is a binary relation or a rule which links or plots or pictures each and every component of set M with a component in set N. The purpose of this chapter is to make you learn about various types of functions so that you can become acquainted with the types. You will also come to know that each type has its own individual graphs. Examples of the different types of functions are shown below.

 

The denotation of function in Mathematics

A function from set M to set N is denoted by:

F: M→N

We chiefly use F, G, H to denote a function

We can also denote a Mathematical class of any function using the following method:

• Tabulation Method

• Graph method

• Arrow Diagram method

 

A function is defined as a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Let A and B be two non-empty sets, mapping from A to B will be a function only when every element in set A has one end and only one image in set B. It can be defined that a function is a special relation which maps each element of set A with one and only one element of set B. Both sets A and B must not be empty. A function will define a particular output for a particular input. Therefore, f: A → B is a function such that for a ∈ A there is a unique element b ∈ B such that (a, b) ∈ f. For every Mathematical expression if it has an input value and a resulting answer can be presented as a function. 

Types of Functions in Mathematics with Examples

Types of functions are generally classified into four different types: Based on Elements, Based on Equation, Based on Range, and Based on Domain.

1. Based on Elements:

2. Based on Equation:

• Identity Function

• Linear Function

• Quadratic Function

• Cubic Function

• Polynomial Functions

3. Based on the Range:

4. Based on the Domain:

 

Types of Function – Based on Elements

1. One-To-One Function.

A Mathematical function is said to be a One-To-One Function if every component of the Domain function possesses its own and unique component in Range of the Function. That being said, a function from set M to set N is considered a One-To-One Function if no two or more elements of set M have the same components mapped or imaged in set N. Also, that no two or more components refined through the function provide the similar output.

For Example:

When f: M→N is described by formula y= f (x) = x³, the function “f” is stated to be a One-To-One function since a cube of different numbers is always different itself.

 

 

2. Onto Function.

A Function is Onto Function if two or more components in its Domain have the same component in its Range.

For Example:

If set M= {M, N, O} and set N= {1,2}

And “f” is a function by which f: M→N is described by:

Then the function “f” is regarded as Onto Function.

 

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3. Into Functions

A function is said to be an Into function in which there is an element of co-domain Y and does not have a pre-image in domain X.

Example:

Take into account, P = {P, Q, R} 

Q = {1, 2, 3, 4}   and f: P→ Q in a way

f = {P,1, Q,2, R,3}

In the function f, the range i.e., {1, 2, 3} ≠ co-domain of Y i.e., {1, 2, 3, 4} 

 

 

4. One – One Into Functions

The function f is said to be one-one into a function if there exists different components of X and have distinctive unique images of Y.

Example: Prove one-one into function from below set

X = P,Q,R

Y = [1, 2, 3, and 4} and f: X → Y in a way

f = {P,1, (Q,3, R,4}

X = P,Q,R 

Y = [1, 2, 3, and 4} and f: X → Y in a way

f = {P,1, (Q,3, R,4}

Thus, function f is a one-one into function

 

 

5. Many-One Functions

The function f is many-one functions if two or more different elements in X have the same image in Y.

Example: Prove many-one function

Taken, X = 1,2,3,4,5

Y = XYZ

X,Y,Z and f: X → Y

Thus and thus f = {1,X,2,X,3,X,4,Y,5,Z} 

Hence, function f is a many-one function

 

 

6. Many-One Into Functions

 The function f is a many-one function only if it is—both many ones and into a function.

 

 

7. Many-One Onto Functions

The function f is many-one onto function only if is –both many ones and onto.

 

 

8. Constant Function

A constant function is one of the important forms of a many to one function. In this domain every element has a single image. 

The constant function is in the form of 

f(x) = K, where K is a real number.

Types of Function – Based on Equation

Identify Function: The function that has the same domain and range.

Constant Function: The polynomial function of degree zero.

Linear Function: The polynomial function of degree one.

Quadratic Function: The polynomial function of degree two.

Cubic Function: The polynomial function of degree three.

 

Types of Functions – Based on Range

Modulus Function

The modulus function is the type of function that gives the absolute value of the function, irrespective of the sign of the input domain value. 

The modulus function is defined as f(x) = |x|. 

The input value of ‘x’ can be a negative or a positive expression.

Rational Function

A Rational Function is the type of function that is composed of two functions and expressed in the form of a fraction X.

