vertices of a pyramid

A vertex of a polyhedron is the point of intersection of three or more faces of the structure.

A pyramid is a polyhedron with a polygonal base connected to a single point called the apex.

The number of vertices of a pyramid varies with the different types of pyramids.


How to determine the number of vertices

In a pyramid, the side faces (originating from the apex) meet at the base.

pyramid

So basically, for the square pyramid shown above, the four vertices of the square base become four vertices of the pyramid.

In addition, we have the apex, where the slant surfaces meet.

Hence, in a pyramid with a n-sided polygon as base, we have (n+1) vertices.

Some examples of vertices of a pyramid.

Now let us try to find the number of vertices of some common pyramids.

triangular pyramid

This is a triangular pyramid, i.e., the base is a triangle (3-sided polygon).

Hence, the number of vertices would be (3+1) = 4


rectangular pyramid

This is a rectangular pyramid, i.e., the base is a rectangle (4-sided polygon).

Hence, the number of vertices would be (4+1) = 5


pentagonal pyramid

This is a pentagonal pyramid, i.e., the base is a pentagon (5-sided polygon).

Hence, the number of vertices would be (5+1) = 6


Thus, it is quite simple to find the number of vertices of a pyramid given we know the type of polygon it has as its base. The square pyramid (with a square as its base) is the most common type of pyramid found in various monuments all over the world.

Construction of a Regular Pentagon

In this part we are going to see in detail about the introduction to construction of a regular pentagon. It’s easy to construct a regular polygon by simply using the compass and the ruler. And so is the construction of the regular pentagon. Most of the pentagons constructed in this method are done by inscribing it into the circle.

Construction of a regular pentagon Method 1:

Draw a circle of diameter AB centered at O.
Draw a line CD perpendicular to AB. Now mark the mid points of OA and OB as P and Q respectively. Draw circles with center as P and Q and radius OP and OQ respectively.
Draw a line from C to P. This line intersects the circle with center Q at T.
With C as center and CT as radius draw a circle. This circle intersects the main circle at E and F which are the successive pentagon vertices.
With length EF find the other vertices of the pentagon.

Since the pentagon has all its sides equal it will be easy to find the other sides if find any one of the vertices.

Construction of a regular pentagon Method 2:

• Draw a circle with center O and Diameter XX’.

• Then draw a line AQ perpendicular to the line XX’.

• Now Mark a point P such that it bisects the line OX’.


• With P as center and length AP draw an arc such that it cuts the line OX. Mark the point G.

• Then with AG as radius and A as center draw an arc such that it cuts the circle at B and C



• With AB or AC as radius find the points D and E.

• Join the points ABCDE to get pentagon.

Study Calculating Median

Study Median:

Median is the middle value of the element in their ascending order. If the size of the elements is even, then the average value of two middle numbers is the median. In the sequence of elements, divide equally the sequence of elements are on one side, half on the other, then the average value of middle elements is the median.

Study Calculating Median Formulas:

If ‘n’ is the total number of elements and ‘n’ is the ‘odd’, then the formula for finding the middle position after the arrangement of elements in ascending order is: (n+1)/2.

If ‘n’ is the total number of elements and ‘n’ is the ‘even’, then you have to find the two middle positions after the arrangement of elements in ascending order. Formulas for finding the two middle positions are: (n/2) and (n/2)+1. After finding the two middle elements, find the average that is the median.

Study Calculating median - Example Problems:

Study Calculating median Problem 1:

Calculate the median of 5, 4, 1, 8, and 7.

Solution:

Count the elements in the sequence has given.

n = 5(odd)

Arrange the elements in ascending order.

1, 4, 5, 7, 8

Formula for finding the middle position is (n+1)/2

(5+1)/2 = 6/2 = 3

Median = 3.
Study Calculating Median Problem 2:

Calculate the median of 8, 6, 1, 2, 3, 4, 7, and 5

Solution:

Count the elements in the sequence has given.

n = 8(even)

Arrange the numbers in ascending order.

1, 2, 3, 4, 5, 6, 7, 8

Find the two middle positions using n/2 and (n/2)+1

Positions are n/2= 8/2 = 4 and 5

4th and 5th position numbers are 4 and 5

The average is (4+5)/2 = 4.5

Median = 4.5

Study Calculating Median Problem 3:

Calculate the median of 2, 4, 6, 3, 1, 6, 3, 7, 4, and 6

Solution:

Count the elements in the sequence has given.

n = 10(even)

Arrange the numbers in ascending order.

