Slope As A Derivative

In this page we are goimng to discuss about slope as a derivative concept . The trigonometrically tangent of the angle that a line makes with the positive direction of x – axis in anticlockwise direction is called the slope of the line.

The slope of a line in general is denoted by m. Thus m = tan Ø, where Ø is the angle which the line makes with the positive x-axis in anticlockwise direction.

The application of derivatives to plane geometry is related to slopes. Let us consider a curve whose equation is y = f(x).

On this curve we take a point P(x1, y1). Let the tangent at this point not be parallel to the co-ordinate axes, we can get the equation of tangent line at P.

The derivative of the function f(x) is the slope of the tangent.

dy /dx at P(x1,y1) is tan α = slope of the tangent at P, where α is the angle which the tangent at P(x1,y1) makes with the positive direction of x-axis.

The equation of straight line with slope m and passing through the point (x1,y1)is given as

y-y1 = m(x –x1).

For the curve y = f(x) we have  dy /dx =f’(x)

For the tangent line, slope = m = f’(x1)= dy / dx at (x1,y1).

The equation line of the tangent line will be of the form

y-y1 = f’(x1)(x –x1).

If the tangent at P is parallel to the x axis then α =0, `rArr`  tan α =0, `rArr`  slope =dy /dx =0.

If the tangent at P is perpendicular to the x –axis, or parallel to the y-axis then,

α = `\pi/{2}`, `rArr`  tan α =1,  `rArr` slope =dy / dx =1.

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Slope of the tangent  at P(x1,y1) on the curve y = f(x)  is the value of dy/dx at (x1,y1)
Slope of the Normal at P  is perpendicular  to the tangent  and is negative 1/ slope of the tangent.

Slope as a Derivative Examples

Below are the examples on slope as a derivative -

Example 1:
Find the slopes of tangent and normal to the curve y = x4 - 3x2 at (1, -2)
Solution:
Step 1 : Let us write the function y = x4 - 3x2
Step 2 : Then slope is dy/dx = 4x3 - 6x
Step 3 : Slope at (1,-2 ) is 4(1)3 - 6(1) = 4 - 6 = -2  Hence slope of the tangent is m= -2

Step 4 : Slope of the normal = -1/ slope of the tangent = - (-1/2) = 1/2. Hence  slope of the normal = 1/2
Let us find the slope and normal to a function which involves the time element also.

Example 2 :
Find the slope and normal to the curve x = ct and  y = c/t at t
Solution:-
Step 1 : Let us write the equation x = ct

Step 2 : Then the slope  of the function is dx/dt = c
Step 3 :  Let us find dy/dt which is equal to -c/t2
Step 4 :  We have to find dy/dx. Hence here we use chain rule . That is dy/dx = dy/dt ÷ dx/dt = -c/t2 ÷ c = -c/t2 * 1/c
= -1 /t2
Step 5 : Hence slope of the tangent to the curve x =ct , y = c/t is -1/t2
Step 6 : Slope of the normal  at 't'is the reciprocal of the tangent with sign change =  t2
Hence  slope of the normal at 't' is t2
Answer ; Slope of the tangent  at 't' is  -1/t2 and slope pf the normal at 't' = t2

Integrals is the opposite of differential equations.  Hence if slope of an equation is given , then we can find the equation using integrals.
Example 3: Find the equation of the curve whose slope at the point (x,y) is 3x2 + 2, if the curve is passing through the point (1,-1)

Solution:-
Step 1 : We have the slope dy /dx = 3x2 + 2
Step 2 : That gives us dy = (3x2 + 2) dx
Step 3 : Let us integrate both sides ∫dy  = ∫ (3x2+ 2) dx
y  =   3 x3 + 2x + c  = > x3 + 2x + c
3
Step 4 : This passes through (1,-1) Hence we get -1 = (1)3 + 2(1) + c
-1  =  1 +2+c
-1  =   3 + c
That is           c = -4
Step 5 : Let us now write the equation as y = x3 + 2x - 4

Answer :- Equation of the curve whose slope at a point (x,y) is 3x2 + 2 and the point is (1,-1) is y = x3 + 2x - 4
We have used  differential equation  to  find slope of the tangent and slope of the normal to the curve and  we have used integrals  to find the equation of the curve when slope is  given.

Basic Information on Algebra

Introduction for basic information on algebra:

Algebra is the division of mathematics connecting to the study of the regulations of operations and associations, and the organization and ideas occurrence from them, jointly with terms, polynomials, equations and algebraic structures. Collectively with geometry, examination, topology, combinatory, and number theory, algebra is one of the mainly significant divisions of pure mathematics.

Some Basic Information of Algebra:

The following information is very useful for basic algebra.

1. Make a revision of the basic formula with deference to the signs:

(plus +)    +  (plus +)    =   (plus+ )
(minus -)  +  (minus- )  =   (minus -)
(plus +)    *   (plus +)    =   (plus+ )
(minus -)  *   (minus- )  =   (plus +)
(minus -)  *   (plus + )   =   (minus -)
(plus +)    *   (minus- )  =   (minus -)


2. Practicing with the basic formulas:

(x + y) 2 =x2 + y2 + 2xy.
(x - y) 2 =x2 + y2 – 2xy.
x 2 – y 2 = (x + y) (x - y).


