Definition Non Linear

In math, a nonlinear system is one of the system which is not linear, that is, a system which does not assure the superposition principle, or whose result is not directly proportional to its input. The equation used to solve the non linear problem is f(x)=C. This function is called as linear if f(x) is a non linear. This article gives some of the examples about the non linear by using this definition.


Examples of Definition Non Linear:

By using the definition of non linear solve the following examples.

Example problem1:

Use the following function rule to find f(7).

`f(x)=2x^2`

Solution:

Given `f(x)=2x^2` ; Find f(7).

Plug x = 7 into the function and simplify.

`f(x) = 2x^2`

`f(7) = 2(7)^2`            Plug in x = 7

f(7) = 2(49)                  Square

f(7) = 98                       Multiply

Which is the required solution.

Example problem2:

Use the following function rule to find f(8).

`f(x) = (x - 4)^2`

Solution:

Given:

`f(x) = (x - 4)^2`

Find: f(8)

Plug x = 8 into the function and simplify.

`f(x) =(x - 4)^2`

`f(8) = ( 8 - 4)^2`            Plug in x = 8

`f(8) = (4)^2`                  Subtract

f(8) = 16                          Square

Which is the required solution.

Example problem3:

Use the following function rule to find f(1).

f(x) = 5 – 4|x|

Solution:

Given: f(x) = 5 – 4|x|; Find f(1)

Plug x = 1 into the function and simplify.

f(x) = 5 –4|x|

f(1) =5 – 4|1|                  Plug in x = 1

f(1) =5 – 4(1)                 Take the absolute value

f(1) =5 – 4                       Multiply

f(1) = 1                            Subtract

Which is the required solution.

Practice Problems of Definition Non Linear:

Problem 1:

Use the following function rule to find f(7).

f(x) = -9|x| + 4

Solution:

f(7) = -59

Problem 2:

Use the following function rule to find f(-8).

`f(x) = 3x^2`

Solution:

f(-8) = 192

Relevance and Application of Exponential Functions in Real Life Situations

In this article relevance and application of exponential functions in real life situations , we will see how exponential function used in real life. Exponential function is used in more real life application like compound interest, problem based on population, problem based on radioactive decay, mortgage problems. Let us work out some problem to make understand that application of an exponential function.

Exponential Function Growth:

Exponential function growth`g=c(p)^t where,`

c-Number at initial

p-growth factor (p =1+r,here r is given growth percentage)

t-time

Example 1 -  Relevance and application of exponential functions in real life situations

A group of 1000 people increase by 5% in an hour near to accident place. How many people will be in the crowd after 3 hour?

Given:

c=1000

p =1+r =1+0.05=1.05

t =3 hour

Solution:

Exponential function growth g=c(p)^t

Substitute the given data in the formulae

g=1000(1.05)^3

g=1000(1.1576)

g=1157.6

g=1158 people

Exponential Function Decay:

Exponential function decay `d=c(p)^t`

where,

c-Number at initial

p-growth factor

(p=1-r,here r is given decay percentage)

t-time

Example 2: Relevance and application of exponential functions in real life situations

The price of a violin is  $1,000 which decreases at a rate of interest of 3%.What is the price of a violin after 2 year?

Given:

c=1,000

p=1-r=1-0.03=0.97

t=2 year

Solution:

Exponential function decay `d=c(p)^t`

`d=1,000(0.97)^2`

d=1,000(0.9409)

d=940.9

d=941

The price of a violin after 2 year=$941

Example 3: Relevance and application of exponential functions in real life situations

John invests $50,000 for 2 year with the interest of 4% compounded half yearly .Find out the compound interest for his investment at a given rate of interest?

Given:

P=$50,000

R=4%

N=2 year

Solution:

Compound interest =` P[1+(R/2)/100]^(2n)`

`=50,000[1+(4/2)/(100)]4`

`=50,000[1+(2/100)]4`

`=50,000xx(102/100)xx(102/100) xx(102/100) xx(102/100)`

=$50,000(1.0824)

=$54121.6

Compound interest=Total amount-principle

=54,121.6-50,000

=$4,121.6

Example 4: Relevance and application of exponential functions in real life situations

Calculate how long blood clotting cell will take to produce 20,400

Solution:

We need to consider  assume function to calculate population based on problems

`f(t)=2^t`

`20,400=2^t`

Find the natural logarithm on both side

`ln(20,400)=ln(2^t)`

ln(20,400)= t ln(2)

`t= ln(20,400)/ ln(2)`

`t=(9.923)/0.693=14.29`

After 14.32 min the blood clotting cell can produce 20,400  cell.

Angle Addition Property

In this article we are going to discuss Angle Addition Property , the terms related to Angle Addition Property and some solved problems on Angle Addition Property .

Introduction to angle addition property :

The angle addition postulate states that if a point is within an angle and you add the two angles that are made by drawing a line through the point that the total will equal the large angle. ...




