Elementary Calculus Problems

Introduction to elementary calculus problems:

Elementary calculus problems deal with solving basic problems in calculus which help to explain the elementary calculus clearly. Calculus is the defined as the process of finding the rate of change of the given function with respect to the change in the given function. In order to find the rate of change, calculus is divided into differential calculus and integral calculus. The following are the example problems in elementary calculus.Please express your views of this topic Mean Value Theorem Example by commenting on blog.

Elementary Calculus Example Problems:

Example 1:

Solve the elementary function by differentiation.

f(n) = 2n 3 – 5n 6  + 7n

Solution:

The given function is

f(n) = 2n 3 – 5n 6  + 7n

Perform differentiation operation for the given function

f '(n) = 2(3 n 2) – 5(6n 5 ) + 7

Solving the above terms we get,

f '(n) = 6n 2 – 30n 5 + 7   is the answer.

Example 2:

Solve the elementary function by differentiation.

f(n) = 4n2 – 7n 3 – 4n 6  + 5

Solution:

The given function is

f(n) = 4n2 – 7n 3 – 4n 6  + 5

Perform differentiation operation for the given function

f '(n) = 4(2n) – 7(3n 2 ) – 4( 6n 5) + 0

Solving the above terms we get,

f '(n) = 8n – 21n 2 – 24n 5  is the answer.

Example 3:

Solve the elementary function by integration.

`int ` f(n) dn = 4+4n2+ 6n3 + 5n4 dn

Solution:

The given function is

`int ` f(n) dn = 4+4n2+ 6n3+ 5n4 dn

`int ` f(n) dn = `int ` (4+4n2+ 6n3 + 5n4) dn

`int ` f(n) dn = `int ` 4dn + `int ` 4n2 dn+ `int ` 6n3 dn + `int ` 5n4 dn

Perform integration operation for the given function

We get

`F(n) = (4n) + (4n^3)/3 + (6n^4)/4 + (5n^5)/5`

Solving the above function we get

`F(n) = 4n+ (4n^3)/3 + (3n^4)/2 + n^5` is the answer.

Example 4:

Solve the elementary function by integration.

`int ` f(n) dn = n5+ 3n2+ 4n dn

Solution:

The given function is

`int ` f(n) dn = n5+3n2+4n dn

`int ` f(n) dn  = `int ` (n5 + 3n2+ 4n) dn

`int ` f(n) dn = `int ` n5 dn + `int ` 3n2 dn + `int ` 4n dn

Perform integration operation for the given function

We get

`F(n) = (n^6)/6 + (3n^3)/3 + (4n^2) /2`

Solving the above function we get

`F(n) =(n^6)/6+ n^3 +2n^2` is the answer.Is this topic how to solve math problems hard for you? Watch out for my coming posts.

Elementary Calculus Practice Problems:

1) Solve the elementary function by integration.

`int ` f(n) = 4n2+ 6n + 5 dn

Answer: `F(n) = (4n^3)/3 + 3n^2 + 5n`

2) Solve the elementary function by differentiation.

f(n) = 13n +5n 3 – 7n 4

Answer: f '(n) = 13 + 15n 2 – 28 n 3

Elementary Math Tutorial

Introduction to elementary math tutorial:

The elementary math work consists of mathematics topics taught in the primary and secondary schools. The most important in elementary mathematics are arithmetic, geometry, algebra, number work and special functions. In secondary school the topics are trigonometry, Calculus etc. In elementary math tutorial, the basic operations and problems are clearly taught by the tutors with the examples solved in steps. Now we are going to see about elementary math tutorial.

About Elementary Math Tutorial:

Now we see the elementary mathematics with the help of the tutorial problems.

1. Number system:

The number system consists of the natural, decimal, fractional, rational numbers etc. Let us see one example for the rational number in number system.

Tutorial Example:

Find the fraction for the given decimal number 0.66

Solution:

The given decimal can be simplified as follows,

0.66 = 6 tenths + 6 hundredths

= 6 × `(1/10)` + 6 × `(1/100)`

= `(6/10)` + `(6/100)`

= `66/100`

= `33/50` .

2. Arithmetic:

In arithmetic, we can see about the housing finance, speed, time, distance, ratio etc.

