Tangent Line Formula

Introduction To Tangent Line Formula:

Normally Tangent Line is defined as the line which locally just touches a curve particularly at one and only one point .Specifically We have to know that there is no intersection at all in any other points in a curve.Tangent Line Formula is used to represnt the tangent line in an efficient manner.In this article we will see about the tangent line formula with some practice example problems.

Tangent Line Formula:
The tangent line  to a curve at a given point is the straight line that just touches the curve particular point. The point of tangency means that the tangent line is going in the same direction with the curve, and  it is the best straight-line approximation to the curve at that point.

Normally To find the formula for the Tangent line to the curve y=f(x) at point `p(x_o,f(x_0))` ,We need to find out the slope of the curve . Slope of the curve is also called as the gradient of the curve.

Slope of the curve  calculated by using the following two steps:

1.First we need to take the derivative of the curve equation,

`dy/dx=f'(x)`

2. Then evaluate the above equation at the point p(x_o,f(x_0))

`dy/dx=f'(x_0)=slope`

The Equation for the line if we are knowing the slope and point

`y-y_1=m(x-x_1)`

Here the points are`(x_0,f(x_0)` and slope is `f'(x_0)`

So the Tangent line formula is

`y-f(x_0)=f'(x_0)(x-x_0)`

Tangent Line Formula Example1:

Find the equation of the tangent line to the curve y = x4 at the point (2, 8).

Solution:

`dy/dx= 4x^2`

Slope of the Curve when  x = 2 is

4 × 22 = 16.
So apply the slope and points in the slope point formula of the line.

`y-y_1=m(x-x_1)`

`y - 8 = 16(x - 2)`

y = 16x – 24

This is the equation of the tangent line

Tangent Line Formula Example2:

Find the equation of the tangent line to the curve `y = 5x^3-4y^2`  at the point (1, 2).

Solution:

`dy/dx= 15x^2-8y`
Slope of the Curve  when the points are (1,2) is

`dy/dx=15-16`

`m =-1`

So apply the slope value and x and y values into the slope point formula of the line.

`y-y_1=m(x-x_1)`

` y - 2 = -1(x - 1)`

`y-2 = -x +1`

y+x=3

This is the equation of the tangent line

Define Loss Ratio

Introduction for define of loss ratio:

A relation is getting form the comparison of two quantities like in some wisdom is called a ratio. Ratio will be show by the terms of fraction, that is, a: b is equal to `a/b` for example 25 is `1/4` of the hundred; Therefore, the ratio of 25 to 100 is `1/4.` We can write it in ratio as 1:4. The loss ration is calculated as the total money paid out is divided by the sum of the money is taken for the premium is multiplied by the 100.

Loss ratio = `"total money paid out"/"sum of money taken for premium"` `xx` 100

Example for Define of Loss Ratio:

Example:

A company has paid out `$` 65000 in claims of a previous year to john. But the company has taken in a total of  `$` 100000 in premiums during a particular year. Calculate the loss ratio to define.

Solution:

Total money paid out = `$` 65000

Sum of money taken for premium = `$` 100000

Loss ratio = `"total money paid out"/"sum of money taken for premium"` `xx` 100

= `65000/100000` `xx` 100

= `65/100` `xx` 100

= .65 `xx` 100

= 65%

Therefore the loss ratio for the company is 65% .

Practice Problems for Definition of Loss Ratio:

Problem 1:

A company has paid out `$` 70000 in claims of a previous year to Martin. But the company has taken in a total of `$` 150000 in premiums during a particular year. Calculate the loss ratio to define.

Solution: The loss ratio for the company is 46.67%

Problem 2:

A company has paid out `$` 75000 in claims of a previous year to Paul. But the company has taken in a total of  `$` 200000 in premiums during a particular year. Calculate the loss ratio to define.

Solution: The loss ratio for the company is 37.5%

Problem 3:

A company has paid out `$` 80000 in claims of a previous year to Sarra. But the company has taken in a total of `$` 250000 in premiums during a particular year. Calculate the loss ratio to define.

Solution: The loss ratio for the company is 32%

Percent Word Problem Solving

Introduction to percent word problem solving:

In math, percent is an expression, expressing the fraction as a percent (that is "per" "cent" means "per" "hundred"). The percentage is represented as "%" sign. The percentage of the number is expressed as 7%. Percentage expresses whether one quantity is larger or smaller with respect to the other quantity. The word problem is expressed in which it fits the equation.

There are three steps to solving math word problems:

Translate into equations
Solve the equations and
Check the answer
In math percent word problems, we have to translate the simple English statements into mathematical expressions. For example “of” indicates multiplication (times). Percentage word problems can be easily solved by using the rule, If suppose,

"(x) is (some percentage) of (y)", which translates to "(x) = (some decimal) × (y)"



Formula for Percent Word Problem Solving:
Formula for percentage:

The formula for percentage is the following and it must be easy to use:

`"is"/"of" = "%"/"100"`

Example 1:

What percent of 40 is 50?