A rational function is of the form 

f(x)/g(x), and g(x) ≠ 0. 

Signum Function

The signum function is the type of function that helps to know the sign of the funct
ion and does not give the numeric value or any other values for the range.

Even and Odd Function

The even and odd function are the type of functions that are based on the relationship between the input and the output values of the function.

Periodic Function

The function is said to be a periodic function if the same range appears for different domain values and in a sequential manner. 

Inverse Function

The inverse of a function is the type of function in which the domain and range of the given function is reverted as the range and domain of the inverse function.

The inverse function f(x) is denoted by f-1(x).  

Greatest Integer Function

The greatest integer function is the type of function that rounds up the number to the nearest integer less than or equal to the given number.

The greatest integer function is represented as

 f(x) = ⌊x⌋. 

Composite Function

The composite function is the type of function that is made of two functions that have the range of one function forming the domain for another function.

 

Types of Functions – Based on Domain

Algebraic Function

An algebraic function is the type of function that is helpful to define the various operations of algebra. This function has a variable, coefficient, constant term, and various arithmetic operators such as addition, subtraction, multiplication, division.

Trigonometric Functions

The trigonometric function is the type of function that has a domain and range similar to any other function. The 6 trigonometric functions are :

f(θ) = sinθ, f(θ) = tanθ, f(θ) = cosθ, f(θ) = secθ, f(θ) = cosecθ.

Logarithmic Functions

Logarithmic functions are the type of function that is derived from the exponential functions. The logarithmic functions are considered to be the inverse of exponential functions.

 

Solved Example of Functions

1. Find the inverse function of the function f(x) = 5x + 4.

Solution: The given function is f(x) = 6x + 4

It is rewritten as y = 6x + 4 and then simplified to find the value of x.

y = 6x + 4

y – 4 = 6x

x = (y – 4)/6

f-1(x) = (x – 4)/6

Ans: So the answer of this inverse function is f-1(x) = (x – 4)/6

2. For the given functions f(x) = 3x + 2 and g(x) = 2x – 1, find the value of fog(x).

Solution: The given two functions are f(x) = 3x + 2 and g(x) = 2x – 1. 

The function fog(x) is to be found.

fog(x) = f(g(x))

= f(2x-1)

= 3(2x – 1) + 2

= 6x – 3 + 2

= 6x – 1

Ans: Therefore fog(x) = 6x – 1

 

Representation of Functions

The functions can be represented in three ways: Venn diagrams, graphical formats, and roster forms. 

Venn Diagram: The Venn diagram is one of the important formats for representing the function. The Venn diagrams are generally presented as two circles with arrows connecting the element in each of the circles. 

Graphical Form: It is said that every function is easy to understand if they are represented in the graphical form with the help of the coordinate axes. The function in graphical form, helps to understand the changing behavior of the functions if the function is increasing or decreasing.

Roster Form: Roster form is a set of a simple Mathematical representation of the set in Mathematical form. The domain and range of the function in Roster form are represented in flower brackets with the first element of a pair representing the domain and the second element representing the range.

 

Practice Problems

Practice Problem-1:

Alex leaves his apartment at 5:50 a.m. and goes for a 9-mile jog. He returns at 7:08 a.m. to answer the following questions, assuming Alex runs at a persistent pace.

Report the distance D (in miles) Alex jogs as a linear function of his run time ‘t’ (in minutes).

Draw a graph of D

Simplify the sense of the slope.

 

Solution-1:

(i) At time t=0, Alex is at his apartment, thus, D (0) =0

At time t= 78 minutes, Alex completed running 9 mi, thus, D (78) =9.

The slope of the linear function comes about as:-

m=9−0 / 78−0= 3 / 26

The y-intercept is (0, 0), thus, the linear equation for this function is

D (t) =3/26 t

 

(ii) Now, to graph D, execute the fact that the graph cross over the origin and has slope m=3/26

 

Image will be Uploaded Soon

 

(iii) The slope m= 3/26 ≈ 0.115 reports the distance (in miles) Alex runs per minute or his average velocity.

 

Fun Facts

• As per Math processing, there are an infinite number of functions, much more than what you learned in this chapter

• Different Mathematical functions can make us protected in life as being misemployed, deceived or exploited.

The unit price is a measurement used to represent the price of a particular good or service to be exchanged with customers or consumers for money. The unit price refers to the price per unit of measures, such as price per pound, quart, ounce, or other units of weight or volume of the food package. 

