Step by step math solution

Mathematics is used throughout the world and it is an essential tool in many fields like natural science, engineering, medicine, and the social sciences. "Mathematics" is the Greek word its gives the meaning of learning, study, science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times.

Step by Step Math Solutions - Method to Solve:

There are many formulae abounded for solving math problem step by step One of the most important formula is P.E.M.D.A.S. It is nothing but the operations done in the problems solving in math. They are

• Parenthesis
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

Step 1: First perform the operations inside a parenthesis.
Step 2: Now solve the exponents
Step 3: Then multiplication and division is done from left to right
Step 4: Then addition and subtraction is done from left to right

Step by Step Math Solutions - Example Problems:

Step by step math - Problem 1:

Solve the equation step by step

X + 6 = 8

Solution:

It is one step equation

Subtract 6 on both sides

X + ( 6 - 6 ) = (8 - 6)

X = 2

Step by step math - Problem 2:

Solve the equation step by step

4x + 2=18

Solution:

Subtract 2 on both sides

4x + (2 - 2) = 18 - 2

4x = 16

Divide by 4 on both sides

`(4x)/4` = `16/4`

X = 4

Step by step math - Problem 3:

Find the value of x step by step

6x - 8 = 4 x -10

Solution:

Subtract 4x  from both sides of the equation

2x - 8 = - 6

Add 6 to both sides of the equation

2x = 2

Divide both sides by 2:

x = 1

The answer is x =  1

Step by Step Math Solutions - Probability Problems:

Step by step math - Problem 1:

A fair coin is tossed two times. What is the probability of getting at least one heads.

Solution:

Let A = be the event of getting at least two heads

Let S = Sample Space which refers to the total number of probable outcomes.

S = (HH, HT, TH, TT) =4

A = (HT, TH) =2

P (A) = `2/4` = `1/2`

The probability is `1/2` .

Step by step math - Problem 2:

Consider a die is rolled; calculate the probability of in receipt of odd numbers?

Solution:

There are six different outcomes 1,3,5,7,9,10,12,14,16,17,18

n (A) = 6, total number of odd numbers occur 6.

n (S) = 11, total number of outcomes is 11.

Probability of the event A happen = P (A) = n (A) / n(S) = `5 / 11` = 0.45.

Probability of the event A does not happen = P (A') = 1 – P (A) = 1 - 0.45 = 0.55.

Circle Division

A circle is a closed curve. The points on the circle are equidistant from the center of the circle. A circle creates 2 regions, the interior of the circle is one region and the exterior is another region.

A circle can be divided into equal or unequal parts. The division can be accomplished by straight lines or curved lines. With every nth line the circle gets divided into n+1 parts, meaning with one line circle can be divided into 2 parts, with nth line circle can be divided into n+1 parts assuming that the lines do not intersect.

Circle Division Term in Maths:

A circle division can be accomplished by using a straight line or a curved line. As we are not considering the area calculation for this discussion, let us restrict ourselves to straight lines for the purposes of this study.

The straight line that joins a point on the circumference of the circle to another point on the circumference of the same circle is called a chord. A chord divides the circle into 2 parts.

Consider several chords on the circle and suppose some or all of the chords intersect. Then it is found that the number of area regions that the chords cut the circle into is always more than the number of chords passing through the circle.

Here is an example

The above is popularly known as the cake cutting example.

In the first picture, one chord divides the circle into 2 parts.in the second picture, two chords divide the circle into 4 parts, in the third three chords divide the circle into 3 parts and the forth diagram 4 chords divide the circle into 11 parts.

Here you see that the first cut creates 1 new region (1+1)

The second cut creates 2 new regions (2+2)

The third cut creates 3 new regions (4+3) and the 4th cut creates 4 new regions (7+4)

This is mathematically represented as

F (n) = n + f (n-1)

Conclusion for Circle Division:

Circle division is an important study in geometry that gives several clues for practical applications. There are several theorems and postulates that delve deeper into circle division and provide new insights into this area of study.

Frequency and Frequency Table

Frequency table, it comes under the category of statistics. First we see about frequency. The frequency of a given table or data is that how many times the value occurs in table or data. For example, suppose in a class five students are score sixty marks in mathematics then the score of sixty is said to have frequency of five. Frequency means range of values also and the frequency of any data is denoted as (ƒ).

What is a frequency table?
A frequency table contains sets of collected data values. The arrangement is such that the magnitude of collected data values is in ascending order along with the corresponding frequencies. Now frequency table definition in other way is that it is a list of quantity in ascending order and list shows the numbers how many times each value occurs.