Angle Addition Property or postulate says that if there is a line segment SV lies in the interior of the angle TSR then Angle TSV + Angle VSR = Angle TSR

For example if Angle TSR = 40 degree and Angle TSV = 15 degree then the other angle will be of 25 degree.

The same property can be applied if there are two or more lines lies in the interior of an angle.

There are some important terms that are associated with Angle Addition Property

If two or more angle sums to 90 degree they are called complementary angles.



If two or more angles are lies in a straight line then definitely sums to 180 degree and called Supplementary Angle.



Two or more angles sharing same side are called Adjacent Angles.



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Lets Learn more about Angle Addition Property in Supplementary Angles



Supplementary Angles

What is the measure of Angle B if angle A is 120 degree?

Angle B + Angle A = 180

Angle B + 120 = 180

Angle B = 60 degree

Problem Based on Angle Addition Property.

Lets learn more about Angle Addition Property



Suppose Angle CAD is a complementary angle
‹CAB = 40 degree

‹EAD = 40 degree

‹BAE =  ?

Solution Angle CAB + Angle EAD + Angle BAE = Angle CAD
40 degree +40 degree + Angle BAE = 90
Angle BAE = 10 degree
What if I remove the line AB
calculate ‹CAE

Solution

‹CAE = 50 degree (‹CAB + ‹ BAE)

What if the three of the angle are of equal

Solution :Suppose each angle = x degree
X+x+x = 90
3x= 90
x= 30 degree
If Angle EAD is thrice to angle CAB

Angle BAE is 10 more to Angle CAB

Measure each angle

Let Angle CAB = x degree
Angle BAE = x+10 degree
Angle EAD = 3x
x+x+10+3x = 90
5x+10= 90
5x=80
x=16
Therefore Angle CAB ,BAE ,EAD are 16 ,26 ,48 degree respectively.

Determine the Slope and Y Intercept

Slope:

The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. Given two points (x1,y1) and (x2,y2) on a line, the slope m of the line is




Y intercept:

In the coordinate system, the y intercept of a line is a point at which the line cuts the Y-axis. The y-intercept of a line is denoted as (0, y)




Example Problems to Determine the Slope and Y Intercept:

Example problem 1:

Determine the slope and y intercept of y = 7x - 12

Solution:

Step 1: Given equation

y = 7x - 12

Step 2: The slope intercept form of a line equation is given by

y = mx + b

Where,

m → slope

b → y intercept

Step 3: Compare the given equation with the slope intercept form

By comparing the given equation with the slope intercept form, we get

m = 7     and b = - 12

Step 4: Solution

Therefore,  Slope = 7

y intercept = - 12

Example problem 2:

Determine the slope and y intercept of 2x - 5y = 4

Solution:

Step 1: Given equation

2x - 5y = 4 ................... (1)

Step 2: Subtract 2x on both sides of the equation 2x - 5y = 4

2x - 5y - 2x  = 4 - 2x

- 5y = 4 - 2x

Step 3: Divide by (-5) on both sides of the equation

Therefore,

y = `2/5` x - `4/5`

y = 0.4x - 0.8 ............. (2)

Step 4: The slope intercept form of a line equation is given by

y = mx + b

Where,

m → slope

b → y intercept

Step 5: Compare the equation (2) with the slope intercept form of a line equation

By comparing the equation (2) with the slope intercept form, we get

m = 0.4     and b = - 0.8

Step 6: Solution

Therefore,  Slope = 0.4

y intercept = - 0.8

Example problem 3:

Determine the slope and y intercept of 9x + 3y = 0

Solution:

Step 1: Given equation

9x + 3y = 0 ............ (1)

Step 2: Subtract 9x on both sides of the equation 9x + 3y = 0

9x + 3y - 9x = - 9x

3y = - 9x

Step 3: Divide by 3 on both sides of the equation

Therefore,

y = - 3x ............... (2)

Step 2: The slope intercept form of a line equation is given by

y = mx + b

Where,

m → slope

b → y intercept

Step 3: Compare the equation (2) with the slope intercept form of a line equation

By comparing the equation (2) with the slope intercept form, we get

m = - 3     and b = 0

Step 4: Solution

Therefore,  Slope = - 3

y intercept = 0

Practice Problems to Determine the Slope and Y Intercept:

1) Determine the slope and y intercept of y = 4x - 13

2) Determine the slope and y intercept of 5x + y = 5

3) Determine the slope and y intercept of 3/2 + 2y = 4

Solutions:

1) Slope = 4; y intercept = - 13

2) Slope = - 5; y intercept = 5

3) Slope = -0.75; y intercept = 2

Non Consecutive Vertices

Non consecutive vertices mean the vertices which are not in the adjacent. Here we are going to learn about the non consecutive vertices of the planes. Generally we can say opposite vertices of the polygons are called non consecutive vertices. We will see some examples for non consecutive vertices. This will help us to understand the non consecutive vertices.  Basically a vertex in a polygon refers the intersection point of two sides.