The unit price is important for both customers and organizations. An organization cannot sustain itself for a long period by selling products at a lower price. Similarly, customers would not buy the product if the value of the product is less than the price. The researcher suggests that unit price information helps shoppers to save around 17-18% in supermarkets when they are educated on how to use it. 

Hence, customers should calculate the unit price to know what he is paying for a single item or a pound of food in supermarkets, rather than looking at the total quantity of the product he is purchasing.

Unit Price Formula 

The unit price includes fixed cost, the variable cost, overheads, direct labor cost, and a margin of profit to encourage the business activities and organization earning. Hence, the unit price formula is derived as:

Unit Price = Unit Cost + Profit Margin

Let’s now understand what unit cost is.

Electricity Unit Price 

Electricity unit price is the price that you pay to consume one unit of electricity in your household. Electricity unit price varies depending on the number of reasons such as your location, your preferred mode of payment, and what traffic you have chosen. If the tariff you have chosen does not have standing charges( fixed amount you have to pay regardless how much electricity you have consumed), then your electricity unit price will be higher to compensate. 

Unit Cost

The unit cost or cost per unit is referred to as the price spent by the company to produce, store, and sell each unit of a particular product. The unit cost includes all the variable costs and fixed costs involved during production. Unit cost is the minimum cost incurred for buying any standard unit.

Unit Cost Formula

Unit cost formula, also referred to as the cost per unit formula is derived by adding all the fixed cost associated with the product that is the cost that does not change when the value of good and services produce changes and all the variable cost associated with the product that is the cost that changes when the value of good or services produced changes and dividing the value by the total units produced during that period.

Accordingly, the unit cost formula is given as :

Cost Per Unit = [frac{text{Total Fixed Cost+Total Variable Cost}}{text{Total Number of Unit Produced}}]

Variable Cost

Variable costs are the cost that varies as the number of goods or services varies. 

Variable Cost Formula

The total variable cost incurred to produce the product is calculated by using the following formula:

Total Variable Cost – Total Number of Units Produced Variable Cost Per Unit

Variable Cost Per Unit

Variable cost per unit is defined as the cost of production of each unit produced in the company. Variable cost changes as the output or the level of the activity changes in the organization and these are not the fixed cost of the company as these costs occur only if there is a production in the company.

Variable Cost per Unit = [frac{text{Total Variable Cost}}{text{Total Number Unit Made}}]

Fixed Cost 

Fixed cost is defined as the cost or expense that is not affected by the increase or decrease in the number of units produced or sold by the company over a short-term horizon. In other words, fixed cost is not affected by the business activities, rather it is related to the period.

Although total fixed cost remains the same within the relevant changes, the fixed cost per unit decreases as the production increases because the same amount of fixed cost is spread over a large number of units of output.

Fixed Cost per unit of a product can be calculated by using the following formula:

Fixed Cost per Unit =  [frac{text{Total Fixed Cost}}{text{Total Number Unit Made}}]

What is Unit Elastic?

In Economics, unit elasticity is a term that describes the situation in which change in one variable is an equally proportional change in another variable. The unit elasticity in economic theory is frequently used to describe the change in demand or supply curves that are perfectly responsive to changes in price. A curve with an elasticity 1 is termed as unit elasticity. 

Unit Elastic Demand 

In economics, the elasticity demand describes the relationship between the number of products sold and the price of the product ( along with other factors). It is generally seen that customers react when the price of the product changes. If the price of the product increases, the demand decreases and vice versa.

Unit elasticity demand is similar to the elasticity demand in which change in price leads to equally proportional changes in quantity demand. In other words, unit elasticity demand states that the percentage change in quantity demanded is exactly the same as the percentage change in price. This generally happens where consumers have substitute goods available in the market to meet their needs. 

This concept also applies to the unit elasticity supply because when the supplier has substitute products to produce goods, the percentage change in price leads to equally proportional changes in quantity supplied.

As the percentage change in price leads to equally proportional changes in quantity demand and quantity supplied, the unit elasticity of demand is equal to -1 and the unit elasticity of supply is equal to 1..

Solved Examples

1. XYZ Company has a Total Variable cost of $ 60000 and total Fixed Cost of $ 40000 in February, which it incurred while producing $ 20000 Gadgets. Calculate Cost per unit of the Gadget produced by the Company.