3. Attempt the quadratic equation with regard to x and y and solve correspondingly

Example Problems for Basic Information on Algebra:

Example 1 for basic information on algebra:

1) (s + 4) (s – 5)

Solution:

= (s + 4) (s – 5)

= s2 -5s + 4s– 20

= s2 - 1s- 20

Example 2 for basic information on algebra:

2) Solve this algebra expression: 5(a -3) + 4b - 5(a -b -3) + 12

Solution:-

Using Multiply Factors

= 5a - 15 + 4b -5a + 5b + 15 + 12

Groups like terms

= 9b + 12

Example 3 for basic information on algebra:

3) Using algebra equation solve: 3(-5x - 2) - (x - 2) = -2(8x + 4) + 4

Solution:-

Multiplying factors:

-15x – 6 - x + 2 = -16 x - 8 + 4

Group like terms:

-16x – 4 = -16x – 4

Add 16x + 4  to both sides and using the equations as fallows

0 = 0

Example 4 for basic information on algebra:

4) Calculate the value of the following algebra problem: `103^2`

Solution:

Given 1032

= (100+3)2

= 1002+ 2(100) (3) + 32

= 10000 + 600 + 9

= 10609.

Solving Basic Statistics Problems

Introduction to solving basic statistics problems:

Everyday we come across a wide variety of information’s in the form of facts, numerical figures, tables, graphs, etc. These figures are the numerical data collected with a definite purpose, are called data. Data is a word in a plural form of the Latin word datum. Every part of our lives utilizes data in one form or the other. This extraction of significant information is studied in a branch of mathematics called Statistics.

Solving Basic Statistics Problems-statistics Definition

Basically the word ‘statistics’ appears to have been taken from the Latin word ‘status’ meaning ‘a state (related to a political)’. In its origin, statistics is simply the collection of data on different aspects of the life of people, useful to the State. Basic Statistics deals with collection, organization, analysis and Interpretation of data. The word ‘statistics’ has held different meanings in different contexts, so that the calculation process is based on the meaning and data’s.Please express your views of this topic Definite Integrals by commenting on blog.

Solving Basic Statistics Problems
Example for basic stastistics problems:

Consider the marks obtained by 10 pupils in a mathematics test as given below:

55 36 95 73 60 42 25 78 75 62

The data in this form is called raw data.

By looking in this way, can you find the highest and the lowest marks?

Did it take you some time to search for the maximum and minimum marks? Wouldn’t it be a lesser amount of time consuming if these scores were arranged in ascending or descending order? So let us arrange the marks in ascending order as

25 36 42 55 60 62 73 75 78 95

Now, we can clearly watch that the lowest marks are 25 and the highest marks are 95.

The difference of the highest and the lowest marks in the basic stastistics data is called the range of the data. So, the range in this case is 95 – 25 = 70.

Arrangement of data in ascending or descending order can be quite time consuming, particularly when the number of observations in an experiment is large.

Define Rational Function

Introduction to define rational function:

Rational numbers is define as an integer that holds the value which is not equal to 0 and the value form of `x/y` , where x, y is two integers and` y != 0` . Get-together of each and every rational integer(numbers) is denoted by y. A rational integer(numbers) form is said to be proper form, when x and y having no similar factor other than 1. All integers(numbers) are said to be rational numbers. Rational function is referred as f(x). Let us discuss about the rational function in breif.

Define Rational Function:
Rational function is defined as a form that is,

`f(x) = ((p(x))/(q(x)))`

A rational function is generally a division of two polynomial functions mentioned as p(x) and q(x). That is, it is a polynomial divided by an another polynomial. Thus the rational function is defined with the definition and the form of the function is shown. Now let we have some example problems for clear understanding.

Example - Define Rational Function:

Example - Define rational function 1:

Determine the domain for the given rational function f(x) = `(x - 5)/(x^2 + 4x - 12)`

Solution:

Step 1: Given rational function is,

f(x) = `(x - 5)/(x^2 + 4x - 12)`

Step 2: Set the denominator of the given function f(x) = `(x - 5)/(x^2 + 4x - 12)` to zero and solve for x

x2 + 4x - 12 = 0

By rewriting the above equation,

x2 + 6x - 2x - 12 = 0

By separating common terms, we get

x(x + 6) - 2(x + 6) = 0

(x - 2)(x + 6) =0

By factoring each term, we get

x - 2 = 0

x = 2

x + 6 = 0

x = -6

Thus, the domain value of the given function is all real numbers except  for x, x = 2 or - 6

Example - Define rational function 2:

Define the domain of the function f(x) = `(7x)/(18x - 9)`

Solution:

Step 1: Given rational function is,

f(x) = `(7x)/(18x - 9)`

Step 2: Set the denominator of the given function f(x) = `(7x)/(18x - 9)` to zero and solve for x

18x - 9 = 0

By adding 9 on both sides, we get

18x =  9

Divide 18 on both sides, therefore

x = `9/18 `

x = `1/2`

x = 0.5

Therefore, the domain of a given rational function is all real numbers except for x = 0.5.

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