Solving Basic Exponential Function

Introduction:

Exponential function can be represented as f(x) = bx where b is real number and x is a variable. Domain of any function f(x) is a set of all real values on X-axis for which function is defined and exponential function is defined for all real value of x on X-axis so domain of exponential function is all real number. For function f(x) = bx value of f(x) will be always positive if b is positive and always negative if b is negative so in range of function there will no negative number if b is positive and there will be no positive number in range of function f(x) if a is negative. Exponential functions will be strictly increasing or strictly decreasing depending on value of b. If b >1 then function will be strictly increasing and if b < 1 function will be strictly decreasing for example function y=5x is strictly increasing and function y = (1/5) x is strictly decreasing function. Inverse of exponential function is logarithmic function, logarithmic function can be written as y = logxy where x and y both are positive numbers. Exponential function can also be written as y =ex where e is a real number.



More on Exponential Functions:
For exponential function f(x) = bx , b is called base of exponential function. If two equal exponential functions have same base then their power will be also same for example if bx = by then we can write as x = y, to find inverse of exponential function just inverse the position of x and y in function.Looking out for more help on college algebra solver in algebra by visiting listed websites.

Basic properties of exponential functions:

`e^x e^y = e^(x+y)`
`(e^x)^p = e^(px)`
`de^x/dx = e^x`
`de^(ax)/dx = ae^(ax)`
`d^n e^(ax)/dx^n = a^n e^(ax)`
`e^x/e^y = e^(x-y)`

Example Problem:

Example 1: What will be value of x if 2x = 8?

Solution: 2x = 8

2x = 23

Base is same for both side of equation so

x = 3

Example 2: Find inverse of exponential function y = 5x

Solution: Given function is y = 5x

To find inverse of this function substitute y for x and x for y

x = 5y

log5 x = log5 5y

= y log5 5

= y

So, Inverse of given function is y = log5 x

Example 3: Find the exponential function y = 7x and also graph the function.

Solution: A set of ordered pairs for the equation is shown below.

Putting the value x = -0.5 we get

`y = 7^-0.5`

= 0.3779

(x,y) = (-0.5, 0.3779)

Putting the value x = -0.25 we get

`y = 7^-0.25`

= 0.614

(x,y) = (-0.25, 0.614)

Putting the value x = 0 we get

`y = 7^0`

= 1

(x,y) = (0,1)

Putting the value x = 0.25 we get

`y = 7^0.25`

= 1.626

(x,y) = (0.25, 1.626)

Plotting the given points to get the graph of the exponential function,



This graph shows the part of the exponential function between the co-ordinate points. So, it might not look like a curve but if we plot all the points over the real domain, we get the exact curve.

Proportional Relation

Introduction to proportional relation:

If two quantities vary uniformly such that when one increase the other also increase or when one decrease the other one also decrease then the they are said to be in direct proportion. If two quantities are very that is one increase the other one is decrease then they are said to be inverse proportional relation. In this article we shall discuss proportional relation with example problem.I like to share this Dispersion Relation with you all through my article.

Proportional Relation Example Problem

Example:

A man takes 4 hours to travel 350 miles. How many miles the man travels in 6 hours?

Solution:

Step 1:

Consider the problem as If the person takes 4 hour then the he travel 350miles. If he takes 6 hours then how many miles he travels?

Step 2:

The proportional relationship direct:

4=>350

6=>`6/4 ` x350

=>525

Answer:

Therefore the person traveled 525 miles in 6 hours

Example:

A man purchases 25 articles at `$` 6 each. How many articles can be purchased at the same amount of `$` 50 each?

Solution:

Here the proportions relation is inverse proportion

Cost per article                  Number of article

6                                                            25

50                                                            ?

As cost increase number if article decreases.

Here 6: 50 = x: 25

50 × x = 6 × 25

x = 6 × `25/50` =3

Therefore number of article for the same amount     = 3

Example:

It takes 5 workers 9 hours to repair a machine. How long will it take 12 workers to do the same work they work at the same rate?

Step 1:

Consider the problem as If 5 workers then take 9 hours to complete a job. If 12 workers then how many hour need to repair the same machine?

Step 2:

The inverse proportional relationship:

5 workers => 9 hours

12 workers=>`5/12` x 9

=>`45/12`

Therefore 3.75 hours need to repair a machine

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Proportion Relation Practice Problem

Problem:

It takes 6 workers 9 hours to repair a machine. How long will it take 10 workers to do the same work they work at the same rate?

Answer:

Therefore 5.4 hours need to repair a machine

Problem:

A man takes 6 hours to travel 450 miles. How many miles the man travels in 8 hours?

Answer:

The person traveled 600 miles in 8 hours

Solving a Ratio for a Triangle

Introduction  of solving a ratio for a triangle:

This topic about the ratio in the mathematical concept.