Solution: The cost per unit of gadgets can be calculated by using the following cost per unit formula: 

Cost Per Unit Formula = [frac{text{Total Fixed Cost+Total Variable Cost}}{text{Total Number of Unit Produced}}]

= [frac{text{40000+60000}}{text{20000}}]

= $5 per unit

Therefore, the cost per unit of the gadget is $ 5

2. A certain product is made by the ABC incorporation. The Product has the following expenses and the Company wants to earn a profit of 20% over the Cost of the Production. Find price per unit of the Product.

Solution: Following are the steps to calculate the price per unit of the product.

Step 1 : Calculate the total cost of the product. To calculate the total cost, we need to add all the expenses incurred to make the product ready for sale.

Accordingly,

Total Cost –  Raw Material + Direct Labor + Overhead Expenses

                          =  1000 + 500 + 300

                          =  1800

So, total cost of the product = $ 1800

Step 2 : Calculate the unit cost of the product 

Unit Cost Formula = [frac{text{Total Cost}}{text{Total Number of Unit Produced}}]

= [frac{text{1800}}{text{100}}]

  = $18 per unit

Step 3: Calculate the profit requirement of the product 

Profit Requirement =  Total cost per unit Profit Margin

             =   $18 x 20%

              = $3.6

Step 4 : Calculate the price per unit of the product 

Price Per Unit =   Cost Per Unit +  Profit Requirement

= $18 + $3.6

= $ 21.6

So, the price per unit of the product is $ 21.6

The time it took to multiply integers by numerous digits was substantially reduced because of logarithms, which were established in the 17^th century to speed up calculations. For nearly 300 years, they were indispensable in numerical work until the invention of mechanical calculating machines in the late 1800s, and computers in the 1900s rendered them obsolete for large-scale calculations. With applications in physical and biological models, the natural logarithm (with base e 2.71828 and written ln n) is still one of the most useful functions in mathematics.

Value of Log e

The power to which a number should be raised to get the specified number is called the logarithm of that number. For example, the logarithm to the base 10 of 1000 is 3 because 10 raised to the power 3 is 1000. Logarithmic function is the inverse Mathematical function of exponential function. The logarithmic function log ax = y is equal to x = a^y. 

There are two types of logarithms generally used in Mathematics. They are common logarithms and natural logarithms. 

1. Common logarithm is any logarithmic function with base 10. It is generally represented as y = log x or y = log ₁₀x.

2. Natural logarithms are the logarithmic functions which have the base equal to ‘e’. Natural logarithms are generally represented as y = log ex or y = ln x . ‘e’  is an irrational constant used in many mathematical calculations. The value of ‘e’ is 2.718281828…

Value of ‘e’

The number ‘e’ is an irrational Mathematical constant and is used as the base of natural logarithms. The number ‘e’ is the only unique number whose value of natural logarithm is equal to unity. The value of ‘e’ was calculated in 1683 by Jacob Bernoulli. This mathematical constant finds its importance in various fields of Mathematics including:

Value of log e can be calculated in two different cases. The two cases are finding the natural logarithmic value of e and the common logarithmic value of ‘e’.

Case 1: Value of Log e to the Base ‘e’ (Natural Logarithm of ‘e’):

By definition, any logarithmic function is the inverse function of an exponential function. 

So, if log _ee = y, it can be written as e = e^y.

log _ee = y

e = e^y

Since the bases of the exponential functions on both sides are the same, powers should also be identical according to the properties of exponential functions. Therefore, it can be inferred that the value of ‘y’ is equal to one.

Initially, it was assumed that log ee = y. So, we can conclude that log ee = 1. Natural logarithm of ‘e’ is equal to unity.

Case 2: What is the Value of Log e Base 10 (Common Logarithm of ‘e’):

It is a fact that the common logarithm of a function whose natural logarithm value is known can be determined by dividing the value of natural logarithm by 2.303. (The natural logarithm of any function is divided by 2.303 to obtain the common logarithmic value because the natural logarithm of 10, i.e., log 10 base e, is calculated as 2.303).

In case 1, the value of log e to the base e calculated is 1. Log e base 10 is obtained by dividing 1 by 2.303.

[ log_{10}x =  frac{ In x}{2.303} ]

[ log_{10} e =  frac{ In e}{2.3033} ]

[ log_{10}e = frac{1}{2.303} ]

[ log_{10}e = 0.43421 ]

Therefore, the value of log e base 10 is equal to 0.43421 up to five decimal places.