Tutorial example:

Find the ratio of 5 Kg to 850 g

Solution:

The ratio can be determined and explained below

5 Kg = 5 × 1000 = 5000 g

The ratio can be given as

Required ratio = 5000: 850

= 500: 85

= 100: 17

Thus, the ratio is given as 100: 17

Other Elementary Math Tutorials:

1. Algebra:

Algebra is the part of mathematics where it consists of many expressions such as variables, constants and coefficients. Now we see an example

Tutorial example:

Determine and expand the given expression: (2a + 2b + 2c)2

Solution:

Let us take the equation with ( x + y + z)2

x = 2a, y = 2b, z = 2c

(2a + 2b + 2c)2 = (2a)2 + (2b)2 + (2c)2 + 2(4ab + 4ac + 4bc)

= 4a2 + 4b2 + 4c2 + 8ab + 8ac + 8bc

2. Geometry:

Geometry is about the learning of various size, shape and properties of the geometrical solid figures.

Tutorial example:

Find the diameter of the circle where the circumference of circle is about 8 cm.

Solution:

The circumference of the circle determined as follows,

C = π × d

8 = 3.14 × d

d = `8/3.14`

d = 3.18cm.

The diameter of the circle is determined as 2.54 cm.

Ratio Problems with Solutions

Introduction to ratio problems with solutions.
Two values can be compared by (i) Subtraction (ii) Division.

If the weights of two quantities A and B are given by 8 kg and 32kg, then the weight of B is 32 – 8 = 24 kg more than the weight of A.

Also `32 / 8` = 4. The weight of B is in 4 times the weight of A.

Therefore, ratio is a relationship between two quantities which expresses how many times one quantity is of the other.

The ratio of two quantities x and y can be expressed as follows:

(i) as a fraction `x / y` .

(ii) as a division x ÷ y

(iii) also as x : y

The ratio can be obtained by dividing the first quantity by the second quantity.

Now let us see few problems on this topic ratio problems with solutions.

Example Problems on Ratio Problems with Solutions.

Ex 1: A plot is 100 m long and 20 m wide. Find the ratio between the following:

(i) Wide and length

(ii) Length and perimeter

Soln: Here wide = 40m, Length = 100 m

Therefore perimeter = 2 (wide + length)

= 2 (40 + 100) = 280 m.

(i) The ratio of wide and length = `40 / 100` = `2 / 5`

Therefore the ratio is 2 : 5.

(ii) The ratio of length and perimeter = `100 / 280` = `5 / 14` .

Therefore the ratio is 5 : 14.

Ex 2: Paul secured 45 marks out of 50 in English and 25 out of 30 in French. In which subject did he perform better?

Soln: Let us find the ratios of the marks.

In English, it is `45 / 50` = `9 / 10` `=>` 9 : 10

In French, it is `25 / 30` = `5 / 6` `=>` 5 : 6.

Hence Paul performed well in English as 9 : 10 is greater than 5 : 6

I like to share this How to do Ratios with you all through my article.

More Example Problems on Ratio Problems with Solutions.

Ex 3: Divide 1250 dollars between two persons in the ratio 12 : 13.

Soln: Here the total ratio is 12 + 12 = 25.

Therefore the value of 1 ratio = `1250 / 25` = 50

Therefore two persons will get 12 * 50 = 600 dollars and 13 * 50 = 650 dollars.

Ex 4: Two numbers are in the ratio 2 : 7. If the second number is 378, find the first number.

Soln: Given: `(First number) / 378` = `2 / 7` `=>` First number = `2 / 7 xx 378` = 108

Therefore the first number is 108.

Real Valued Function

Introduction to real valued function:

A function whose range is within the real numbers be assumed to be a real function, moreover called a real-valued function. During math, a real-valued function is a function to associates near each part of the domain a real number within the image. f be a function as of set A toward a set B but all element x within A be able to be related through a unique element within B. It can be written as, `f:A->B`

Examples of Real Valued Functions:

A real valued function f : A to B or just a real function 'f ' be a rule which associates near all feasible real number xε A, a unique real number f(x)εB, while A with B are subsets of R, the set of real numbers.

