Sent from my GT-S5310I using MyElimu mobile app]]>

Sent from my GT-S5310I using MyElimu mobile app]]>

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where h and k are given in terms of coefficients a , b and c .

Let us start with the quadratic function in general form and complete the square to rewrite it in standard form.

- Given function f(x)

f(x) = ax^{ 2}+ bx + c - factor coefficient a out of the terms in x
^{ 2}and x

f(x) = a ( x^{ 2}+ (b / a) x ) + c - add and subtract (b / 2a)
^{ 2}inside the parentheses

f(x) = a ( x^{ 2}+ (b/a) x + (b/2a)^{ 2}- (b/2a)^{ 2}) + c - Note that

x^{ 2}+ (b/a) x + (b/2a)^{ 2} - can be written as

(x + (b/2a))^{ 2} - We now write f as follows

f(x) = a ( x + (b / 2a) )^{2}- a(b / 2a)^{ 2}+ c - which can be written as

f(x) = a ( x + (b / 2a) )^{2}- (b^{ 2}/ 4a) + c - This is the standard form of a quadratic function with h = - b / 2a k = c - b
^{ 2}/ 4a

where h and k are given in terms of coefficients a , b and c .

Let us start with the quadratic function in general form and complete the square to rewrite it in standard form.

- Given function f(x)

f(x) = ax^{ 2}+ bx + c - factor coefficient a out of the terms in x
^{ 2}and x

f(x) = a ( x^{ 2}+ (b / a) x ) + c - add and subtract (b / 2a)
^{ 2}inside the parentheses

f(x) = a ( x^{ 2}+ (b/a) x + (b/2a)^{ 2}- (b/2a)^{ 2}) + c - Note that

x^{ 2}+ (b/a) x + (b/2a)^{ 2} - can be written as

(x + (b/2a))^{ 2} - We now write f as follows

f(x) = a ( x + (b / 2a) )^{2}- a(b / 2a)^{ 2}+ c - which can be written as

f(x) = a ( x + (b / 2a) )^{2}- (b^{ 2}/ 4a) + c - This is the standard form of a quadratic function with h = - b / 2a k = c - b
^{ 2}/ 4a

Welcome to the newly established Topic, titled how did i get an "A" in mathematics,

Am, Emanuel, a third year students at Ardhi University former UCLAS pursuing bachelor of science in Land Management and Valuation, also among the permanent students of myElimu.

I obtained my Ordinary level and Advanced level Education at Kibasila Secondary School in Dar es Salaam from 2006 to 2012.

Following many questions that i have come accross, have decided to share with you how did i make it to the Top in my class and obtained an "A" on my Basic Mathematics ordinary level.

Will then higlight 6 major points that is think where had greatly contributed to the acheivement.

1. Primarily, had a friend who was good in mathementics and i choose him to be my friend , ask him questions on different matters concerning the topics.

2. I had a group dicscussion of serious people where we were not greater than 7 people discussing together different subjects and when it comes to Math discussion we make sure everyone solves atleast a single question.

3. I had my own library which had compises of past papers, notes from other schools , tuitions and different pamplets on mathemantic and moreover, i had a well kept counter book where i kept class notes that i carefully wrote by my own hands.

4. Everyday i had a tendency to make sure that i have solved atleast 5 questions of mathematics randomly aside form my normal study routine.

5. I had a challenge to follow my own study timetable so i decided to make a rough timetable that in a praticular period i must've covered a certain subject or topic, the same applied during exams period i gave special attention on how i attain my timetable whic then is connected to the exams timetable.

6. Inspite of having satisfactory notes and other materials i never missed class periods listerning to the respective subject teacher, this had made me to score highier even in other exams as in quiz, tests, midterm and terminal examinations.

It is possible to score an "A" in Maths in your national exams and that "A" should start form your desire, you should aim to get it and you shall get it.Never Give up.

I would like to welcome additions if you have any questions, recommendations or anything to make an "A" happen.