Properties of Logarithmic Functions

Property 1: Product Rule

The natural logarithm of a product of two numbers is equal to the sum of natural logarithms of individual numbers. 

ln (x.y) = ln (x) + ln (y)

Property 2: Quotient Rule

Natural logarithm of fraction of two numbers is equal to the subtraction of natural logarithm of denominator from the natural logarithm of numerator.

ln[ frac{x}{y} ]= ln (x) − ln (y)

Property 3: Power Rule

Natural logarithm of a number raised to the power of another number is equal to the product of the power and the natural logarithm of the base.

ln (xy) = y ln x

Property 4: Derivative of Natural Logarithm

The first order derivative of a natural logarithmic function is equal to the reciprocal of that function.

If y = ln (x), then y’ = 1/x.

Property 5:

Natural logarithm of any number less than zero (negative numbers) is undefined.

Property 6:

Natural logarithm of 1 is equal to zero.

Property 7:

Natural logarithm of infinity is infinity.

Property 8:

Natural logarithm of -1 is a constant known as Euler’s constant. 

ln (-1) = i (Euler’s Constant)

Example Problems:

1. Evaluate the value of natural logarithm of √e.

Solution:

ln √e = ln e½ 

= ½ ln e (Power rule of logarithmic function)

= ½ x 1  (Natural value of log e is unity)

= ½

2. Evaluate ln[ frac{sqrt{x-1}}{e} ] .

Solution:

In [frac{sqrt{x-1}}{e} ] = In [frac{(x-)^{frac{1}{2}}}{e} ]

= ½ ln (x – 1) – ln e (Power rule and quotient rule)

= ½ ln (x – 1) – 1  (Recall what is the value of log e to the base e)

Fun Facts:

• Derivative of the natural logarithm of ‘e’ is equal to zero because the value of log e to the base e is equal to one, which is a constant value. The derivative of any constant value is equal to zero. 

• The logarithmic value of any number is equal to one when the base is equal to the number whose log is to be determined. Example: Log e base e is equal to 1, whereas log 10 base e is not equal to one. 

• Common logarithm of one is equal to zero.

• The value of log 10 base e is equal to 2.303.

In Mathematics, calculus refers to the branch which deals with the study of the rate of change of a given function. Calculus plays an important role in several fields like engineering, science, and navigation. Usually, calculus is used in the development of a mathematical model for getting an optimal solution. You know that calculus is classified into two different types which are known as differential calculus and integral calculus. However, you might not be aware of vector calculus. In these vector calculus pdf notes, we will discuss the vector calculus formulas, vector calculus identities, and application of vector calculus. Let us first take a look at what is vector differential calculus in these vector calculus notes.

 

Vector Calculus Definition

Vector calculus is also known as vector analysis which deals with the differentiation and the integration of the vector field in the three-dimensional Euclidean space. Vector fields represent the distribution of a given vector to each point in the subset of the space. In the Euclidean space, the vector field on a domain is represented in the form of a vector-valued function which compares the n-tuple of the real numbers to each point on the domain.

 

Vector analysis is a type of analysis that deals with the quantities which have both the magnitude and the direction. Vector calculus also deals with two integrals known as the line integrals and the surface integrals.

 

1. Line Integral

According to vector calculus, the line integral of a vector field is known as the integral of some particular function along a curve. In simple words, the line integral is said to be integral in which the function that is to be integrated is calculated along with the curve. You can integrate some particular type of the vector-valued functions along with the curve. For example, you can also integrate the scalar-valued function along the curve. Sometimes, the line integral is also called the path integral, or the curve integral or the curvilinear integrals.

 

2. Surface Integral

In calculus, the surface integral is known as the generalization of different integrals to the integrations over the surfaces. It means that you can think about the double integral being related to the line integral. For a specific given surface, you can integrate the scalar field over the surface, or the vector field over the surface.

 

Vector Calculus Formulas

Let us now learn about the different vector calculus formulas in this vector calculus pdf. The important vector calculus formulas are as follows:

From the fundamental theorems, you can take,

F(x,y,z)=P(x,y,z)i+Q(x,y,z)j+R(x,y,z)k

 

1. Fundamental Theorem of the Line Integral

Consider F=▽f and a curve C that has the endpoints A and B.