Non Consecutive Vertices:

Basically non consecutive vertices of a polygon mean the vertex which does not lies next to the other. In other words we can say the opposite vertices or the other vertices which is not lies very next to the vertex. The distance between the two vertices will give us the height or width of the polygon. If we find the distance we can calculate the area and volume of the polygon.

If we are having a polygon with n number of vertices then each polygons vertex is having n – 3 non consecutive vertices. For example if we take a square it is having 4 sides. So for each vertex it has n – 3 = 4 – 3 = 1 non consecutive vertices. We can calculate the diagonals of the square using the non consecutive vertices.




Examples for Non Consecutive Vertices:

Example 1 for non consecutive vertices:

Find the number of non consecutive vertices in the following polygon.



Solution:

In the above diagram we are having the number of sides as 6.

We know for a single vertex we have n – 3 numbers of vertices.

So for a six vertices we have 6 (n – 3) vertices = 6 (6 – 3) = 6 x 3 =18 non consecutive vertices.


Example 2 for non consecutive vertices:

Find the number of non consecutive vertices in the following polygon.



Solution:

In the above diagram we are having the number of sides as 8.

We know for a single vertex we have n – 3 numbers of vertices.

So for a six vertices we have 8 (n – 3) vertices = 8 (8 – 3) = 8 x 5 =40 non consecutive vertices.

Negative Cosine Graph

Basically if we draw the cosine graph mean the graph will like a normal graph. Negative cosine graph is nothing but the negative sign in front of the cos.  A negative cosine equation will be like the following

Y = - a cos (bx +c)

Hare – sign is in front of the cosine graph. If we draw the negative cosine it will produce the reflection about the x – axis. So the negative cosine graph will be opposite to the positive graph.

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Comparison between Positive and Negative Cosine Graphs:

Let us any cosine function y = ± Cos ((2πx + 3))

Solution:

We want to draw the graph for both positive and negative cosine function.

Here phase shift = -c / b = -3 / 2π = -`1.5 / pi`

Period = 2π / b = 2π / 2π = 1

So the positive cosine graph will be like the following



And the negative cosine graph will be like the following



To know the difference we have to combine these two graphs. So we get



From the above we know the positive cosine graph will reflect the negative cosine graph. Here the red line graph indicates positive cosine graph and the black line graph indicates the negative cosine graph.

Example for Negative Cosine Graph:

Example 1:

Draw the graph for the following function y = - 5 Cos x

Solution:

Here the amplitude is 5 and this is having a negative sign with it. So it is a negative cosine graph. Here the period is 2π and there is no  phase shift. So the graph will be



Example 2:

Draw the graph for the following function y = - 0.5 Cos (π x / 6)

Solution:

Here the amplitude is 0.5 and this is having a negative sign with it. So it is a negative cosine graph. Here the period is 2π / (π / 6) = 12 and there is no phase shift. So the graph will be

Quadrilateral Trapezoid

A quadrilateral is a two-dimensional figure created by connecting four segments endpoint to endpoint with each segment intersecting exactly two others. It also has four sides and four angles. Sum of interior angle of any polygon is (n-2) *180o. Here quadrilateral has four sides so interior angle is (4-2)*180o = 360o.Exterior angle of any polygon is 360o so the exterior angle of a quadrilateral is 360 degree.

Trapezoid


                         Trapezoid

Characteristics of Trapezoid :-

The trapezoid is a type of a quadrilateral.
A pair of opposite equal sides is known as trapezoid.
A trapezoid has unequal sides.
Normally the trapezoid has no lines of symmetry.
A trapezoid is an irregular shape.
Two trapezoids can be used to form a Parallelogram.
There will be one pair of parallel lines in a trapezoid
Perimeter of a trapezoid is to sum all the length’s of the trapezoid
Perimeter = a + b + c + B
Area of a trapezoid is ½ h ( B + b )
The sum of the adjacent angles are equal to 180o
The sum of all the interior angles are equal to 360o
Isosceles trapezoid is a type or trapezoid

Isosceles Trapezoid


                   Isosceles trapezoid

Characteristics of  Isosceles Trapezoid :-

The Isosceles trapezoid is a type of trapezoid.
If non-parallel pair of opposite sides of a trapezium is equal then it is called as isosceles trapezium.
The angles on the both sides of the base are equal
The sum of the adjacent angles are equal to 180o
There will be one pair of parallel lines in a Isosceles trapezoid
There will be one pair of opposite sides which are equal
In a Isosceles trapezoid diagonals are equal
The sum of two adjacent angles are equal to 180o
The sum of all the interior angles are equal to 360o