USAD Math

General Mathematics, Geometry

Introduction to Differential Calculus

General Mathematics

Simple and Compound Interest

Permutation and Combination

Probability

Geometry

Right Triangle

Coordinate Geometry

Plane and Solid Figures

Circles

Differential Calculus

Polynomial

Limits and Continuity

Derivative

General Mathematics

Interests

Single Interest

The principal balance do not change

Single Interest

Formula

A=P(1+rt)
A

Final Value

P

Principal balance

Rate

r
t

Month, Year

Example

Borrow $500 from the bank with simple interest 5% per week

How much do you need to pay back after a month?

Example

Borrow $500 from the bank with simple interest 5% per week

How much do you need to pay back after a month?

500 + 4 (500\times0.05) = 600

Compound Interest

The amount of interest earned would be added

in the principal amount of money

For example

 

For example

 

You save $10000 in a bank with a rate 1%

compounded annually

How much money would you get after 10 years?

For example

 

You save $10000 in a bank with a rate 1% compounded annually

How much money would you get after 10 years?

Solution

10000(1 + 1 \%) = 10100 \rightarrow 10100(1+1\%) = 10201

do the upper calculation 10 times

10000(1+1\%)^{10} = 11046

Compound Interest

Formula

A=P(1+\frac{r}{n})^{nt}
A

Final Value

P

Principal balance

Rate

(Monthly, Annual)

r
t

Month, Year 

 

number of

interests applied in a time period

n

For example

 

You save $10000 in a bank with a rate 1%

compounded monthly

How much money would you get after 10 years?

For example

 

You save $10000 in a bank with a rate 1% compounded monthly

How much money would you get after 10 years?

Solution

10000(1 + \frac{1 \%}{12}) = 10008 \rightarrow 10100(1+\frac{1\%}{12}) = 10016

do the upper calculation 10 times

10000(1+\frac{1\%}{12})^{10\times12} = 11051

Compound Continuously

The value compound continuously

Compound Continuously

The value compound continuously

\displaystyle{A = P\lim_{n\to\infty}(1+\frac{r}{n})^{nt}}
\displaystyle{A = P\lim_{n\to\infty}(1+\frac{r}{n})^{nt}}
\displaystyle{A = P\lim_{n\to\infty}(1+\frac{r}{n})^{nt}}
\displaystyle{\lim_{n\to\infty}(1+\frac{1}{n})^{n}}

Type 

in your calculator

\displaystyle{A = P\lim_{n\to\infty}(1+\frac{r}{n})^{nt}}
\displaystyle{\lim_{n\to\infty}(1+\frac{1}{n})^{n}}

Type 

in your calculator

We will get a value

2.718...
\displaystyle{A = P\lim_{n\to\infty}(1+\frac{r}{n})^{nt}}
\displaystyle{\lim_{n\to\infty}(1+\frac{1}{n})^{n}}

Type 

in your calculator

We will get a value

2.718...

This number, we use a symbol to represent it

e
\displaystyle{A = P\lim_{n\to\infty}(1+\frac{1}{n})^{nt}}

We rewrite this formula

A = Pe^{rt}

For example

 

You save $10000 in a bank with a rate 1%

compounded continuously

How much money would you get after 10 years?

For example

 

You save $10000 in a bank with a rate 1% compounded continously

How much money would you get after 10 years?

Solution

10000e^{0.01\times10} = 11051

Permuation & Combination

What does these two word means?

What does these two word means?

Four people are waiting in a row, how many possible ways can they form?

We want to choose three people from a class

to participate in a contest

How many ways can we choose?

What does these two word means?

Four people are waiting in a row, how many possible ways can they form?

We want to choose three people from a class of ten

to participate in a contest

How many ways can we choose?

P
C

Four people are waiting in a row, how many possible ways can they form?

ABCD BACD CABD DABC
ABDC BADC CADB DACB
ACBD BCAD CBAD DBAC
ACDB BCDA CBDA DBCA
ADBC BDAC CDAB DCAB
ADCB BDCA CDBA DCBA

Four people are waiting in a row, how many possible ways can they form?

ABCD BACD CABD DABC
ABDC BADC CADB DACB
ACBD BCAD CBAD DBAC
ACDB BCDA CBDA DBCA
ADBC BDAC CDAB DCAB
ADCB BDCA CDBA DCBA

Four people are waiting in a row, how many possible ways can they form?

24

Four people are waiting in a row, how many possible ways can they form?

A
B
C
D
B
C
D
C
D
D
4\times3\times2\times1
24

n people are waiting in a row, how many possible ways can they form?

n people are waiting in a row, how many possible ways can they form?

n\times(n-1)\times(n-2)\times...\times1

n people are waiting in a row, how many possible ways can they form?

n\times(n-1)\times(n-2)\times...\times1
n!

We define ! as the symbol of the calculation

We call it "Factorial"

And define

0! = 1

There are 6 people and 3 seats

How many ways can they form?

There are 6 people and 3 seats

How many ways can they form?

6\times5\times4

There are 6 people and 3 seats

How many ways can they form?

6\times5\times4
\displaystyle{P^{6}_{3}}

There are n people and r seats

How many ways can they form?

\displaystyle{P^{n}_{r}}
\displaystyle{P^{n}_{r}=\frac{n!}{(n-r)!}}

Choose r people from a group of n people

to permute

What does these two word means?

Four people are waiting in a row, how many possible ways can they form?

We want to choose three people from a class of ten

to participate in a contest

How many ways can we choose?

P
C

We want to choose three people from a class of ten

to participate in a contest

How many ways can we choose?

We want to choose three people from a class of ten

to participate in a contest

How many ways can we choose?