Then you would get

[int cF .dr = f(B) -f(A)]

2. Circulation Curl Form

According to the Green’s theorem,

[iint_{D}left ( frac{partial Q}{partial x} right )- left ( frac{partial P}{partial y} right )dA = oint CF. dr]

According to the Stoke’s theorem,

[iint_{D}bigtriangledown times F.ndsigma = oint CF. dr]

Here, C refers to the edge curve of S.

3. Flux Divergence Form

According to the Green’s theorem,

[iint_{D}bigtriangledown .F dA = oint CF. nds]

According to the Divergence theorem,

[int int int_{D}triangledown .FdV] = ∯  SF. ndσ

Vector Calculus Identities

Let us learn about the different vector calculus identities. The list of the vector differential calculus identities is given below.

1. Gradient Function

1. [vec{bigtriangledown}(f+g) =  vec{bigtriangledown}f + vec{bigtriangledown}g]

2. [vec{bigtriangledown}(cf) =  cvec{bigtriangledown}f], for a constent c

3. [vec{bigtriangledown}(fg) =  fvec{bigtriangledown}g + gvec{bigtriangledown}f] 

4. [vec{bigtriangledown}(frac{f}{g}) =  frac{gvec{bigtriangledown}f-fvec{bigtriangledown}g}{g^{2}}] at the point [vec{x}] where g [(vec{x}) neq 0]

5. [vec{bigtriangledown} (vec{F}.vec{G}) = vec{F}times (vecbigtriangledowntimesvec G  )- (vecbigtriangledowntimesvec F  )times vec G + (vec G .vec{bigtriangledown})vec F + (vec{F}.vec{bigtriangledown})]

 

2. Divergence Function

1. [vec{bigtriangledown} (vec{F}+vec{G}) = vec{bigtriangledown}.vec{F} + vec{bigtriangledown}.vec{G}]

2. [vec bigtriangledown.(cvec{F)} = c vec{bigtriangledown .vec F }]

3. [vec bigtriangledown.(fvec{F)} = f vec{bigtriangledown .vec F }+ vec F .vec bigtriangledown]

4. [vec bigtriangledown.(vec{F}times vec{G}) = vec{G}. (vec{bigtriangledown times vec{F}})-vec{F}.(vec{bigtriangledown times vec{G}})]

 

3. Curl Function

1. [vec bigtriangledowntimes (vec{F}+vec{G}) = vec{bigtriangledown times vec{F}}+ vec{bigtriangledown }times vec{G}]

2. [vec bigtriangledowntimes (cvec{F)} = cvec{bigtriangledown } times vec{F}], for a constant c

3. [vec bigtriangledowntimes (fvec{F)} = fvec{bigtriangledown } times vec{F} + vec{bigtriangledown }ftimes vec{F}]

4. [vec bigtriangledowntimes (vec{F}times vec{G}) = vec{F}.(vec{bigtriangledown . vec{G}})-(vec{bigtriangledown }vec{F})vec{G} + (vec{G}. vec{bigtriangledown })vec{F} -(vec{F}.vec{bigtriangledown }  )]

4. Laplacian Function

1. [vec{bigtriangledown ^{2}}(f+g) = vec{bigtriangledown ^{2}}f + vec{bigtriangledown ^{2}}g]

2. [vec{bigtriangledown ^{2}}(cf) = cvec{bigtriangledown ^{2}}f], for a constant c

3. [vec{bigtriangledown ^{2}}(fg) = fvec{bigtriangledown ^{2}}g + 2vec{bigtriangledown f}.g + g vec{bigtriangledown ^{2}}]

 

5. Degree Two Function

1. [vec{bigtriangledown }.(vec{bigtriangledown times vec{F}})] = 0

2. [vec{bigtriangledown }times (vec{bigtriangledown f})] = 0

3. [vec{bigtriangledown }.(vec{bigtriangledown  f}timesvec{bigtriangledown  g} ) = 0]

4. [vec{bigtriangledown }.(fvec{bigtriangledown  g}- gvec{bigtriangledown f}) = f vec{bigtriangledown ^{2}}g – g vec{bigtriangledown ^{2}}f]

5. [vec{bigtriangledown }times (vec{bigtriangledown times vec{F}}) = vec{bigtriangledown } (vec{bigtriangledown . vec{F}}) – vec{bigtriangledown ^{2}}]