Arithmetic Alge Calculus

8/6/2019 Arithmetic Alge Calculus

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From Arithmetic to Algebra to Calculus

Kwong Chung-Ping

CUHK

Kwong Chung-Ping (CUHK) From Arithmetic to Algebra to Calculus 1 / 45


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4/45Kwong Chung-Ping (CUHK) From Arithmetic to Algebra to Calculus 4 / 45


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Numbers: Integers

3



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Arithmetic: Addition

I bought 3 apples yesterday and I shall buy 2 more today:

3 + 2 = 5 . (addition)

I bought 3 apples yesterday and I shall buy 2 monkeys today:

3 + 2 = nonsense



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Arithmetic: Subtraction

I had 3 apples yesterday and I have just consumed 2 of them:

3 2 = 1 . (subtraction)

I had 3 apples yesterday and I have just consumed 3 of them:

3 3 = 0 . = the idea of zero

Note:

3 = 1 + 2 . = addition is the inverse of substraction,subtraction is the inverse of addition.



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Arithmetic: Subtraction

Professor Kwong had no money for lunch and he borrowed from

his wife $20. Professor Kwong owes his wife $20:

0 20 = 20 . = the idea of negative numbers



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Arithmetic: Multiplication/Division

An apple costs $3, how much for 4 apples? Answer:

3 + 3 + 3 + 3 = 3 4 = 12 . (multiplication)

I have $18, how many apples can I get? Answer:

183

= 6 . (division)

Note:

18 = 3 6 . = multiplication is the inverse of division,division is the inverse of multiplication.



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Arithmetic: Power

Sometimes we multiply a number by itself. Example: the area of a square with length 4 for each side is

4 4 = 16 .

The volume of a cube constructed by six equal squares eachwith area 4 is

4 4 4 = 64 .

For convenience we write

4 4 = 4 2 (4 to the power 2) .

and4 4 4 = 4 3 (4 to the power 3) .



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Algebra

Muhammad ibn M us a al-Khw arizm i

al-Jabr (Arabic) = Algebra (English): Move a negative term of an equation from one side to another.



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Algebra: Variables

Yesterday I bought 3 apples and this morning I bought 2applesthe number of apples I bought changes from 3 to 2.

Yesterday the temperature at 8am was 10 and was 15 todaythe same timethe temperature at 8am changes from 10 to

15 .We write the number of apples as a for short ; a = 3yesterday and a = 2 today.We write the temperature as T for short ; T = 10 yesterdayand T = 15 today.a and T are called variables (vary change).



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Algebra: Algebraic Expressions

Let x and y be variables. Depending on what x and y represent,their values can be integers or fractions, or other numbers weskip this time.

Since we can perform arithmetic operations ( + , , , , power)over these numbers, the following example expressions makesense:

x + y , x y , xy (short form of x y ) , x y

, x 3 ,

4 y , 2 x 2

+ 3 x 4 . These expressions are called algebraic expressions .

x 3 is an algebraic expression for the volume of a cube.



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Algebra: Valuation of Expressions

An apple costs $3. Let a be the number of apples. Then

3 a

is the money we have to pay for a apples.By substituting a = 2 into 3 a , we obtain the cost of 2 apples as$6. By substituting a = 3 into 3 a , we obtain the cost of 3 applesas $9, and so on.

The above process is called the valuation of an algebraicexpression.



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Algebra: Functions

The value of the expression 3 a depends only on a . We say 3 a is afunction of a , written f (a ) = 3 a . The value of the expression 2 x 2 + 3 x 4 depends only on x . Wesay 2 x 2 + 3 x 4 is a function of x , written f ( x ) = 2 x 2 + 3 x 4.

The value of the expressionx y depends both on x and y . We say

x y is a function of both x and y , written f ( x , y ) =

x y .

Hence an algebraic expression can also be called a function andthe valuation of an expression becomes the valuation of the

corresponding function. Example:

f ( x ) = 2 x 2 + 3 x 4 = f (2 ) = 2 (2 )2 + 3 (2 ) 4 = 10 .



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Algebra: Graph of a Function

We can show visually how the value of a function changes byplotting its graph . Example: The area of a square given by thefunction f ( x ) = x 2 has the graph:

0 1 2 3 4 50

5

10

15

20

25

x

f ( x

)



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Algebra: Algebraic Equations

An equation is an expression to show a state of being equal by

using the symbol = .An algebraic equation is an equation constructed by algebraicexpressions.