1, 2, 3, 3, 4, 4, 6, 6, 6, 7

Calculate the two middle positions using n/2 and (n/2)+1

Positions are n/2= 10/2 = 5 and 6

4th and 5th position numbers are 4 and 4

The average is (4+4)/2 = 4

Median = 4

Study median practice Problems:

Study median practice problem 1:

Calculate the median of 4, 3, 1, 2, and 6.

Answer: 3
Study median practice Problem 2:

Calculate the median of 7, 9, 1, 4, 3, 6, 2, and 8

Answer: 5
Study median practice Problem 3:

Calculate the median of 5, 4, 6, 2, 3, 1, and 9

Answer: 5

Composition and Invertible Function

An invertible function for ƒ is a function from B to A, with the property that a round trip (a composition) from A to B to A returns each element of the first set to itself. A function ƒ that has an inverse is called invertible; the inverse function is then uniquely determined by ƒ and is denoted by ƒ−1. Function composition is the applications of one function to the results of another. For instance, the functions f: X → Y and g: YZcomprised by computing the output of g when it has an argument of f(x) instead of x.  (Source. Wikipedia) .

Examples for invertible function:

To define the invertible of a function f i.e. f−1 (read as ‘f inverse’), the function f must be one-to-one and onto.

Example 1:

Let A = {1, 2, 3}, B = {a, b, c, d}. Consider a function f = {(1, a), (2, b),(3, c)}. Here the image set or the range is {a, b, c} which is not equal to the co domain {a, b, c, d}. Therefore, it is not onto.


For the inverse function f−1 the co-domain of f becomes domain of f −1.

i.e. If f : A → B then f−1 : B → A . According to the definition of domain, each element of the domain must have image in the co-domain. In f−1, the element‘d’ has no image in A. Therefore f −1 is not a function.

Example 2:

f = {(1, a), (2, a), (3, b)} where A = {1, 2, 3}, B = {a, b}

Here the two different elements ‘1’ and ‘2’ have the same image ‘a’.

Therefore the function is not one-to-one.

The range = {a, b} = B. ∴ The function is onto.

f(1) = a

f (2) = a

f(3) = b

Here all the elements in A has unique image

f −1 (a) = 1

f −1 (a) = 2

f −1 (b) = 3


The element ‘a’ has the two images 1 and 2. It violates the principle of the function that each element has a unique image. This is because the function is not one-to-one.

Thus, ‘f −1 exists if and only if f is one-to-one and onto’.

Examples for composition function:

Example :

Let A = {1, 2}, B = {3, 4} and C = {5, 6} and f : A → B and g : B → C such that f(1) = 3, f(2) = 4, g(3) = 5, g(4) = 6. Find gof.

Solution:

gof is a composition function from A → C.

Identify the images of elements of an under the composition function gof.

(gof) (1) = g(f(1)) = g(3) = 5

(gof) (2) = g(f(2)) = g(4) = 6

i.e. image of 1 is 5 and image of 2 is 6 under gof

∴ gof = {(1, 5), (2, 6)}

How to Calculate Median

To find the median in a set of data, intial step is rearrange the set of data in ascending order or descending order. The mid-term value is the median for thae set of data.Half the numbers in the record are fewer, and half the numbers are greater. Establish the Median, put the numbers are known in value arrange and locate the middle number. But there are two middle numbers (as happens when there is an even amount of numbers) then average those two numbers.


Single middle number:

Example 1:

Establish the median for the following listing of value:

8, 13, 4, 7, 5

Solution:

Find the Median of: 8, 13, 4, 7, and 5 (Odd amount of numbers)

Line up your numbers: 4, 5, 7, 8, and 13 (smallest to largest)

The Median is: 7 (The number in the middle)

Example 2:

Establish the median for the following listing of value:

3, 8, 3, 4, 3, 6, 4, 1, 3

Solution:

Find the Median of: 3, 8, 3, 4, 3, 6, 4, 1, and 3 (Odd amount of numbers)

Line up your numbers: 1, 3, 3, 3, 3, 4, 4, 6, and 8 (smallest to largest)

The Median is: 3 (The number in the middle)


Double middle numbers:

Example 1:

Establish the median for the following listing of value:

8, 3, 4, 7, 2, and 6

Solution:

Find the Median of: 8, 3, 4, 7, 2, and 6 (Even amount of numbers)

Line up your numbers: 2, 3, 4, 6, 7, and 8 (smallest to largest)

Add the 2 middles numbers and divide by 2:

= (4 + 6) / 2

= 10 / 2

= 5

The Median is 5.