In further words, functions whose domain with co-domain be subsets of R, the place of real numbers, be call real valued functions.Is this topic Properties of Real Numbers hard for you? Watch out for my coming posts.

Problem 1 for real valued function

Solve the field of the real valued square root function f known .

f(x) = sqrt(4x)

Solution :

The known function be as follow

f(x) = sqrt(4x)

This collected square root function. The domain be establish through Finding  the inequality

4x > = 0.

The answer set for the exceeding dissimilarity be the domain along with is specified through the interval

(0 , +infinity)

Problem 2 for Real Valued Function :
The known function is a rational

f(x) = (x - 1) / (x - 7)

Its field is the put of all real numbers except for individual’s value of x to create the denominator zero. Therefore the field is known through the interval

(-infinity , 7) U (7 , +infinity)

Comparing Fractions Answers

Introduction for Comparing Fractions Answers:

A fraction (from the Latin fractus, broken) is a number that can represent part of a whole. The earliest fractions were reciprocals of integers: ancient symbols representing one part of two, one part of three, one part of four, and so on. A much later development were the common or "vulgar" fractions which are still used today (½, ?, ¾, etc.) and which consist of a numerator and a denominator.

Source – Wikipedia.

Comparing Fractions Answers - which are Smaller:

The examples show the fractions compare the smaller.

Comparing  fractions which are smaller: `1/7` or `2/9` ?

Solution:

The LCD for the denominators 7, 9 is 63.

Multiply the numerator `(1*9)/63` and `(2*7)/63` .

The solution for the fractions is `9/63` and `14/63`

When comparing the answers `1/7` is smaller.



Comparing the fractions which are smaller: `4/3` or `12/7` ?

Solution:

The LCD for the denominators 3, 7 is 21.

Multiply the numerator `(4*7)/21` and `(3*12)/21` .

The solution for the fractions is `28/21` and `36/21`

When comparing the answers `4/3 ` is smaller.

Between, if you have problem on these topics how to solve proportions, please browse expert math related websites for more help on how to find the volume of a cube.

Comparing the fractions which are smaller: `17/9` or `21/8` ?

Solution:

Convert the given fractions in to decimal form.

Divide the fractions as (17÷9) and (21÷8).

The solution for the fractions is `17/9` = 1.88 and `21/8` = 2.62

When comparing the answers `17/9` is smaller.

Comparing Fractions Answers - which are Greater:

The examples show the fractions compare the greater.

Comparing the fractions which are greater: `13/7` or `5/3` ?

Solution:

The LCD for the denominators 7, 3 is 21.

Multiply the numerator `(13*3)/21` and `(7*5)/21` .

The solution for the fractions is `39/21` and `35/21`

When comparing the answers `13/7` is greater.

Comparing the fractions which are greater: `9/3` or `12/5` ?

Solution:

The LCD for the denominators 3, 5 is 15.

Multiply the numerator `(9*5)/15 ` and `(3*12)/15` .

The solution for the fractions is `45/15` and `36/15`

When comparing the answers `9/3` is greater.

Comparing the fractions which are greater: `23/8` or `33/6` ?

Solution:

Convert the given fractions in to decimal form.

Divide the fractions as (23÷8) and (33÷6).

The solution for the fractions is `23/8` = 2.875 and `33/6` = 5.5

When comparing the answers `33/6 ` is greater.

Definition of Mathematical Function

Introduction to definition of mathematical function:

A mathematical function is equations that assigns  only answer to every possible query. A variable indicates to unknown value in the function's equation. The function's key variable is generally denoted by an x. This is the self-determining variable to the purpose. The function's output is a variable generally denoted by y. Let us see about the topic definition of mathematical function are given some example with solution are shown below content.

Example Problem for Definition of Mathematical Function

Example problem 1:-Using definition of mathematical function
To calculate the equation x =16y+1

Solution:

Step: 1 x is a function of y, because of every value of y; there is only one value of x.

Step: 2 If we substitute y=4, we get x=65 and no other value.

Step: 3 The values of x only depend on the values selected for y.

Step: 4 So from this we obtain, independent variable is y and needy variable is x.

Example Problem 2:-using Definition of Mathematical Function

To calculate the equation A =9B+4

Solution:

Step: 1 A is a function of B, because of every value of B; there is only one value of A.