The ratio of width to height of typical computer displays. In mathematics, a ratio expresses the magnitude of quantities relative to each other. Specifically, the ratio of two quantities indicates how many times the first quantity is contained in the second and may be expressed algebraically as their quotient

Example: For every Spoon of sugar, you need 2 spoons of flour (1:2)

Solving a Ratio for a Triangle Explanation:

In this explanation the ratio of two triangle of same type is the relative between their measures and determines how many times the one triangle is greater than or less than other triangle.

Thus ratio of x and y is the fraction of x / y, and is generally written as x: y



triangle

Here will explain the ratio of the two triangles Based on the value of the each side.

The triangle one value of each side here given,

First triangle AB=4;      BC=5;      CA=7;

Second triangle value is AB=8;  BC=10;   CA=14, for solving this triangle ratio problem.

The Ratio of the two triangle is = first triangle sides / second triangle sides

AB1/AB2;     BC1/BC2;         CA1/CA2;

Final answer of the two triangle is 4/8;       5/10;           7/14;

Ratio=1/2; 1/2; 1/2;

So, Total ratio of two triangle is ½ or 1:2.

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Solving a Ratio for Triangle Problems:
Ex 1:  To solve this Problem the ratio for the two triangles the values are given below?

The triangle one value of each side here given

First triangle AB=3;     BC=5;        CA=6;

Second triangle value is AB=9;   BC=15;        CA=18;

Sol :  By solving ratio of triangle rule,

The Ratio of the two triangle is = first triangle sides / second triangle sides

This formula used for solving this problems.

AB1/AB2;      BC1/BC2;        CA1/CA2;

Final answer of the two triangle is 3/9;      5/15;         6/18;

Ratio=1/3; 1/3; 1/3;

So, ratio of two triangles is ½ or 1:3

Ex 2: To solve this problem the ratio for the two triangles the values are given below?

The triangle one value of each side here given

First triangle AB=2; BC=3; CA=4;

Second triangle value is AB=8; BC=12; CA=16;

Sol : By solving ratio of triangle rule,

The Ratio of the two triangle is = first triangle value / second triangle

This formula used for solving this problems.

AB1/AB2;   BC1/BC2;      CA1/CA2;

Final answer of the two triangle is 2/8; 3/12; 4/16;

Ratio=1/4; 1/4; 1/4;

So, ratio of two triangles is ½ or 1:4

Numerical Data Table

Introduction of Numerical data table:

Numerical data Table shows the measurement of numerical scale. Statistical methods are using for analyzing the numerical data for numerical data table. Values in the numerical data table is using for charts, histogram and graphs. Numerical data table is associating with collection, organization and interpretation of numerical data. In numerical data table, data should be number.

Description of Numerical Data Table:

Numerical data table is incorporated to statistics. The numerical data table is described with frequency distribution, percent and average. Frequency distribution is used for bar graph, Percent is used for pie chart and Average is used for normal graph.

Ordering data in Numerical Data table:

Ordering the data in the numerical data table will be done by qualitative numerical data and quantitative numerical data.

Ordering qualitative numerical data

It shows the numerical data with some quality parameters.

ordering quantitative data.

It shows the numerical data in terms on quantity.

Qualitative data can be shown in:

Pie charts: Another name of pie chart is circle graph, which is partitioned into sector by sector.

Bar graph: Bar graph consists the rectangular bars, which are representing the values.

Quantitative data can be shown:

Statistical tables: This table contains only  statistical data.

Rank order: This is arranging the data by its rank.

Frequency distribution: Frequency distribution denotes data of  intervals

Understanding numerical data table:

Graphs and number understand the values in the numerical data table. We have already seen the different graphs and number above.Is this topic the prime numbers from 1 to 100 hard for you? Watch out for my coming posts.

Example Problems - Numerical Data Table:

1.compute the average of the numbers, 25, 15, 28, 30, 27and 20 on numerical data table.

Solution:

Mean = 25+15+28+30+27+20 / 6

= 145 / 6

= 24.6             

2.compute the Median of the numbers, 15, 20, 28, 25 and 27 on numerical data table.
Solution:

Median = 15, 20, 28, 25, 27 = 27

3.compute the mode of the numbers, 22, 15, 22, 21, 15 and 20 on numerical data table.

Solution:

Here 15 & 22 repeated twice.

So, the modes are 15, 22.

Delta Mathematical Expression

Introduction for delta mathematical expression:

Delta (uppercase Δ, lowercase δ) is the fourth letter of the Greek alphabet. In the system of Greek numerals it has a value of 4. It was derived from the Phoenician letter Dalet. In the Ancient Greek language, it represented a voiced dental plosive /d/, while in Modern Greek it represents a voiced dental fricative, hence it is pronounced as "th" in "that" or "this". it is Romanized as either "D" or "Dh"

(Source: Wikipedia)

Delta Mathematical Expression - Definition Delta and Symmetric Delta:

Definition for delta:

Delta, Δ, and procedures the rate of change of option value with respect to changes in the fundamental asset's price. Delta is the initial derivative of the value V of the derivative with respect to the fundamental instrument's price S.

