Wow, I Had Started To Write A Math Instruction Book a Long Time Ago... Should I Finnish?

 math review for physics:

I)
measurements, units, comparing quantities, decimals, fractions and scientific notation

oy

1) can you write down approximately the length, surface area, volume, mass, density, length of time of a coupla dozen things around the house?

give these without measuring, don't think too hard:
2) how many meters long is the house?
3) how many cm long are you
4) how many grams is a sugar cube
5) how many meters long is your foot
6) how many meters from the house to houston road
7) how many mm long is your index finger
8) what's the surface area of the floor in your room?
9) what's the surface area of your body
10) how many ccs of water fit in your cupped palm
11) how many liters fills the bath tub
12) how many mililiters in a gallon milk jug?
13) how many grams is a liter jug of soda
14) how many mm from fingertip to fingertip in outstrethced hands
15) how many kg are you
16) how many grams is a gallon of milk?
17) how many kilograms is a car
18) whats the density of water
19) what's the density of balsa wood
20) what's the density of a quartzite pebble
21) what's the density of a steel butter knife?
20) what's the density of the air in a balloon
22) how many seconds does it take for a pen to fall off the table
23) how many seconds in a hour
24) how many seconds in a day
25) how many seconds in a year
26) how many km is it around the earth?
27) how many miles is it?
28) how many km to the moon?
29) how many kg is the earth?
30) how many meters per second is 60miles per hour?  (this one you should calculate and memorise)

conversions: look up and memorise to 1 dec place:
how many cm in an inch
how many inches in a meter
now many feet in a mile
now calculate how many meters in a mile
and which is smaller a km or a mile
how many of the smaller one in the larger?  1.something...
how many grams in a cubic cm of water?
how many grams in a lb?
now you know how to convert lbs and kg



ALL ABOUT MEASUREMENTS

(memorize the conversions i give (starred) and also memorize how big each unit is (starred)

We will measure, length, area, volume, mass (related to weight), density, time, speed, energy, power

Length:

length is measured in inches, feet, miles, and in metric:
all sorts of combinations of meters, lets start:

an inch is about the width of two fingers***
and there are

12 inches in a foot. ***
 3 feet in a yard. ***

so how many inches in a yard?

there are 5280 feet in a mile. ***

if you use google maps you can see how far down the road 200 feet is, 500ft ('ft' is an abreviation for feet), 1000ft and a mile.

in american carpentry and machining we measure to 1/2 inch and 1/4" and 1/8" (the two ticks mean inches while one tick: 3' means feet), 1/16th and 1/32th of an inch.  you can see 1/2 through 1/16th inch on a ruler or tape measureer.

Metric:
a meter is a little longer than a yard, 3ft.  ***
more accurate a meter is closer to 39 inches.  ***
learn to hold your fingers a yard or a meter apart. ***

at the time of the french revolution this system of measurement was invented to make measurement based on powers of 10 which makes alot of things easy.  (it's 1/10,000th  the distance between two degrees of lattitude?  check this).

now we base all our metric units on this.  there are 100 cents in a dollar so there are

100 centimeters in a meter***

your finger is a little more than a cm thick ***
so a cm is a little less than a half an inch!  there are close to
2.5cm in an inch  ***

so 10finger widths is 10cm ***

10centimeters is about 4 inches.  ***

so how many cm in a foot?  (30).  now 10 of these makes 100cm and that's a meter.

now visualize your meter with your fingers, divide it in half with your fingers and visualize that there are 5 of these 10cm lengths in it.

1/10th of a centimeter (cm) is a millimeter (milli a thousandth, i.e. in french mille is 1000), so:

one millimeter is about the thickness of your fingernail ***

10mm in a cm ***

1000mm in a meter ***

some single celled protozoa are a mm long.  most cells are smaller than this so we divide by 1000 again and get

1000 micrometers in a millimeter***

1000 nanometers in a micrometer***

many cells are 10s to 100s of micrometers.

DNA and proteins  are on the order of nanometers

GOING THE OTHER WAY:

1000 meters makes a kilometer (km)****

NOTE:
sometimes you will get confused is it 10mm per cm or 10cm per mm.  there is NEVER any need to get confused because you should always be visualizing things:

cm is finger width, mm is fingernail width.  can't be 10 fingerwidths in 1 fingernail width.