With love,

Emanuel,

+255 718 880 607

enjavike@myelimu.com

enjavike@gmail.com

Ardhi University Tanzania

]]>

Welcome to the newly established Topic, titled how did i get an "A" in mathematics,

Am, Emanuel, a third year students at Ardhi University former UCLAS pursuing bachelor of science in Land Management and Valuation, also among the permanent students of myElimu.

I obtained my Ordinary level and Advanced level Education at Kibasila Secondary School in Dar es Salaam from 2006 to 2012.

Following many questions that i have come accross, have decided to share with you how did i make it to the Top in my class and obtained an "A" on my Basic Mathematics ordinary level.

Will then higlight 6 major points that is think where had greatly contributed to the acheivement.

1. Primarily, had a friend who was good in mathementics and i choose him to be my friend , ask him questions on different matters concerning the topics.

2. I had a group dicscussion of serious people where we were not greater than 7 people discussing together different subjects and when it comes to Math discussion we make sure everyone solves atleast a single question.

3. I had my own library which had compises of past papers, notes from other schools , tuitions and different pamplets on mathemantic and moreover, i had a well kept counter book where i kept class notes that i carefully wrote by my own hands.

4. Everyday i had a tendency to make sure that i have solved atleast 5 questions of mathematics randomly aside form my normal study routine.

5. I had a challenge to follow my own study timetable so i decided to make a rough timetable that in a praticular period i must've covered a certain subject or topic, the same applied during exams period i gave special attention on how i attain my timetable whic then is connected to the exams timetable.

6. Inspite of having satisfactory notes and other materials i never missed class periods listerning to the respective subject teacher, this had made me to score highier even in other exams as in quiz, tests, midterm and terminal examinations.

It is possible to score an "A" in Maths in your national exams and that "A" should start form your desire, you should aim to get it and you shall get it.Never Give up.

I would like to welcome additions if you have any questions, recommendations or anything to make an "A" happen.

With love,

Emanuel,

+255 718 880 607

enjavike@myelimu.com

enjavike@gmail.com

Ardhi University Tanzania

]]>

differentiate lntanXsecX

dh ]]>

differentiate lntanXsecX

dh ]]>

- Integers (the whole numbers, plus negative whole numbers)
- Number properties (where we learn about imaginary and real numbers, rational and irrational numbers, prime numbers and reciprocals)
- Order of operations (why 3 + 2 × 4 does not equal 20)
- Powers, roots and radicals (indices)
- Approximate numbers (significant digits)
- Scientific notation (for really large and really small numbers)
- Ratio and proportion (comparing two or more quantities)
- Pi is a special number in mathematics.

Engineering Mathematics Numbers.pdf |
||

File Type: |
| |

Downloaded: | 4 times | |

Size: | 438.55 KB |

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- Integers (the whole numbers, plus negative whole numbers)
- Number properties (where we learn about imaginary and real numbers, rational and irrational numbers, prime numbers and reciprocals)
- Order of operations (why 3 + 2 × 4 does not equal 20)
- Powers, roots and radicals (indices)
- Approximate numbers (significant digits)
- Scientific notation (for really large and really small numbers)
- Ratio and proportion (comparing two or more quantities)
- Pi is a special number in mathematics.

Engineering Mathematics Numbers.pdf |
||

File Type: |
| |

Downloaded: | 4 times | |

Size: | 438.55 KB |

]]>

A sequence is a set of numbers in which the numbers have a prescribed order.

Example of sequence

- 1, 3, 5, 7, 9, 11, 13 …
- 2, 4, 8, 16, 32, 64 …

A series is a sum of all the terms of sequence.

Example of series

- 1+3+5+7+9+11+13+…
- 2+4+8+16+32+64+…

TERMS OF A SEQUENCE/SERIES

These are the numbers that form a sequence or a series and they are denoted by , where ……….

- = the first term
- = the second term
- = the third term
- = the last term or n
^{th}-term or general term

TYPES OF SEQUENCE/SERIES

In this level, we only observe two types of sequence/series namely:-

- Arithmetic Progression (A.P)
- Geometric Progression (G.P)

ARITHMETIC PROGRESSION (A.P)

Arithmetic progression (A.P): where the difference between successive terms is constant/common, and is denoted by (this is a key property of A.P)

The terms of A.P sequence/series are denoted by , where …..