Solution:

The given statement is

Percent * 40 = 50

Therefore,

x % `xx` 40 = 50

`x/100` `xx` 40 = 50

`(40x)/100` = 50

Now cross multiply to get

40x = 50*100 = 5000

x = `5000/40`

x = 125

Therefore, Fifty is 125% of 40

Example 2:

25% of what is 30?

Solution:

The statement is

25% `xx` x = 30

Therefore,

`25/100` `xx` x = 30

`(25x)/100" ` = 30

x = `3000/25`

= 120

Answer: 30 is 25% of 120

Examples of Percent Word Problem Solving:

Example 3:

In a college, 30% of students are studying computer science. If their total number is 120, how many students are studying in that college totally?

Solution:

There are 30% of students are studing computer science. We understood that 30% of all students in college are equal to number of students studying computer science.

Therefore, the equation is

30% of x = 120

`30/100`` xx x` = 120

30 x = 120 * 100

30 x = 12000

x = 12000/30

x = 400

Therefore, we have 400 students in the college.

Practice Percentage Word Problems

Practice problem 1:

What percent of 15 is 20?

Answer key: 133.33

Practice problem 2:

40% of what is 70?

Answer key: 175

Practice problem 3:

In a test there are 30 questions in total. If john gets 70% correct, how many questions did john missed?

Answer key: John missed 9 questions

Equations with Variables on both Sides Calculator

Introduction for equations with variables on both sides calculator:

Equation with variable are the important part of algebra, generally the equation with same variable are said to be equation with  variables on both sides. Fro example x+2x+4 = x+8 here x is the variable in this equation we need to find the value of variable using calculator. In this article we will discuss about equation with variables on both sides calculator with suitable example problem.

Problem on Equations with Variables on both Sides Calculator:

Simplify the following equations with variables on both sides calculator

Problem(i) : The equation  is 2x+4 = 4x+6+8.

Solution

The given equation  is 2x+4 = 4x+6+8.

Step 1: In this given equation collect the variable

2x-4x+4 = 6+8

Step2: Collecting the constant

2x-4x+4 = 14

2x-4x = 14-4.

Step 3: simplifying the equation

-2x = 10

Step 4: Dividing both sides by -2

`(-2x)/-2` =`10/-2`

x= -5

Step 5: The variable of x is -5.

Problem(ii) : The equation  is 3x= 4x+6+1.

Solution

The given equation  is 3x = 4x+6+1.

Step 1: In this given equation collect the variable

3x-4x = 6+1

Step 2: Collecting the constant

3x-4x = 7

3x-4x = 7.

Step 3: simplifying the equation

-x = 7

Step 4: The variable of x is -7.

Problem on Equations with Variables on both Sides Calculator:

Simplify the following equations with variables on both sides calculator

Problem (i) The equation is -5x+7+8 = -9x +15+10

Solution

The given equation  is -5x+7+8 = -9x +15+10

Step 1: In this given equation collect the variable

-5x+9x+15 = 15+10

Step2: Collecting the constant

-5x+9x+15 = 25

-5x+9x = 25-15.

Step 3: simplifying the equation

4x = 10

Step 4: Dividing both sides by 4

`(4x)/4` = `10/4`

x = `5/2`

Step 5: The variable of x is `5/2` .

Problem (ii) The equation is -2x+7+13 = 4x +14

Solution

The given equation  is -2x+7+13 = 4x +14

Step 1: In this given equation collect the variable

-2x-4x+7+13 = 14

Step2: Collecting the constant

-2x-4x+20 = 14

-6x = 14-20

Step 3: simplifying the equation

-6x = -6

Step 4: Dividing both sides by -6

`(-6x)/-6` =`(-6)/(-6)`

x = 1

Step 5: The variable of x is 1.

Simple Regression Formula

Introduction to simple regression formula:

In mathematics, one of the most important topics in statistics is regression. Regression is determining the relationship between two variables. Regression math are used to analysis the several variables. Regression is one of the statistical analysis methods that can be used to assessing the association between the two different variables.

Example of Simple Regression Formula:
Here we study about the simple regression formula are,

Formula for regression analysis:

Regression Equation (y) = a + bx

Slope `(b) = (NsumXY-(sumX)(sumY))/(NsumX^2-(NsumX)^2)`

Intercept`(a) = (sumY-b(sumX))/N`


Where,
x and y are the variables.
b = the slope of the regression line is also defined as regression coefficient
a = intercept point of the regression line where is in the y-axis.
N = Number of values or elements
X = First Score
Y = Second Score
`(sumXY)` = Sum of the product of the first scores and Second Scores
`(sumX)` = Sum of First Scores
`(sumY)` = Sum of Second Scores
`(sumX^2)` = Sum of square First Scores.