We have

P^{10}_3

for choosing 3 people in 10 to permute

We can get the combination by not permuting the people

We want to choose three people from a class of ten

to participate in a contest

How many ways can we choose?

We have

P^{10}_3

for choosing 3 people in 10 to permute

We can get the combination by not permuting the people

\frac{P^{10}_3}{3!}

We want to choose three people from a class of ten

to participate in a contest

How many ways can we choose?

We have

P^{10}_3

for choosing 3 people in 10 to permute

We can get the combination by not permuting the people

\frac{P^{10}_3}{3!}
120

We choose r people from n people

How many ways are there?

We choose r people from n people

How many ways are there?

\frac{P^n_r}{r!}

We choose r people from n people

How many ways are there?

\frac{P^n_r}{r!}
C^n_r = \frac{P^n_r}{r!} = \frac{n!}{(n-r)!r!}=\dbinom{n}{r}

Problems

 

Permuation

1) There are A,B,C,D,E,F six people choosing seats

A wants to sit with B

C wants to sit with D

How many ways can they sit

2) There are A,B,C,D,E,F six people choosing seats

C doesn't want to sit with D

How many ways can they sit

3) There are A,B,C,D,E,F six people choosing seats

E and F want to sit on sides

How many ways can they sit

4) We have 2, 4, 6, 8 four numbers

How many six digits number can they form?

5) We have a word COMBINATION

How many ways can you rearrange the letters

6) There are A,B,C,D,E,F six people choosing seats

C want to sit at the right side of D

How many ways can they sit

Combination

1) In a class of twenty people

Two people will be choose

How many ways can they choose

2) In a class of twenty people

We want to divide it to three groups

Each group contains 10, 6, 4 people

How many ways are there?

3) In a class of twenty people

We want to divide it to three groups

Each group contains 10, 5, 5 people

How many ways are there?

4) There are five favor of ice cream

Each person can choose two (can be same)

How many ways can one choose?

5)

6) Row three dices

How many different results

are there?

x+y+z = 10

How many positive (x, y, z)

are there?

11/13 Review

Factorial

n!

11/13 Review

Factorial

n!

The ways 

n

people in a row can form

11/13 Review

Factorial

n!

The ways 

n

people in a row can form

P^n_r

Permutation

11/13 Review

Factorial

n!

The ways 

n

people in a row can form

P^n_r

Permutation

The ways        people choose 

n
r

position

11/13 Review

Factorial

n!

The ways 

n

people in a row can form

P^n_r

Permutation

The ways        people choose 

n
r

position

Combination

C^n_r

The ways to choose 

n

things from 

r

What are we learning today?

What are we learning today?

Probability

Probability

What is probability?

 

What is probability?

 

The chance of an event to happen

What is probability?

 

The chance of an event to happen

How many times

All

P(Event)=

Example

Rowing a dice

What is the probability of showing 6

Example

Rowing a dice

What is the probability of showing 6

\frac{1}{6}

Example

Rowing two dices

What is the probability that

the sum adds up to 7

Example

Rowing two dices

What is the probability that

the sum adds up to 7

1,6
2,5
3,4
4,3
5,2
6,1

Example

Rowing two dices

What is the probability that

the sum adds up to 7

1,6
2,5
3,4
4,3
5,2
6,1
\frac{6}{6\times6} = \frac{1}{6}

Example

Rowing five dices

What is the probability that

5 will show at least once

Example

Rowing five dices

What is the probability that

5 will show at least once

(Probability of five showing at least once)

=

1 - (Probability that five do not show)

Example

Rowing five dices

What is the probability that

5 will show at least once

Answer

1-(\frac{1}{2})^5 = \frac{15}{16}

Problems

1) In an online game, when you kill a monster

You have a probability of 10% to get $10

What is the probability to get $10

if you killed 10 monsters

2) What is the probability that

a four digit number

contains a 9?

3) What is the probability that

a four digit number

contains two 9?

4) You need to write USAD Essay

in December

It will have 3 subjects from 5 for you to choose

However, you only studied 2 subjects of the 5

What is the probability that you have studied

one of the three subject

5) You and your friend are playing a game

You will choose a number from 1~10

Your friend will choose a number from 1~15

What is the probability that the number

you chosen is less than your friend

Binomial Theorem

(x+1)^1 = x+1
(x+1)^2 = x^2+2x+1
(x+1)^3 = x^3+3x^2+3x+1
(x+1)^n = ?
1
1
1
1
1
2
1
1
3
3
1
4
6
4
1
\dbinom{1}{0}
\dbinom{1}{0}
\dbinom{1}{1}
\dbinom{1}{0}
\dbinom{2}{0}
\dbinom{2}{1}
\dbinom{2}{2}
\dbinom{3}{0}
\dbinom{3}{1}
\dbinom{3}{2}
\dbinom{3}{3}
\dbinom{4}{0}
\dbinom{4}{1}
\dbinom{4}{2}
\dbinom{4}{3}
\dbinom{4}{4}
C^n_r = C^{n-1}_{r-1} + C^{n-1}_{r}
(x+1)^n = \dbinom{n}{0}x^n + \dbinom{n}{1}x^{n-1}+...

Example 

x^3

What is the coefficient of

                              in

(2x+5)^{10}

Binomial Probability

Example

Sam is doing a test with 20 problems. However, he guessed the answer on 5 problems. He has 25% to guess the answer of the question. What is the probability that he guessed the correct answer for 3 problems

Example

Sam is doing a test with 20 problems. However, he guessed the answer on 5 problems. He has 25% to guess the answer of the question. What is the probability that he guessed the correct answer for 3 problems

C^{5}_3(0.25)^3(0.75)^2

Geometry

Vector

(Not in textbook)

What is vector?

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