100 meters (arms breadth) can't = 1cm (finger width)!  km are how far down the road, meter is arm's breadth so can't be 1000km in a meter!  etc...

ALWAYS VISUALIZE THIS BEFORE USING CONVERSIONS



CONVERSIONS

1) which is bigger, a km or a mile? well?  how many feet in a km?

km=1000meters

now to do conversions we simply multiply by fractions and get one unit to cancell out and a new unit to appear, so:

1km *1000meters/1km (put the unit on the bottom in this case to make km to cancell out:

1000meters *3feet/meter=
3000feet.

(REMEMBER: it's 3feet per meter and not 3meters per foot, cause you can see before you 3meters is 3yards is across the room and a foot is only as long as your forarm!)

but a mile is 5280 feet.  so miles are longer (hence 60miles per hour is faster than 60km/hr!)

now lets convert between km and miles:

we can write out many conversion fractions one after another:

100miles * 5280ft/mile * meter/3ft * 1km/1000meters=

first of all see how all the units cancell out leaving km.

now cancell out numbers: divide 5280/3 to get approx 1420 and we have 100*1420/1000.  that's easy, cancell out 100 and we get 1420/10 and cancell out 10s and get 142.  so 100 miles is 142 km.  if you need more accuracy, use 39inches is 1 meter instead and multiply in more conversions:

12inch/foot * 1meter/39inches...

2) how many mm is a 1/4 of an inch?

(1/4) inches *

how do you do it?  write 1 inch / 4 on the left side of the paper.  write mm/1 on the right side of the paper.  you have to get from left to right.  so first turn inches to cm?  the conversion is 2.5cm per inch.  we can mult by 2.5cm/inch or 1 inch/2.5cm.  which one will cancell out the inches?

1 inch/4 * 2.5cm/inch

now we got cm, we want to get to mm so use the conversion 10mm/1cm.  which way to write it?

1 inch/4 * 2.5cm/inch *10mm/cm.  now we multiply and divide:  25mm/4= about 6mm.

in physics, we will sometimes always use meters or always centimeters.  so for instance if we have to do physics with something 1/4 inch long, we have to convert 6mm to meters.  Again, remember to visualize: 6mm is tiny, meters are big, so if i want to write how many meters are in a tiny 6mm, it's going to have to be a FRACTION of a meter, or in decimal, something like 0.001 meter, right?

convert:

6mm * 1meter/1000mm=

so now what is 6/1000?  use our decimal techniques: three decimal places to the left:
6.
0.6
0.06
0.006

so it's 0.006meters.  that's a tiny fraction of a meter.


AREA:

you can measure the area of your room in square feet or square yards.

NOTE ON UNITS AND DIMENSIONS:

1 foot is as long as your forearm
1 square foot is a SQUARE with 1ft on each side, like a floor tile
1 cubic foot is a BOX 1 ft on each side, or a square tile on the bottom and 1ft high.  or a box made of 6 sqare tiles (bottom, 4 sides and top)

math:

you can add 1 foot and 1 foot and get 2 feet long.

if you have a floor 10feet long and 8 feet wide it's got 10*8 square tiles so it's 80 sqare feet (abbreviated: 80 sq ft or 80 ft^2)

now if you have a box with the bottom 2ft by 3ft, the bottom has an area of 6 square feet.  if the height is 2feet then it's 2feet * 3feet* 2 feet=12 cubic feet or 12 cu ft or 12 ft^3.

notice you can multiply 6 ft^2 by 2ft and get 12 ft^3.

it makes no sense to ADD 6 sq ft and 2 ft and get 8 WHAT? ok.

***

so, lets convert areas:  1 foot by 1 foot is a tile 1ft^2
2ft x 3 feet is 6 tiles or 6ft sq.