Example: Suppose an apple costs $3 and an orange costs $2.

Then 3 x will be the cost of buying x apples and 2 y will be thecost of buying y oranges. We can set up an equation forcalculating the total cost z (in dollars):

z = 3 x + 2 y .

The above equation is valid for any number of apples or oranges.For example the money we have to pay for 4 apples and 5oranges is (3 4 ) + ( 2 5 ) = $ 22.



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Algebra: Algebraic Equations

Example:

x

x

L

L 2 x

(box) Volume of box: V = x (L 2 x )2 .



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Algebra: Solutions of Algebraic Equations

We may draw an analogy between an equation and a balance:

In order that the bar of a balance stays horizontal, the contentsof the two dishes must have the same weight.Given an algebraic equation like that for the total cost of applesand oranges:

(apple-orange) z =

3 x +

2 y .

In order that the left-hand side of the equation to be equal to itsright-hand side, the values of the variables in the equation areusually not arbitrary. The values satisfying the equality arecalled the solutions of the equation.


l b l f l b


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Example: If I tell you I have spent $18 to buy 4 apples and someoranges, how many oranges have I bought?

The solution is easy. Putting z = 18 and x = 4 intoEquation ( apple-orange ), we obtain

18 = ( 3 4 ) + 2 y .

Subtracting 12 from both sides (the balance will not bedisturbed) gives

6 = 2 y .

Finally dividing both sides by 2 gives y = 3. Hence I have bought3 oranges.


Al b S l i f Al b i E i


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On the other hand, if I ask how many apples and oranges can Iget if I spend the same $18, then we have more than oneanswer .

Lets start with 2 apples, i.e., x = 2. We have

18 = ( 3 2 ) + 2 y .

Solving in the same way gives y = 6, i.e., 6 oranges.Next, try 3 apples, i.e., x = 3. We have

18 = ( 3 3 ) + 2 y .

We nd the solution y = 4 .5. Therefore we can also buy 3 applesand 4 and a half oranges using $18. The list goes on.



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Al b Q d ti E ti


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Algebra: Quadratic Equations

The following algebraic equation is very imporant:

(quadratic) ax 2 + bx + c = 0 .

In this equation a , b , and c are any numbers depending on whatthe equation is describing. The equation has only one variable x and there is a term associated with its second (but not higher)power x 2 . For this reason the equation is named quadraticequation .

Example: 4 x 2 8 x 12 = 0 is a quadratic equation witha = 4 , b = 8, and c = 12.


Algebra: Sol tions of Q adratic Eq ations


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Algebra: Solutions of Quadratic Equations

Consider the example 4 x 2 8 x 12 = 0. If we divide both sidesof the equation by 4, we obtain

x 2 2 x 3 = 0 .

This operation does not disturb the balance of the original

equation and hence solving 4 x 2

8 x 12 = 0 is the same assolving x 2 2 x 3 = 0. Therefore, even we are given a quadratic equation in the form of Equation ( quadratic ), we can simplify it by dividing its two sidesby a to give

x 2 +ba

x +ca

= 0 , or x 2 + px + q = 0

by setting p = ba and q =ca .




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It is not difcult to nd the solution of

x 2 + px + q = 0

where p and q are any two numbers.

First, we move q to the right side:

x 2 + px = q .

Then add p2

4 to both sides:

x 2 + px +p 2

4=

p 2

4 q .

Since

x 2 + px +p 2

4= x +

p2

2

(easy to verify by expanding the right-hand side),




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we have x +

p2

2=

p 2

4 q .

Taking square root on both sides gives

x + p2

= p24 q . Thus nally we obtain two solutions for x 2 + px + q = 0 as

() x 1 = p2+ p24 q and x 2 = p2 p24 q .




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Example: For the equation x 2 2 x 3 = 0, p = 2 and q = 3. The solutions are, according to Formula ( ):

x 1 = 22

+ ( 2 )24 + 3 = 3and

x 2 = 22

( 2 )24 + 3 = 1 .If we let x = 3 or x = 1, we will nd the left-hand side of theequation becomes 0, which equals to the right-hand side. Hence3 and 1 are indeed the solutions.




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Formula ( ) tells us interesting properties of the solution of quadratic equations:

1 The equation has two identical solutions x 1 = x 2 = p2 whenever p2 = 4 q .

2

The equation has two distinct solutions x 1 and x 2 whenever p2 > 4 q .3 The equation has no solution whenever p 2 < 4 q since we cannot

take square root of a negative number (unless we allow complexnumbers as solutions).

Example: The equation x 2 + 8 x + 16 = 0 has two identicalsolutions because p 2 = 4 q = 64.


Algebra: Algebraic Equations of Higher Order


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Algebra: Algebraic Equations of Higher Order

A quadratic equation consists of terms in x 2 , x 1 , and x 0 = 1 (theconstant term). The equation is of 2nd- order because thehighest power of x is 2.

A 3rd-order algebraic equation is of the form

x 3

+ a 2 x 2

+ a 1 x + a 0 = 0 ,and a 4th-order algebraic equation is of the form

x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 = 0 ,

and so on. However, their solutions are more complex.

Example: Equation ( box ): x (L 2 x )2 V = 0 is a 3rd-orderalgebraic equation.


Algebra: Graphical Solutions of Algebraic Equations


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Example: The graph of f ( x ) = x 2 2 x 3 is

2 1 0 1 2 3 4 54

2

0

2

4

6

8

10

12

x

x 2

2 x

3

The solutions of x 2 2 x 3 = 0 are the two points on the x -axis atwhich f ( x ) = 0, i.e., x = 1 , 3.




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Example: The graph of V = x (L 2 x )2 when L = 10 is

2 0 2 4 6 8400

300

200

100

0

100

200

300

V ;

L =

1 0

x

The solutions of V = x (10 2 x )2 (a 3rd-order algebraic equation) arethe two points on the x -axis at which V = 0, i.e., x = 0 , 5 , 5.


Calculus


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Calculus

Two major inventors of calculus: Isaac Newton (Left) andGottfried Leibniz (right):

Calculus = Differential Calculus + Integral Calculus


Integral Calculus: Areas of Shapes


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g p

Area of a rectangle:

x

f ( x )

h

0 a

f ( x ) = h , 0 x a ;

Area = ha .




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g p

Area of a triangle:

x

f ( x )

h

0 a

f ( x ) = hx a

, 0 x a ;

Area =ha2

.




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g p

Approximate the triangle by 3 rectangles:

x

f ( x )

h

0 aa4

Approximate Area =h4

a4

+2 h4

a4

+3 h4

a4

=3 ha

8

ha2

.


Integral Calculus: Denite Integrals


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g g

Approximate the area of an arbitrary shape:

x

f ( x )

0 x 1 = a b x 2 x n x

Approximate Area = f ( x 1 ) x + f ( x 2 ) x + + f ( x n ) x

=n

i= 1

f ( x i) x .


Integral Calculus: Denite Integrals


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Smaller x = Better approximationWhen x is arbitrarily small:

x dx ; f ( x i) f ( x );

;

n

i= 1

f ( x i) x b

af ( x ) dx .

b

af ( x ) dx is the denite integral of f ( x ) from a to b .


Differential Calculus: Slope


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Two staircases with different slopes :

x x y 1

y 2

y 2 x

> y 1 x

.


Differential Calculus: Slope


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Given a function:

x 1 x 1 + x x

f ( x 1 )

f ( x )

f ( x 1 + x )

The slope at x = x 1 is approximately equal tof ( x 1 + x ) f ( x 1 )

x .


Differential Calculus: Derivatives


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When x 0, the value to whichf ( x 1 + x ) f ( x 1 )

x tends tois called the derivative of f ( x ) at x 1 , labeled

f ( x ) x = x 1 ordf dx x = x 1

:

f ( x 1 + x ) f ( x 1 ) x

x 0 f ( x ) x = x 1 =

df dx x = x 1

.

We can just write

f

( x ) ordf dx

for a general x .




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Example: f ( x ) = x 2 :

f ( x + x ) f ( x ) x

=( x + x )2 x 2

x

=x 2 + 2 x ( x ) + ( x )2 x 2

x = 2 x + x .

2 x + x x 0 f ( x ) =

df dx

= 2 x .




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0 1 2 3 4 50

5

10

15

20

25

x

x2

2x


Differential Calculus: An Application of Derivatives


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The graph of V = x (10 2 x )2 (the volume of the box whenL = 10):

2 0 2 4 6 8400

300

200

100

0

100

200

300

V ;

L =

1 0

x

The maximum and the minimum of V can be found by solving

dV dx

= 0 .


Relationship Between Derivatives and Integrals


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Observe:

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10

15

20

25

x

x2

2x

Derivatives and Integrals are inverse of each other.



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Arithmetic Alge Calculus