Example Problem for Simple Regression Formula:

Problem for simple regression formula:

Example 1:

Find the regression slope coefficient, intercept value and create a regression equation by using the given table.

X Values   Y Values

10            11

20            22

30            33

40            44

50            55


For the given data set of data, solve the regression slope and intercept values.

Solution:

Let us count the number of values.
N = 5
Determine the values for XY, X2

X Value  Y Value    X*Y      X*X

10        11        110      100

20        22        440      400

30        33        990      900

40        44       1760     1600

50        55       2750     2500


Determine the following values `(sumX), (sumY), (sumXY), (sumX^2).`
`(sumX) = 150`
`(sumY)= 165`
`(sumXY)= 6050`
`(sumX^2) = 5500`



Plug values in the slope formula,


Slope `(b) = (NsumXY-(sumX)(sumY))/(NsumX^2-(NsumX)^2)`


`= (5 xx(6050)-(150)xx(165))/((5)xx(5500)-(150)^2)`


`= (30250 - 24750)/(27500-22500)`


`= 5500/5000`

`b= 1.1`

Plug the values in the intercept formula,


Intercept `(a) = (sumY- b(sumX))/N`


`= (165-(1.1xx150))/5`


`= (165 - 165)/5`


`= 0/5`


`a = 0`

Plug the Regression coefficient values and intercept values in the regression equation,
Regression Equation(y) = a + bx
= 0 + 1.1x

Answer:

Slope (or) Regression coefficient (b) = 1.1

Intercept (y) = 0

Regression equation y = 0 + 1.1x

Number of Factors of an Integer

Introduction to number of factors of an integer:

In math, the natural numbers are form the integer and another name of an integer is whole number. The factor is divisor of a given number. This divisor is divides the given integer without any remainder. The factors may be two or more in an integer. Now we are going to see about number of factors of an integer.

Explanation for Number of Factors of an Integer

Some notes about integer and factors in math:

An integer may be positive and negative. We can’t calculate the number of factors of decimal or fraction.

The integer has one or more divisors. We can classify the factors into two. The names of factors are prime and composite. The prime factor is defining the prime number that is prime number has only two factors. The composite factor is defining the composite number that is the composite number has more than two factors.

We can count the number of factors of integer by listing method. We can separate the normal factors and prime factors.

More about Number of Factors of an Integer

Example problems for number of an integer in math:

Problem 1: Count how much number of factors and prime factors are present in given integer.

94

Answer:

Given integer are 94.

The given integer 94 has 4 factors as 1, 2, 47, 94 and 2 prime factors are present as 2 x 47.

Problem 2: Count how many number of factors present in given integer?

112

Answer:

Given integer are 112.

The given integer has 10 factors. They are 1, 2, 4, 7, 8, 14, 16, 28, 56 and 112.

It has 5 prime factors. They are 2 x 2 x 2 x 2 x 7.

Exercise problems for number of factors of an integer:

1. How many numbers of factors in integer 75?

Answer: The integer 75 has 6 factors as 1, 3, 5, 15, 25, and 75.

2. How many numbers of factors in integer 19?

Answer: The integer 19 has 2 factors.

The Various Types of Sets in Set Theory

One of the important theories in modern mathematics is the set theory. This theory has been present now for a very long time. This was developed during the 1870’s itself. There are various operations in the set theory. Set is basically a collection of objects. When it comes to sets in mathematics, the objects must be related to mathematics. There is the presence of a universal set in set theory. This acts as the reference set. Other sets are compared with the same. Set theory is best explained with the help of Venn diagrams. The operation on sets like union, intersection, and compliment and so on can be well represented with the help of Venn diagrams. The subset R contains some of the elements present in R and not all the elements.’ R’ acts as the universal set in this case. The concepts sets and subsets are closely related. Without the presence of one the other doesn’t exist. The definition of subset is that it is a set which contains some of the elements present in the original set.

An example can be used to explain the concept. A set has elements {x, y, z} and another one contains {x). The latter set is the subset of the former one. There can be a number of subsets of the same set. If there was another set containing the element {y}, then it also becomes the subset of the given set.  The subset notation is used to convey that a particular set is the subset of the other set. The other set is called the super set. It contains all the elements present in the subset and the subset contains some of the elements of the super set. This is nothing but the subset is a part of the super set.

Set theory can be very helpful in solving mathematical problems. The Venn diagrams give a clear picture on the sets and their operations. By the Venn diagrams problems can be solved. So, lot of arithmetical calculations can be avoided. The process also becomes very simple and easy to understand. Once one is thorough with the concepts of the set theory, the problems can be very easily solved. One of the important operations in set theory is that of intersection of two sets. The intersection of two sets yields a new set which contains the common elements of both the sets.