Example 2:

Establish the median for the following listing of value:

8, 8, 8, 9, 9, 9, 11 and 10

Solution:

Find the Median of: 8, 8, 8, 9, 9, 9, 11 and 10 (Even amount of numbers)

Line up your numbers: 8, 8, 8, 9, 9, 9, 10 and 11 (smallest to largest)

Add the 2 middles numbers and divide by 2:

= (9 + 9) / 2

= 18 / 2

= 9

The Median is 9


Example 3:

Establish the median for the following listing of value:

7, 8, 6, 9, 8, 7, 12 and 11

Solution:

Find the Median of: 7, 8, 6, 9, 8, 7, 12 and 11 (Even amount of numbers)

Line up your numbers: 6, 7, 7, 8, 8, 9, 11 and 12 (smallest to largest)

Add the 2 middles numbers and divide by 2:

= (8 + 8) / 2

= 16 / 2

= 8

The Median is 8

Derivative Differentiation

what is differentation?

Differentiation is all about finding rates of change of one quantity compared to another. We need differentiation when the rate of change is not constant.

What does this mean constant rate change ?

the distance from the starting point increases at a constant rate of 60 km each hour, so after 5 hours we have travelled 300 km. And the slope (gradient) is always 300/5 = 60 for the whole graph. There is a constant rate of change of the distance compared to the time. The slope is positive all the way (the graph goes up as you go left to right along the graph.)

What does this mean when rate of change is not constant ?

Now let's throw a ball straight up in the air. Because gravity acts on the ball it slows down, then it reverses direction and starts to fall. All the time during this motion the velocity is changing. It goes from positive (when the ball is going up), slows down to zero, then becomes negative (as the ball is coming down). During the "up" phase, the ball has negative acceleration and as it falls, the acceleration is positive. Notice this time that the slope of the graph is changing throughout the motion. At the beginning, it has a steep positive slope (indicating the large velocity we give it when we throw it). Then, as it slows, the slope get less and less until it become 0 (when the ball is at the highest point and the velocity is zero). Then the ball starts to fall and the slope becomes negative (corresponding to the negative velocity) and the slope becomes steeper (as the velocity increases).

derivative differentiation -The Derivative

The concept of Derivative is at the core of Calculus and modern mathematics. The definition of the derivative can be approached in two different ways. One is geometrical (as a slope of a curve) and the other one is physical (as a rate of change). Historically there was (and maybe still is) a fight between mathematicians which of the two illustrates the concept of the derivative best and which one is more useful. We will not dwell on this and will introduce both concepts. Our emphasis will be on the use of the derivative as a tool.

the physical concept of derivatives

This approach was used by Newton in the development of his Classical Mechanics. The main idea is the concept of velocity and speed. Indeed, assume you are traveling from point A to point B, what is the average velocity during the trip? It is given by

Average velocity = distance from A to B / time to get from A to B.

If we now assume that A and B are very close to each other, we get close to what is called the instantaneous velocity. Of course, if A and B are close to each other, then the time it takes to travel from A to B will also be small. Indeed, assume that at time t=a, we are at A. If the time elapsed to get to B is $\Delta t$, then we will be at B at time $t=a + \Delta t$. If $\Delta s$ is the distance from A to B, then the average velocity is

\begin{displaymath}\mbox{Average velocity} = \frac{\Delta s}{\Delta t}\cdot\end{displaymath}

The instantaneous velocity (at A) will be found when $\Delta t$get smaller and smaller. Here we naturally run into the concept of limit. Indeed, we have
\begin{displaymath}\mbox{Instantaneous Velocity (at A)} = \lim_{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t}\cdot\end{displaymath}

derivative differentiation - formulas

General Derivative Formulas:
1)  Where  is any constant.
2)  It is called Power Rule of Derivative.
3)
4)  Power Rule for Function.
5)
6)
7)
8)
9)  It is called Product Rule.
10)  It is called Quotient Rule.