How to make a frequency table?
Here we understand the procedure of preparing A frequency table. For this we follow some steps. In step one; we make a table with three columns. In these three columns, first columns shows the marks and marks are arranged in ascending order means start from the lowest value. The second column shows tally marks. It means put corresponding tally in front of marks. When all values listed then make horizontal lines for all the values. Third column shows frequency, count the value of frequency for each mark and write in third column. Finally the frequency table is constructed.

The frequency table is different for different types of problems. Sometimes in problems range values is given with no of students and cumulative numbers. When we make frequency table for this type of problem then in first column we write range value in ascending order. In second column we write number of student which comes under the corresponding range. In third column we write cumulative numbers then we solve the problem.
Always i find probability distributions is very hard for me. If you do feel the same watch out for my coming posts.

How to do a frequency table?
We start with simple example. Suppose we have a frequency table with two columns. In first column marks(20-30, 30-40,40-50,50-60, 60-70) is given and in second column frequency(2,4,3,5,7) is given. Frequency column represent the number of students who scored marks in particular range of frequency. We have to calculate the number of student who gets fifty plus marks. Fifty plus marks comes in 50-60 and 60-70 range and their corresponding number of student is 5 and 7. So the answer is 12 students.

Integral properties and definition

In this article first we define Integration, in mathematics it is an important concept. Its inverse definition is also equally important. Integration is one of the main operations from two basic operations of calculus. In simple form we can define that integration means to calculate area. Now we define mathematically, suppose we have a given function (f) with real variable (x) over an interval [a, b] for a given real line, then we expressed this function as ∫f(x) dx. Integration means calculation of area of the region in XY-plane, which is bounded by the graph of function (x). Area above from the X-axis adds the total value and area below the X-axis subtracts from the total value.

The term integrals also known as antiderivatives. Suppose we have a given function is (F) and derivative of this function is (f). In this case it is known as indefinite integral and can be expressed as ∫f(x) dx. The notion of antiderivative are basic tools of calculus. It has many applications in science and field of engineering. Integral is an infinite sum of rectangles of infinite width. Integral is based on limiting procedure of area. Line integral means function with two or three variables where closed interval are replaced by any curve. Curve may be made in any plane or space. In surface integral in place of plane a short piece of surface is used.

There are various integral properties; integral properties are for the definite notion of antiderivative based on the certain theorems. First theorem is, suppose M(x) and N(x) are two defined functions. They are also continuous function in interval [a, b], then we have linearity property for the notion of antiderivative which can be expressed as
∫ [M(x) +N(x)] dx= ∫M(x) dx + ∫N(x) dx
∫a. M(x) dx= a∫M(x) dx. a is an arbitrary constant and we carry out the constant term from the function.
Second theorem is, suppose function f(x) is defined which is continuous in closed interval [a, b], then we have some special property of integral such as…
∫f(x) dx= 0, when limit are same.
∫f(x) dx= ∫f(x) dx+∫f(x) dx, when limits are divided between interval s like a to c and c to b.
∫f(x) dx= -∫f(x) dx, when upper limit becomes lower limit and lower limit becomes upper limit.

There are many integral types such as definite integral in which function is continuous and define in a closed interval. Other types are indefinite integrals, surface integrals, double integral known as Green’s theorem, triple integrals known as Gauss divergence theorem and line integrals

Graphing Trig functions

Sine graph equation:
The general form of a sine function is like this:
Y = sin x.
As we already know, sine is a periodic function. The period of a sine function is 2 pi. That means each value repeats itself after an interval of 2𝛑 on the x axis. The range of the sin function is from -1 to 1. So the value of sin x would not exceed 1 and would not go below -1 at any point. To be able to plot the graph of the function, let us make a table of values of the sine function.

The graph of the above table would look like a wave.

Cosine graph equation:
Just like the sine function, the cosine function is also periodic. The parent cosine function would be like this:
y = cos x
Similar to the sine function, the range of the cosine function is also from -1 to 1. The period of the cosine function is also 2𝛑. That means that the value of the function repeats itself after an interval of 2𝛑. To be able to plot the cosine function now let us make a table of values of cos.

The graph of the function would look like this:

How to graph tangent functions?

The tangent function is also a periodic function. However the period of the tangent function is 𝛑. That means the values repeat itself after an interval of ?? on the x axis. The range of the tangent function is –inf to inf. That means that the tangent function can have any real number value. Just like how we did for the sine and the cosine functions, for plotting the tangent function also we shall make a table of values.