1meter X 1meter is 1meter^2 tile
3mX4m is 12 of those squares 12m^2

one thing you can do is keep track of diff between area and PERIMETER.  for instance you draw the 3m X 4m recangle and see: 3x4=12 squares inside and the perimeter is 3m +4M + 3m +4m = 14meters (not sqare meters)

is the perimeter a similar number to area?  hmm... try some different rectangles 10X10, 5X20, 2X50, 1X100, 0.5 X200...

now convert:  if you have a square 3ft X 3ft that's 9 square tiles and 9ft^2, but it's also a yard by a yard so 1 sq. yd.  hmm

if you have a lawn 10yds X 20yds, that's 200sq yds, but each sq yd has 9sq feet so it's 200X9=1800sq ft!

using our conversion fractions:

200yd^2 X 3ft/yd doesn't work, we need yd^2 in the denomenator, so we do this:

200yd^2 X (3ft/yd)^2 that comes out to:

200yd^2 X 9ft^2/yd^2 and the yd^2 cancells out and we get 200X9ft^2 or 1800ft^2

we can do all kinds of things:

how many ccs in a cubic foot?

1ft^3 X ?

12inch=1ft so: put ft on the bottom of the conversion:

1ft^3 X 12in/1ft

but cube it:

1ft^3 X (12in/ft)^3=
1ft^3 X 12X12X12 in^3/ft^3 ft cancell out and then ...

ah. 2.5cm in 1inch, put inches on bottom:

1ft^3 X 12X12X12 in^3/ft^3 X (2.5cm/in)^3=

1ft^3 X 12X12X12 in^3/ft^3 X 2.5X2.5X2.5 cm^3/in^3

cancell out the units and we have

12X12X12X 2.5 X 2.5 X 2.5 ccs

at least 10X10X10X8=8000 probably much more, use a calculator

onward:

convert 60mi/hr to meters per second!

60mi/hr X 5280ft/mi X 1m/3ft X 1hr/3600 s=

well calculate it!  now when they give you a problem with a baseball flying at 40m/s you can relate to how many mi/hr it's going

find a bunch of these in the physics book and do them.







Exercises:
1) how many meters is the circumference of the earth?  if it's 8000miles across, then it's pi times that for circumference, so 3*8000=24000miles and we convert to feet, it's 5280*24,000= 120,000,000feet.  /3 to get meters, 40,000,000meters.  1/4 is from pole to equator: 10,000,000meters, and divide by 90degrees of latitude from equator to pole and we get about 100,000meters.  is this correct?



QUANTITIES
next we got to do a bunch of these
do them without calculating

1) which is bigger, 1/2 or 3/4
2) which is bigger, 5/4 or 8/9
3) which is bigger 1/9 or 0.9
4) 1/3 or .3
5) what's half of 1.5?
6) what's 2/3 of 6?
7) what's half of 1/4
8) which is bigger 2/3 of 10 or 1/2 of 7
9) what's half of 0.25
10) what's 8/4=
11) what's 8/2=
12) what's 8/1=
13) what's 8/(1/2)= (don't calculate, give the obvious answer)
it's like 8 times 2 yes?
14) what's 200/100=
15) what's 200/10=
16) what's 200/1=
17) what's 200/0.1=
18) which is bigger 1.001 or 0.998?
on and on it goes, to be able to see at a glance whether a number makes sense or not

be able to convert between fractions, mixed numbers, top heavy fractions and decimals

decimals
1) what's a tenth of 100?
2) what's a hundredth of 2100
3) what's 0.1 times 30?
4) which is bigger 1.5 X 2.5 or 5
6) what's 100/100
7) what's 100/10
8) what's 100/1
9) what's 100/0.1?
how many of those tiny 0.1s fit in 1?  fit in 10?  fit in 100?
it's like 100X10, yes?
10) what's 100/0.01  even MORE teeny tiny 0.01s fit in 100



DAMM I LOST PAGES OF WORK.  GOTTA GET NEW COMPUTER!

onward

in decimal:

what's 100/10=
10/10=
1/10=
0.1/10=?  smaller right?  what's 1/10th of a 1/10th?
0.01/10?

or 0.01/100?  really small right?  do everything by 10 at a time to get the decimal place to move right

so 13.2/10=1.32 yes?

and 13.2/0.1 like (9) above  
it's like 13.2 X 10, yes?

so multiplying by 10 makes bigger moves decimal place to the right
dividing by 0.1 makes bigger, moves dec place to right
mult by 0.1 makes smaller, moves dec place to left
divide by 10 makes smaller, moves dec place to left

so dividing by 0.001 is like multiplying by 10, 100, 1000?  so move decimal point right by 3

etc..