Step: 2 If we substitute B=2, we get A=22 and no other value.

Step: 3 The values of A only depend on the values chosen for B.

Step: 4 So from this we obtain, independent variable is B and dependent variable is A.

Between, if you have problem on these topics functions and linear equations and inequalities, please browse expert math related websites for more help on math answers to problems.

Example Problem 3:-using Definition of Mathematical Function

To calculate the given equation 4x2+6x+4

Solution:

Step 1: Given 4x2+6x+4

Step 2: It is a different example and it can be write in a function like,

Step 3: f(x)= 4x2 + 6x + 4

Step 4: Function is nothing but just substitution.

Step 5: Simply, f(x) =f (a) when x=a

I.e if x=2 means like

y=4(22) +6(3) +4

f (2)=4(4)+18+4

=16+18+4

=38.

So when x=2, the value of the function f(x) =38.

Non Negative Real Numbers

Introduction to non negative real numbers:Let us study about non negative real numbers. Real numbers are commonly defined as the combined form rational and irrational numbers.
Smallest numbers in the ratio form as `a/b` , where ‘b’ should not be equal to ‘0’ at any situation are termed to be as rational numbers.
Numbers that cannot be represented in the ratio forms are termed to be as irrational numbers. Examples for non negative real numbers are below.

Non Negative Real Numbers:

Example 1:

Joseph is having 3 fruits, `1/3` chocolates and` 2/7` nuts, Jennifer is having `1/3` fruits, 3 chocolates and `3/5 ` nuts and Nimmi is having `2/5` fruits, `2/3` chocolates and 2 nuts. Find the total number of fruits, chocolates and nuts?

Solution:

Given:
Joseph having 3 fruits, `1/3` chocolates and `2/7` nuts

Jennifer is having `2/3` fruits, 3 chocolates and `3/5` nuts

Nimmi is having `1/3` fruits, `2/3` chocolates and 2 nuts

To calculate the total number of fruits, chocolates and nuts follow the steps as below:
Fruits:
= 3 + `2/3` + `1/3` (add all the fruit numbers from three of them)

= 3 `(3/3)` + `2/3` + `1/3 ` (multiply and divide ‘3’ with the number 3 in order to make it into fraction)

=` 9/3` + `2/3` + `1/3`

= `(9 + 2 + 1)/3` (take the number ‘3’ as common divisor)

= `12/3`

= 4 (total non negative real numbers of fruits)

Chocolates:
= `1/3 ` + 3 + `2/3`

= `1/3` + 3 `(3/3)` + `2/3` (multiply and divide ‘3’ with the number 3 in order to make it into fraction)

= `1/3 ` + `9/3` + `2/3`

= `(1 + 9 + 2)/3` (take the number ‘3’ as common divisor)

= `12/3`

= 4 (total non negative real numbers of chocolates)

Nuts:
= `2/7 ` + `3/5` + 2

= `(2/7)` `(5/5)` + `(3/5)` ` (7/7)` + 2` (35/35)` (multiply and divide the number ‘5’ with `2/7` , ‘7’ with `3/5` and ‘35’ with 2 in order to get the common divisor as ‘35’ for evaluation)

= `10/35 ` + `21/35` + `70/35`

= `(10 + 21 + 70)/35` (take the number ’35’ as common divisor)

= `101/35` (total non negative real numbers of nuts)

Having problem with One to One Correspondence keep reading my upcoming posts, i will try to help you.

Example 2:

If sita is completing 7 puzzle squares per hour then how hour she will take to complete 119 squares?

Solution:

7 puzzles 1 hour
Then to calculate the total hours required to complete 119 squares are as follows:
=`119/7`

= 17 hours (total non negative real numbers hours to complete 119 squares)


Exercises:

Rani is having 3 meters of fabrics, Srikanth is having `1/2 ` meters of fabrics and Mani is having `1/5` meters of fabrics. Find the total meters of fabrics they have with them? (Answer: `37/10 ` meters)
If Meena is having `2/5` liters of coke and Nakul is having `3/5` liters of coke. Find the total liters of coke they have with them? (Answer:1 liter)