Properties and derivatives of Logarithm functions

Definition of logarithmic function: For any positive number a is not equal to 1, log base a x = inverse of a^x. The graph of y = log x can be obtained by reflecting the graph of y = a^x across the line y = x. since log x and a^x are inverses of one another, composing them in either order gives the identity function.

We can have some observations about the logarithmic functions. From the graph, we can see that Logarithmic function is defined for positive values only and hence its domain is positive real numbers. The range is set of all real numbers. The graph always passes through (1, 0). The graph is increasing as we move from left to right. In the fourth quadrant, the graph approaches y-axis (but never meets it). It is also clear that the graphs of log base a x and a^x are mirror images of each other if y = x is taken as a mirror line.

Properties of Logarithmic Functions:
For any number x > 0 and y > 0, properties of base a logarithms are:
Product rule: log xy = log x + log y,
Quotient rule: log x / y = log x – log y,
Reciprocal rule: log 1 / y = -log y,
Power rule: log x^y = y log x.

Derivative of Logarithmic Functions: We use the following properties in the differentiation of logarithmic functions.
d/dx(e^x) = e^x
d/dx (log x) = 1/x.

Inverse of logarithmic functions : Since ln x and e^x are inverses of one another, we have  e^ln x = x ( all x > 0 ) ln ( e^x ) = x( all x ).

Logarithmic functions examples: Suppose we have to  Evaluate d/dx log base 10 ( 3x + 1 ).
d/dx log base 10 ( 3x + 1 ) = ( 1 / ln 10 ) .
( 1 / ( 3x + 1 ) ) d / dx ( 3x + 1 ) = 3 / [ ( ln 10 ) ( 3x + 1 ) ].

Evaluate integral log base 2 x /x dx.
Solution: integral log base 2 x / x dx = ( 1 / ln 2 ) integral ln x / x = ( 1 / ln 2 ) integral u du = ( 1 / ln 2 ) ( u^2 / 2 ) + C
= ( 1 / ln 2 ) [ ( ln x )^2 / 2 ] + C = ( ln x )^2 / 2 ln 2 + C.

Instantaneous rate of change of a function

Suppose we have a linear function such as y = 2x+3. The graph of this function is as follows:

Assume that to be a graph of the distance traveled by a car from home base, such that at time x = 0, the car is 3 miles from home. Now consider two points on the graph P(1,5) and Q(2,7). If we wish to find the rate of change of distance between these two points, we use the formula:
Rate of change = (y2-y1)/(x2-x1) = (7-5)/(2-1) = 2/1 = 2
That was fairly simple. That is because our graph was a straight line. Now suppose if the graph is not a straight line. And it is a curve instead.

The above graph as we see is not a straight line, but it is a curve. This time the co-ordinates of the points P and Q are (1,7) and (2,13) respectively. The average rate of change from P to Q can be found using the same formula above:
Average rate of change = (13-7)/(2-1) = 6/1 = 6. But we call this average rate of change since, it cannot be exact because the line between P and Q is not a straight line.
With this back ground let us now try to understand what is instantaneous rate of change. Now suppose the point Q moves closer and closer to point P, such that the distance between P and Q is infinitesimally small.

Therefore if co-ordinates of P are (x0, f(x0)) then those of Q would be (x0+h, f(x0+h)). Now, as Q moved closer and closer to P, the value of h goes on decreasing till it finally becomes 0 when Q coincides with P. That does not actually happen, h goes on decreasing to an infinitesimally small value. So we say that h tends to 0. Symbolically, h -> 0. Then the rate of change of the function f would be given as:
Rate of change = lim (h->0) [f(x+h) – f(x)]/[x+h – x] = lim(h->0) [f(x+h) - f(x)]/[h]
Stated above is the instantaneous rate of change equation. The term ‘rate of change’ now becomes ‘instantaneous rate of change’. We call it instantaneous because, at the instant when x = x0, the rate of change of the function is given by the limit:
instantaneous rate of change = lim(h->0) [f(x0+h) – f(x0)]/h

Instantaneous rate of change examples:
Find instantaneous rate of change of f(x) = x^2 at x = 4.
Ir = lim(h->0) [f(4+h) – f(4)]/h = lim(h->0)[16+8h+h^2-16]/h = lim(h->0)[8h+h^2]/h = lim(h->0)[h(8+h)]/h = lim(h->0)[8+h] = 8+0 = 8