Suppose that are the terms of A.P, then

GEOMETRIC PROGRESSION (G.P)

Geometric Progression (G.P): where the ratio between successive terms is constant/common (this is a key property of G.P)

The terms of G.P sequence/series are denoted by , where …..

Suppose that are the terms of G.P, then

Example:#000000 solid;margin:15px;width:95%;">

Given the series …………………. Say whether it is an AP or GP. |

Solution:

Case 1:

Assume the series …………………. follows A.P, then

Condition for A.P is a common difference ( :

Case 2:

Assume the series …………………. follows G.P, then

Condition for G.P is a common ratio ( :

Observing the two cases above, the series satisfy the A.P condition, hence it is an A.P.

THE GENERAL TERM OF AN ARTHMETIC PROGRESSION (A.P)

The General term of A.P is given by:

PROOF:

Given that: are the terms of A.P, it follows that;

: :

: :

: :

, Hence proved.

Examples:#000000 solid;margin:15px;width:95%;">

Given the series 100+92+84+……………… find the 20^{th} term. |

Solution:

Check if the series follows an A.P or not

For Arithmetic progression (A.P);

In our case:

It follows that:

The series is an A.P, then 20

,

THE GENERAL TERM OF A GEOMETRIC PROGRESSION (G.P)

The general term of G.P is given by:

PROOF:

Suppose that, are the terms of G.P, it follows that

⤇

⤇

⤇

: :

: :

: :

, Hence proved.

Example:#000000 solid;margin:15px;width:95%;">

If are in G.P, find the values of . |

Solution:

………………….. (i)

………….. (ii)

………… (iii)

………. (iv)

Substituting (i) into (iv), we get

From (ii):

From (iii):

.

THE SUM OF N-TERMS OF ARTHIMETIC PROGRESSION (A.P)

Suppose that the series ……………………. is an A.P, then its sum is given by:-

Or

PROOF:

Given an A.P series …………………. . Then its sum ( ) is

…………………….

……… --------------- (1)

.......... + --------------- (2)

Equation (1) and (2) mean the same thing; it is just a matter of arrangement.

Adding equation (1) and (2), simplifies to

………+

OR ,

Example:#000000 solid;margin:15px;width:95%;">

The sum of the eighth term and tenth term of an Arithmetic progression AP is 28 and the sum of the first ten terms is 35; find the first term and the common difference of this A.P |

Solution:

=28

…………………….…….. (i)

…………………………….. (ii)

Solving simultaneously equation (i) and (ii) for and , we get;

From equation (i)

…………………… (iii)

Substituting equation (iii) into (ii), we get;

From equation (iii):

The first term ( ) = and common difference ( ) =

THE ARITHMETIC MEAN

The arithmetic mean of A.P sequence is given by

PROOF:

Suppose that; are sequence terms of A.P and is an arithmetic mean, it follows that:

…………………………….. (Key property of A.P)

, hence proved.

Example:#000000 solid;margin:15px;width:95%;">

Calculate the arithmetic mean of and . |

Solution:

Given two terms of A.P; and

Arithmetic Mean (A.M);

THE SUM OF THE FIRST N-TERMS OF GEOMETRIC PROGRESSION (G.P)

Suppose that the series is a G.P, then its sum is given by:-

or

PROOF:

Given an G.P series . Then its sum ( ) is

. --------------- (1)

Multiply by the common ratio ® on both sides of equation (1), we get;

. ------------- (2)

Now, consider the two cases:

CASE 1:

Subtract equation (1) from (2), we get

, Hence proved.

CASE 2:

Subtract equation (2) from (1), we get

, Hence proved.

NB: The two formulas are nothing but one, they are set two basically for those who get problems when dealing with negative numbers.

Example:#000000 solid;margin:15px;width:95%;">[td]The sum of the first two terms of a geometric progression is 10 and the sum of the first four terms is 40. Given that all terms of the progression are positive, show that.

- The common ratio is
- The sum of the first n terms is 5( ).

[/td]

Solution:

- From the formula: …………. (1)

When,

=10

…………. (i)

When,

= 40

………….. (ii)

Substituting equation (i) into (ii) we get,

………….. (iii)

Given that all terms of the progression are positive, it follows that;

Hence shown.

- Substituting equation (iii) into (i) we get,

……… (iv)

Substituting (iii) and (iv) into (1) we get,

Hence shown

THE GEOMETRIC MEAN (G.M)

The Geometric mean of a three terms sequence of G.P is given by

PROOF:

Suppose that; are sequence terms of G.P and is a geometric mean, it follows that:

……………………….. (Key property of G.P)

, hence proved.

Example:#000000 solid;margin:15px;width:95%;">

The arithmetic mean and geometric mean of two numbers m and n 17 and 15 respectively. Find two numbers. |

Solution:

Arithmetic Mean (A.M)

…………….. (i)

Geometric Mean (G.M)

………………… (ii)

Solving simultaneously equation (i) and (ii) we get,

………………… (iii)

Substitute equation (iii) into (ii), we get

This is a quadratic equation in , where

Or

Or

When ,

When ,

Or

COMPOUND INTEREST

In a simple interest, the principle is taken constant each year, but under a compound interest the amount obtained at the end of a specific year is taken as a principal to its next year.

- …….. Amount of a compound interest
- …….. The compound interest
- ……… n
^{th}year compound interest

= is the amount collected at the end of n-years.

= Principal

= Duration of interest rate (always a year, then )

=number of times that the money can be collected from where invested.

I = the compound interest.

PROOF OF THE FORMULA:

From a simple interest:

………………… (Amount at the end of first year)

Now

Where

………………. (Amount at the end of second year)

Now P =

Where

……………… (Amount at the end of third year)

Observing the sequence ……………., it follows that;

…………….. (Amount at the end of n-years). Hence proved.

………………… (The compound Interest)

The formula , is valid if and only if the principal is compounded annually.

NB: if the amount is collected more than once in a year, then below formula is valid.

- If compounded semi-annually, then .
- If compounded quarterly , then

On the birth of a child. What sum of money should a family invest at 9% per annum, compounded semi-annually, to provide the child with Tsh 1,000,000/= on his sixteenth birth day? |

……………….. (1)

Apply log on both sides of equation (1), we get

REVIEW QUESTIONS:

- Determine the sum of the first eight terms of the series 1 – ½ + ½ - ⅛ + . . .

- The sum of a number of consecutive terms of an arithmetic progression is –19.5, the first term is 16.5 and the common difference is –3. Find the number of terms of the progression.

- In a geometric progression, the sum of the second and the third terms is 6 and the sum of the third and fourth terms is –12. Find the first term and the common ratio of the progression.
- The arithmetic mean and geometric of two numbers a and b are 17 and 15 respectively. Find the value of a and b.

- Prove that the series is an arithmetic progression whose sum to n terms is
- Find three numbers in geometric progression such that their sum is 39 and their product is 729.
- The nth term of a certain sequence is Find the sum of the first five terms of the corresponding series.

- John borrowed £1200 from Zachary with agreement of 5% compounded quarterly, what amount of money Zachary demand from John after 4 years.

- Find the amount A to which the principal P amounts at R% interest compounded annually for n years is given by the formula An = P(1 + 0.01R)
^{n}. If P = 126, R = 4 and n = 8

- The third, the fourth and eighth terms of an arithmetic progression forms the first three consecutive terms of a geometric progression. If the sum of the first ten terms of the A.P is 85. Calculate:

(i) The first terms of both A.P and G.P

(ii) The common ratio

(iii) The sum of the first four terms of the G.P

]]>

A sequence is a set of numbers in which the numbers have a prescribed order.

Example of sequence

- 1, 3, 5, 7, 9, 11, 13 …
- 2, 4, 8, 16, 32, 64 …

A series is a sum of all the terms of sequence.

Example of series

- 1+3+5+7+9+11+13+…
- 2+4+8+16+32+64+…

TERMS OF A SEQUENCE/SERIES

These are the numbers that form a sequence or a series and they are denoted by , where ……….

