Math core term 3 project

sequences , limits and

differentiations

Done by:

-Fatima Salem -Safeya Mohammed

Section: 12.52

What are sequences? A sequence is a list of numbers. Any time you write numbers in a list

format, you are creating a sequence.

Something as simple as 1, 2, 3, 4, 5, 6, . . is a sequence.

Rather than just listing the numbers, we usually identify it as a sequence with the notation an = 1, 2, 3, 4, 5, 6, . . .

Usually there is some type of pattern to a sequence.

In the sequence above, you are adding one to each term to get the next term.

Terms can be identified by their location. We note the 1st term in a sequence as a1 and we would call the 5th term in the sequence a5.

We described the pattern in the sequence as adding one to each term to get the next term. We can express this as a recursive formula by writing an = an-1 + 1

Types of sequences:

There are two types of sequences:

- Arithmetic Sequences

- Geometric Sequences

An arithmetic sequence : any time you are adding the same number to each term to complete the sequence, it is called an arithmetic sequence

A geometric sequence: When you multiply every term by the same number to get the next term in the sequence, you have a geometric sequence

Task 1Ahmed,who is a grade 09 student, is planning to save money while he is

studying at ATHS. Ahmed has 5,000 Dhs to start with, and he is planning to save 500 Dhs per month.

a) What is the type of the sequence at which the money will grow?

It is an arithmetic sequence.

b)How much money will he save after 4 years?

12 months in a year => 12(4) = 48 months

an = a1 +d(n-1)

a48 = 5000 +500(48-1)

a48 = 28500 AED

c)His brother has borrowed from him some money, his brother agreed to pay 50 Dhs in the first month and 25 Dhs more in each of the following months; how much did Ahmed’s brother borrow knowing that it took him 1 year to pay his debt?

an = a1 +d(n-1)

a12 = 50 +25(12-1)

a12 = 325 AED

Ahmed’s brother borrowed 325 AED.

d) Let the amount saved at the end of the first year be and the amount that was saved at the second year, and the amount that was saved at the end of the fourth year, find a relation between ?

 

- The difference between the first year and the second year is 6000 AED

- The difference between the second year and the Forth year is 12000 AED

- Each year, the amount saved is increasing by 6000AED,

a+6000=b

b+6000+6000=c

First year(a)=a12 = a1 +d(12-1)a12 = 5000 +500(12-1)a12 = 10500 AED

Second year(b)=a24 = a1 +d(12-1)a24 = 5000 +500(24-1)a24 = 16500 AED

Fourth year (c) = a48= 28500 AED

Task 2

Ahmed wanted to invest his savings; he bought shares in a company, the company will offer an interest rate of 8% annually.

a) How much money will be in the account in 4 years?

b) After 4 years the shares were decreased by 20% each year, after how many years Ahmed will lose all of the profits?

1st year = 36201.75(0.8) = 28961.4 

2nd year = 28961.4(0.6) = 17376.84

3rd year = 17376.84 (0.4) = 6950.736

4th year = 6950.736(0.2) = 1390.1472 

5th year = 1390.1472(0.0) = Zero

He will lose all of his profits in 5 years.

Task 31) Ahmed has started his own company. He plans to hire 8 new

employees. He wants to hire males and females, knowing that there are an equal number of males and females applying for the job, use the binomial theorem to find the all combinations of males and females that could be hired.

Total :8, (m+f)8

8C0 m8 = (1)m8 = m8

8C1 m7f1 = (8)m7 f1 =8m7f

8C2 m6f2 = (28)m6 f2 =28m6f2

8C3 m5f3 = (56)m5 f3 =56m5f3

8C4 m4f4 = (70)m4 f4 =70m4f4

8C5 m3f5 = (56)m3 f5 =56m3f5

8C6 m2f6 = (28)m2 f6 =28m2f6

8C7 m1f7 = (8)m1 f7 =8mf7

8C8 f8 = (1) f8 =f8

2) Ahmed established a phone chain in which every staff member calls two other staff member to notify them about company events, the first round of calls begins with Ahmed calling two members. If there are 94 total staff members, how many rounds of calls are there to pass the information to all employees?

an = a1 rn-1

94= 1(2)n-1

94=2n-1 we have to plug in till we reach 94.

When n=7 the answer will be 64

When n=8 the answer will be 128

 

So, there will be 8 rounds.

3) Those employees will be seated around tables attached end-to-end for an event, How many tables are required to seat all the employees?

a1=4

a2=6

a3=8

d=2

 

Between 8 to 9 tables are required to seat all the 94 employees.

Derivatives

The derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's instantaneous velocity.

Applications of derivatives

• Rates of Change

• Critical Points

• Minimum and Maximum Values

• Finding Absolute Extrema

Task 4 The revenue of his company was modeled by ; where is the number

of items sold.

a) Write a Java program that gives the value of to fill the table below:

c) Let the cost to produce items is , find the profit when 20 items are sold.