Derivative of Logarithm Functions:

11)
12)
13)
14)

Derivative of Exponential Functions:

15)
16)
17)
18)
19)

Derivative of Trigonometric Functions:

20)
21)
22)
23)
24)
25)

Derivative of Hyperbolic Functions:

26)
27)
28)
29)
30)
31)

Derivative of Inverse Trigonometric Functions:

32)
33)
34)
35)
36)
37)

Derivative of Inverse Hyperbolic Functions:

38)
39)
40)
41)
42)
43)

Derivative Differentiation

what is differentation?

Differentiation is all about finding rates of change of one quantity compared to another. We need differentiation when the rate of change is not constant.

What does this mean constant rate change ?

the distance from the starting point increases at a constant rate of 60 km each hour, so after 5 hours we have travelled 300 km. And the slope (gradient) is always 300/5 = 60 for the whole graph. There is a constant rate of change of the distance compared to the time. The slope is positive all the way (the graph goes up as you go left to right along the graph.)

What does this mean when rate of change is not constant ?

Now let's throw a ball straight up in the air. Because gravity acts on the ball it slows down, then it reverses direction and starts to fall. All the time during this motion the velocity is changing. It goes from positive (when the ball is going up), slows down to zero, then becomes negative (as the ball is coming down). During the "up" phase, the ball has negative acceleration and as it falls, the acceleration is positive. Notice this time that the slope of the graph is changing throughout the motion. At the beginning, it has a steep positive slope (indicating the large velocity we give it when we throw it). Then, as it slows, the slope get less and less until it become 0 (when the ball is at the highest point and the velocity is zero). Then the ball starts to fall and the slope becomes negative (corresponding to the negative velocity) and the slope becomes steeper (as the velocity increases).

derivative differentiation -The Derivative

The concept of Derivative is at the core of Calculus and modern mathematics. The definition of the derivative can be approached in two different ways. One is geometrical (as a slope of a curve) and the other one is physical (as a rate of change). Historically there was (and maybe still is) a fight between mathematicians which of the two illustrates the concept of the derivative best and which one is more useful. We will not dwell on this and will introduce both concepts. Our emphasis will be on the use of the derivative as a tool.

the physical concept of derivatives

This approach was used by Newton in the development of his Classical Mechanics. The main idea is the concept of velocity and speed. Indeed, assume you are traveling from point A to point B, what is the average velocity during the trip? It is given by

Average velocity = distance from A to B / time to get from A to B.

If we now assume that A and B are very close to each other, we get close to what is called the instantaneous velocity. Of course, if A and B are close to each other, then the time it takes to travel from A to B will also be small. Indeed, assume that at time t=a, we are at A. If the time elapsed to get to B is $\Delta t$, then we will be at B at time $t=a + \Delta t$. If $\Delta s$ is the distance from A to B, then the average velocity is

\begin{displaymath}\mbox{Average velocity} = \frac{\Delta s}{\Delta t}\cdot\end{displaymath}

The instantaneous velocity (at A) will be found when $\Delta t$get smaller and smaller. Here we naturally run into the concept of limit. Indeed, we have
\begin{displaymath}\mbox{Instantaneous Velocity (at A)} = \lim_{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t}\cdot\end{displaymath}

derivative differentiation - formulas

General Derivative Formulas:
1)  Where  is any constant.
2)  It is called Power Rule of Derivative.
3)
4)  Power Rule for Function.
5)
6)
7)
8)
9)  It is called Product Rule.
10)  It is called Quotient Rule.

Derivative of Logarithm Functions:

11)
12)
13)
14)

Derivative of Exponential Functions:

15)
16)
17)
18)
19)

Derivative of Trigonometric Functions:

20)
21)
22)
23)
24)
25)

Derivative of Hyperbolic Functions:

26)
27)
28)
29)
30)
31)

Derivative of Inverse Trigonometric Functions:

32)
33)
34)
35)
36)
37)

Derivative of Inverse Hyperbolic Functions:

38)
39)
40)
41)
42)
43)