`Delta ` = `del`V / `del`S   

Symmetric difference for delta:

In symmetric distinction of two sets is the set of basics which are in one of the sets, but not in together. This procedure is the set-theoretic kin of the exclusive disjunction (XOR operation) in Boolean logic. The symmetric distinction of the sets A and B is often denoted by

A`Delta` B

Delta mathematical expression - slope and application:

Delta from slope formula:

Slope = (y2 – y1) / (x2 – x1)

= Δy / Δx

Where, Δ changes in difference

Delta mathematical expression – Example problems:

Example 1:

Find the slope between two points (4,1) (7,6)

Solution:

Delta from slope = (y2 – y1) / (x2 – x1)

(4,1) (7,6) – (x1,y1) (x2, y2)

= (6 – 1) / (7 – 4)

= (5) / (3)

Δy / Δx = 5 / 3

Example 2:

Find the slope between two points (5,6) (7,8)

Solution:

Delta from slope = (y2 – y1) / (x2 – x1)

(5,6) (7,8) – (x1,y1) (x2, y2)

= (8 – 6) / (7 – 5)

= (2) / (2)

Δy / Δx = 1

Delta Mathematical Expression - Application of Delta in Quadratic Equation:

In the polynomials of quadratic equations, Δ is referred as the discriminant value.

Δ = b2 – 4ac

Where Δ is the discriminant value.

Delta mathematical expression - Example problems:

Example 1:

Find the discriminant value given b = 8, a = 2, c = 6

Solution:

Δ = b2 – 4ac

= 82 – 4 (2)(6)

= 64 – 48

`Delta` = 16

Example 2:

Find the discriminant value given b = 8, a = 2, c = 4

Solution:

Δ = b2 – 4ac

= 82 – 4 (2)(4)

= 64 – 32

`Delta`= 32.

Divide an Angle into Two Equal Angles

Introduction to divide an angle into two equal angles:

Let us assume an angle formed by two rays. The vertex of the angle is the initial points for both the rays. Suppose you need to divide this angle into two equal angles, what is to be done.

One method is to take the help of a protractor to measure the angle. Then you numerically divide that by two, mark a point again using the protractor and join that to the vertex of the angle.

However the better method to divide an angle into two equal angles is by using a compass.

Divide an Angle into Two Equal Angles – Compass Method
The following diagram illustrates the method to divide an angle into two equal angles.



Let OX and OY be the rays making an angle XOY at vertex O.

Set the compass to a convenient radius and strike two arcs from O, one on the ray OX and another on the ray OY. Let the points of intersection of these arcs be A and B respectively. (Fig i)

Now set the compass again for a convenient radius, draw an arc from point A. Without changing the compass setting, draw a similar arc from point B.  Let these arcs intersect at P. Draw a ray connecting the vertex O and the point P. The ray OP divides angle XOY into two  equal angles.  (Fig ii)

The ray OP is called the internal angle bisector of the angle XOY.

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Divide an Angle into Two Equal Angles – Proof for the Method

Join AP and BP (Fig iii)

Considering the triangles  OAO and OBP,

OA = OB  (By construction)

AP = BP  (By construction)

Side OP is common.

Hence the triangles  OAO and OBP are congruent as per SSS postulate.

Therefore , angle AOP = angle BOP (corresponding parts of congruent triangles)

Thus it is proved that the angle XOY is divided into two  equal angles.

Binomial Distribution Normal

Introduction to binomial distribution:
In a specified series of n independent experiments, the discrete probability distribution for the number of successes is called as normal binomial distribution. Bernoulli research is also an experiment of success/failure in normal binomial distribution.Bernoulli experiment is also known as  Bernoulli trial. In normal binomial distribution when n=1, it  is called as Bernoulli distribution.

Mean and Median

Consider the random variable K having the parameters p and n follows the normal binomial distribution. We can write it as K ~ B(n, p).

Probability mass function gives the probability of receiving successes(k) in n trials.

f(k;n,p)=Pr(K=k)=(n k) p^k (1-p)^n-k. for k = 0, 1, 2, ..., n,

where f(k;n,p)=f(n-k;n,1-p)

Mean and variance: If X ~ B(n, p) X  is a binomially distributed variable, then we can say that the expected value of X is E[X]=np. and the variance is Var[X]=np(1-p)

Mode and Median

Normally the mode and the median of a normal binomial B(n,p) distribution is said to be identical to (n + 1)p. However when (n + 1)p is an integer and p is neither 0 nor 1, then the distribution has two modes namely the (n + 1)p and (n + 1)p - 1. However the small special results have been established in the given ratio:

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Mode, median and mean gets coincide for np is the given integer.
Median m should lie in the known interval ?np? = m = ?np?.
For any median m the value cannot recline too far away from the given mean: |m-np| = min {ln2, max {p, 1-p}}.
When p = ½ and n is odd, any m in the given interval ½(n-1) = m = ½(n+1) is said to be a median for the binomial distribution.
If p = ½ and n is the even, then the m = n/2 is an unique median for the given Var(X)=E(X^2)-(E(X))^2. For using the formula need the expected value of X 2: