Thursday, February 20, 2020

Definite Integrals

Definite Integrals

Integration

Integration can be used to find areas, volumes, central points and many useful things. But it is often used to find the area under the graph of a function like this: integral area
The area can be found by adding slices that approach zero in width:
And there are Rules of Integration that help us get the answer.
 integral area dx

Notation

The symbol for "Integral" is a stylish "S"
(for "Sum", the idea of summing slices):
 integral notation
After the Integral Symbol we put the function we want to find the integral of (called the Integrand),
and then finish with dx to mean the slices go in the x direction (and approach zero in width).

Definite Integral

Definite Integral has start and end values: in other words there is an interval [a, b].
a and b (called limits, bounds or boundaries) are put at the bottom and top of the "S", like this:
definite integral indefinite integral
Definite Integral
(from a to b)
 Indefinite Integral
(no specific values)

Derivative Rules

Derivative Rules

The Derivative tells us the slope of a function at any point.
slope examples y=3, slope=0; y=2x, slope=2
There are rules we can follow to find many derivatives.
For example:
  • The slope of a constant value (like 3) is always 0
  • The slope of a line like 2x is 2, or 3x is 3 etc
  • and so on.
Here are useful rules to help you work out the derivatives of many functions (with examples below). Note: the little mark  means "Derivative of", and f and g are functions.
Common FunctionsFunctionDerivative
Constantc0
Linex1
axa
Squarex22x
Square Root√x(½)x
Exponentialexex
axln(a) ax
Logarithmsln(x)1/x
loga(x)1 / (x ln(a))
Trigonometry (x is in radians)sin(x)cos(x)
cos(x)−sin(x)
tan(x)sec2(x)
Inverse Trigonometrysin-1(x)1/√(1−x2)
cos-1(x)−1/√(1−x2)
tan-1(x)1/(1+x2)
RulesFunctionDerivative
Multiplication by constantcfcf’
Power Rulexnnxn−1
Sum Rulef + gf’ + g’
Difference Rulef - gf’ − g’
Product Rulefgf g’ + f’ g
Quotient Rulef/g(f’ g − g’ f )/g2
Reciprocal Rule1/f−f’/f2
Chain Rule
(as "Composition of Functions")
f º g(f’ º g) × g’
Chain Rule (using ’ )f(g(x))f’(g(x))g’(x)
Chain Rule (using ddx )dydx = dydududx

Integration Rules

Integration Rules

Integration

Integration can be used to find areas, volumes, central points and many useful things. But it is often used to find the area underneath the graph of a function like this:
 integral area

The integral of many functions are well known, and there are useful rules to work out the integral of more complicated functions, many of which are shown here.
There are examples below to help you.
Common FunctionsFunctionIntegral
Constanta dxax + C
Variablex dxx2/2 + C
Squarex2 dxx3/3 + C
Reciprocal(1/x) dxln|x| + C
Exponentialex dxex + C
 ax dxax/ln(a) + C
 ln(x) dxx ln(x) − x + C
Trigonometry (x in radians)cos(x) dxsin(x) + C
 sin(x) dx-cos(x) + C
 sec2(x) dxtan(x) + C
   
RulesFunctionIntegral
Multiplication by constantcf(x) dxcf(x) dx
Power Rule (n≠-1)xn dxxn+1n+1 + C
Sum Rule(f + g) dxf dx + g dx
Difference Rule(f - g) dxf dx - g dx
Integration by PartsSee Integration by Parts
Substitution RuleSee Integration by Substitution

Squares and Odd Numbers

Squares and Odd Numbers


Add up odd numbers from 1 onwards and you get square numbers!
  • 1 is a square number (= 1 × 1)
  • 1 + 3 = 4, and 4 is a square number (= 2 × 2)
  • 1 + 3 + 5 = 9, and 9 is a square number (= 3 × 3)
  • etc!
Like this:
Odd
Number
Running
Total
 
11= 1 × 1
34= 2 × 2
59= 3 ×3
716= 4 × 4
925= 5 × 5
1136= 6 × 6
etc...

Is this some kind of magic?

odd square numbers
Not really, just look at this picture:
  • 1 is a square (a 1×1 square)
  • Add 3 and you get a 2×2 square
  • Add 5 and you get a 3×3 square
  • etc
It is good to know this, as it may help you solve things one day.

Cardinal, Ordinal and Nominal Numbers

Cardinal, Ordinal and Nominal Numbers

Cardinal Numbers

A Cardinal Number says how many of something there are, such as one, two, three, four, five.
A Cardinal Number answers the question "How Many?"

It does not have fractions or decimals, it is only used for counting.
How to remember: "Cardinal is Counting"

Ordinal Numbers

An Ordinal Number tells us the position of something in a list.
1st, 2nd, 3rd, 4th, 5th and so on (seeCardinal and Ordinal Numbers Chart for more.)

How to remember: "Ordinal says what Order things are in".

Nominal Numbers

A Nominal Number is a number used only as a name, or to identify something (not as an actual value or position)
How to remember: "Nominal is a Name".

Ratio - Make Some Chocolate Crispies

Ratio - Make Some Chocolate Crispies

To make these chocolate crispies I used:

  • 20 g (grams) of chocolate
  • 15 g of cornflakes or similar

This made 1 cake. Recipe Here

Your mathematics task is to:

1) calculate the ratio of chocolate to cornflakes, and then:
2) work out the amount of ingredients to make 21 cakes.
 

1) We can see that the ratio of chocolate to cornflakes is 20:15
But this can be simplified further. The Greatest Common Factor is 5, so:

If we have 20 parts chocolate to 15 parts cornflake, divide each side by 5 and we get 4:3

This is the ratio of chocolate to cornflakes.

2) We need to work out how much chocolate and cornflakes we need to make 21 cakes.
The recipe is for 1 cake but we want 21, so multiply both ingredients by (21 cakes / 1 cake), or simply by 21:
  • 21×20 = 420
  • 21×15 = 315
So to make our 21 cakes we need:
  • 420 g of chocolate
  • 315 g of cornflakes

Supplementary Angles

Supplementary Angles

Two Angles are Supplementary when they add up to 180 degrees.

supplementary angles 40 and 140
These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°:
Notice that together they make a straight angle.

supplementary angles 60 and 120
But the angles don't have to be together.
These two are supplementary because
60° + 120° = 180°







Complementary vs Supplementary

A related idea is Complementary Angles, they add up to 90°
How to remember which is which? Easy! Think:
  • "C" of Complementary stands for "Corner" right angle (a Right Angle), and
  • "S" of Supplementary stands for "Straight" (180 degrees is a straight line)
You can also think "Supplement" (like a Vitamin Supplement) is something extra, so it is bigger.

Laws of Exponents

Laws of Exponents


LawExample
x1 = x61 = 6
x0 = 170 = 1
x-1 = 1/x4-1 = 1/4
xmxn = xm+nx2x3 = x2+3 = x5
xm/xn = xm-nx6/x2 = x6-2 = x4
(xm)n = xmn(x2)3 = x2×3 = x6
(xy)n = xnyn(xy)3 = x3y3
(x/y)n = xn/yn(x/y)2 = x2 / y2
x-n = 1/xnx-3 = 1/x3
And the law about Fractional Exponents:
x^(m/n) = n-th root of (x^m) = (n-th root of x)^mx^(2/3) = 3rd root of (x^2) = (3rd root of x)^2

Fractions in Algebra

Fractions in Algebra


We can add, subtract, multiply and divide fractions in algebra in the same way we do in simple arithmetic.

Adding Fractions

To add fractions there is a simple rule:
adding fractions rule

Example:

x2 + y5 = (x)(5) + (2)(y)(2)(5)
5x+2y10

Subtracting Fractions

Subtracting fractions is very similar, except that the + is now −
subtracting fractions

Example:

x + 2x  −  xx − 2  =  (x+2)(x−2) − (x)(x)x(x−2)  
=  (x− 22) − x2x2 − 2x
=  −4x2 − 2x

Common Big and Small Numbers

Common Big and Small Numbers


NameThe NumberPrefixSymbol
trillion1,000,000,000,000teraT
billion1,000,000,000gigaG
million1,000,000megaM
thousand1,000kilok
hundred100hectoh
ten10dekada
unit1  
tenth0.1decid
hundredth0.01centic
thousandth0.001millim
millionth0.000 001microµ
billionth0.000 000 001nanon
trillionth0.000 000 000 001picop

Area of Plane Shapes

Area of Plane Shapes


triangle base heightTriangle
Area = ½ × b × h
b = base
h = vertical height
 squareSquare
Area = a2
a = length of side
rectangleRectangle
Area = w × h
w = width
h = height
 parallelogramParallelogram
Area = b × h
b = base
h = vertical height
trapezoidTrapezoid (US)
Trapezium (UK)
Area = ½(a+b) × h
h = vertical height
 circleCircle
Area = π × r2
Circumference = 2 × π × r
r = radius
ellipseEllipse
Area = πab
 sectorSector
Area = ½ × r2 × θ
r = radius
θ = angle in radians