- = the first term
- = the second term
- = the third term
- = the last term or n
^{th}-term or general term

TYPES OF SEQUENCE/SERIES

In this level, we only observe two types of sequence/series namely:-

- Arithmetic Progression (A.P)
- Geometric Progression (G.P)

ARITHMETIC PROGRESSION (A.P)

Arithmetic progression (A.P): where the difference between successive terms is constant/common, and is denoted by (this is a key property of A.P)

The terms of A.P sequence/series are denoted by , where …..

Suppose that are the terms of A.P, then

GEOMETRIC PROGRESSION (G.P)

Geometric Progression (G.P): where the ratio between successive terms is constant/common (this is a key property of G.P)

The terms of G.P sequence/series are denoted by , where …..

Suppose that are the terms of G.P, then

Example:#000000 solid;margin:15px;width:95%;">

Given the series …………………. Say whether it is an AP or GP. |

Solution:

Case 1:

Assume the series …………………. follows A.P, then

Condition for A.P is a common difference ( :

Case 2:

Assume the series …………………. follows G.P, then

Condition for G.P is a common ratio ( :

Observing the two cases above, the series satisfy the A.P condition, hence it is an A.P.

THE GENERAL TERM OF AN ARTHMETIC PROGRESSION (A.P)

The General term of A.P is given by:

PROOF:

Given that: are the terms of A.P, it follows that;

: :

: :

: :

, Hence proved.

Examples:#000000 solid;margin:15px;width:95%;">

Given the series 100+92+84+……………… find the 20^{th} term. |

Solution:

Check if the series follows an A.P or not

For Arithmetic progression (A.P);

In our case:

It follows that:

The series is an A.P, then 20

,

THE GENERAL TERM OF A GEOMETRIC PROGRESSION (G.P)

The general term of G.P is given by:

PROOF:

Suppose that, are the terms of G.P, it follows that

⤇

⤇

⤇

: :

: :

: :

, Hence proved.

Example:#000000 solid;margin:15px;width:95%;">

If are in G.P, find the values of . |

Solution:

………………….. (i)

………….. (ii)

………… (iii)

………. (iv)

Substituting (i) into (iv), we get

From (ii):

From (iii):

.

THE SUM OF N-TERMS OF ARTHIMETIC PROGRESSION (A.P)

Suppose that the series ……………………. is an A.P, then its sum is given by:-

Or

PROOF:

Given an A.P series …………………. . Then its sum ( ) is

…………………….

……… --------------- (1)

.......... + --------------- (2)

Equation (1) and (2) mean the same thing; it is just a matter of arrangement.

Adding equation (1) and (2), simplifies to

………+

OR ,

Example:#000000 solid;margin:15px;width:95%;">

The sum of the eighth term and tenth term of an Arithmetic progression AP is 28 and the sum of the first ten terms is 35; find the first term and the common difference of this A.P |

Solution:

=28

…………………….…….. (i)

…………………………….. (ii)

Solving simultaneously equation (i) and (ii) for and , we get;

From equation (i)

…………………… (iii)

Substituting equation (iii) into (ii), we get;

From equation (iii):

The first term ( ) = and common difference ( ) =

THE ARITHMETIC MEAN

The arithmetic mean of A.P sequence is given by

PROOF:

Suppose that; are sequence terms of A.P and is an arithmetic mean, it follows that:

…………………………….. (Key property of A.P)

, hence proved.

Example:#000000 solid;margin:15px;width:95%;">

Calculate the arithmetic mean of and . |

Solution:

Given two terms of A.P; and

Arithmetic Mean (A.M);

THE SUM OF THE FIRST N-TERMS OF GEOMETRIC PROGRESSION (G.P)

Suppose that the series is a G.P, then its sum is given by:-

or

PROOF:

Given an G.P series . Then its sum ( ) is

. --------------- (1)

Multiply by the common ratio ® on both sides of equation (1), we get;

. ------------- (2)

Now, consider the two cases:

CASE 1:

Subtract equation (1) from (2), we get

, Hence proved.

CASE 2:

Subtract equation (2) from (1), we get

, Hence proved.

NB: The two formulas are nothing but one, they are set two basically for those who get problems when dealing with negative numbers.

Example:#000000 solid;margin:15px;width:95%;">[td]The sum of the first two terms of a geometric progression is 10 and the sum of the first four terms is 40. Given that all terms of the progression are positive, show that.

- The common ratio is
- The sum of the first n terms is 5( ).

[/td]

Solution:

- From the formula: …………. (1)

When,

=10

…………. (i)

When,

= 40

………….. (ii)

Substituting equation (i) into (ii) we get,

………….. (iii)

Given that all terms of the progression are positive, it follows that;

Hence shown.

- Substituting equation (iii) into (i) we get,

……… (iv)

Substituting (iii) and (iv) into (1) we get,

Hence shown

THE GEOMETRIC MEAN (G.M)

The Geometric mean of a three terms sequence of G.P is given by

PROOF:

Suppose that; are sequence terms of G.P and is a geometric mean, it follows that:

……………………….. (Key property of G.P)

, hence proved.

Example:#000000 solid;margin:15px;width:95%;">

The arithmetic mean and geometric mean of two numbers m and n 17 and 15 respectively. Find two numbers. |

Solution:

Arithmetic Mean (A.M)

…………….. (i)

Geometric Mean (G.M)

………………… (ii)

Solving simultaneously equation (i) and (ii) we get,

………………… (iii)

Substitute equation (iii) into (ii), we get

This is a quadratic equation in , where

Or

Or

When ,

When ,

Or

COMPOUND INTEREST

In a simple interest, the principle is taken constant each year, but under a compound interest the amount obtained at the end of a specific year is taken as a principal to its next year.

- …….. Amount of a compound interest
- …….. The compound interest
- ……… n
^{th}year compound interest

= is the amount collected at the end of n-years.

= Principal

= Duration of interest rate (always a year, then )

=number of times that the money can be collected from where invested.

I = the compound interest.

PROOF OF THE FORMULA:

From a simple interest:

………………… (Amount at the end of first year)

Now

Where

………………. (Amount at the end of second year)

Now P =

Where

……………… (Amount at the end of third year)

Observing the sequence ……………., it follows that;

…………….. (Amount at the end of n-years). Hence proved.

………………… (The compound Interest)

The formula , is valid if and only if the principal is compounded annually.

NB: if the amount is collected more than once in a year, then below formula is valid.

- If compounded semi-annually, then .
- If compounded quarterly , then

On the birth of a child. What sum of money should a family invest at 9% per annum, compounded semi-annually, to provide the child with Tsh 1,000,000/= on his sixteenth birth day? |

……………….. (1)

Apply log on both sides of equation (1), we get

REVIEW QUESTIONS:

- Determine the sum of the first eight terms of the series 1 – ½ + ½ - ⅛ + . . .

- The sum of a number of consecutive terms of an arithmetic progression is –19.5, the first term is 16.5 and the common difference is –3. Find the number of terms of the progression.

- In a geometric progression, the sum of the second and the third terms is 6 and the sum of the third and fourth terms is –12. Find the first term and the common ratio of the progression.
- The arithmetic mean and geometric of two numbers a and b are 17 and 15 respectively. Find the value of a and b.

- Prove that the series is an arithmetic progression whose sum to n terms is
- Find three numbers in geometric progression such that their sum is 39 and their product is 729.
- The nth term of a certain sequence is Find the sum of the first five terms of the corresponding series.

- John borrowed £1200 from Zachary with agreement of 5% compounded quarterly, what amount of money Zachary demand from John after 4 years.

- Find the amount A to which the principal P amounts at R% interest compounded annually for n years is given by the formula An = P(1 + 0.01R)
^{n}. If P = 126, R = 4 and n = 8

- The third, the fourth and eighth terms of an arithmetic progression forms the first three consecutive terms of a geometric progression. If the sum of the first ten terms of the A.P is 85. Calculate:

(i) The first terms of both A.P and G.P

(ii) The common ratio

(iii) The sum of the first four terms of the G.P

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