so what's
20/0.25?  well how many of those quaters fit in ONE?  so how many fit in 20?

or this:
0.01/12?  well we know that 0.01 / 10 is smaller move decimal point left 0.001 and dividing by 12 is even smaller.

what's a number a little smaller than 0.001?  0.0012 or 0.0008?

so if you have to divide
0.123/0.023... well 10 .02s fit in 0.20 yes? or 5 0.02s fit in 0.1...

then when you use the rules you won't get mixed up

another way:

you can multiply the top and bottom of a fraction by the same number yes and leave it the same

so if you mult 0.123/0.023 by 10 you get
1.23/0.23, and mult by 10 again
12.3/2.3  and again
123/23 and that's about 6 yes?


SCIENTIFIC NOTATION

SCIENTIFIC NOTATION

(10^2 means 10 squared, 10X10)

10^3=
10^2=
10^1=

from one line to the next what do you have to do to the answer to get to the next answer? so what is

10^0=
and

10^(-1)=
10^(-2)=

another way to see this:

what's 10^3 times 10?
and 10^4 times 10?
so
10^3 times 10^2 =

or do it this way:

10^3 X 10^2=
10 X 10 X 10 X 10 X 10

so anyway it's 10^5 yes?
even 10^1 X 10^3=10^4 yes?

so it looks like 10^a X 10^b = 10^(a+b) yes?

on the other hand, what's
(10^3)^2?

that's
(10 X 10 X 10) X (10 X 10 X 10)

(2X 2) X (2X2X2) is the same as 2 X 2 X 2 X 2 X 2 right? order doesn't matter

so (10^3)^2=10^6

and that's diff than
10^3 X 10^2

HOMEWORK:

WHAT IS:
1) (2^3)*(2^2)
2) (2^2)*(3^3) (CAREFUL, OUR RULES DON'T APPLY HERE
3) (10^2)*(10^4)
4) (10^3)^2
5) (2^3)^3
6) (X^2)*(X^4)
7) (X^2)^4



now negatives.

what's 10^3/10=
that's 10^3/10^1
and it's =1000/10=
=100
=10^2

and what's
10^3/10^2?
and 10^3/10^3

and 10^5/10^3?

so it looks like 10^a/10^b = 10^(a-b), yes?

so what's  10^3/10^4
that's 1000/10000  hmm that's 1/10,  yes?
but it's also 10^-1  by our formula

now that we can do that

if 1/10 =10^-1, then
1/100 =10^-2 etc..

HOMEWORK:
1) 10^4/10^2
2) X^2/X^2
3) X^2/X^4
4) (X^-2)/(X^-3)  CAREFULL, DOUBLE NEGATIVES!
5) (10^5)*(10^-3)
6) (10^2)*(10^-5)





so onto sci notation

which is bigger 0.000012 or 0.00000845?  hard to bloody see so we do this:

12/10=1.2
12/100=0.12
12/1000=0.012 yes?

hmm lets do it this way:

to turn 0.000012 into 12 we multiply by 10 6 times yes?

or to turn 12 into 0.000012 we divide by 10 6 times

so i'ts =12 /10^6

or 12 X 10^-6 will do the same thing, right?

only for scfi notation we make the number part between 1 and 10

so we turn it into 1.2X10^-5

how did i do that?  i made 12 smaller so i had to compensate and make 10^-6 bigger.  10^-5 is BIGGER than 10^-6 right?  10^-20 is really really tiny right?  so we are going in the right direction

where are we?

so the easiest way to do this is:

to make 1,312 into scientific notation we turn it into 1.312, to do that we have to make it smaller by a 1000 or move the decimal point 3 places to the left, so we compensate by saying
1,312=1.312X10^3  (1000? right)

to make 0.00043 into scientific notation we turn it into
4.3 and this is multiplying it by ... hard to think that so just count how much to move the decimal point to the right..  4 places so it's multplying by 10^4  or to compensate, we say

0.00043=4.3X10^-4  

so those are simple rules.


ARITHMETIC WITH SCI NOTATION

multiplying is easiest:

1.5X10^2 X 3X10^6

that's 1.5 X 3 X 10^2 X 10^6

but we know how to do all that!

4.5 and add the exponents right?

4.5X10^8

so 1.5X10^3 X 2.09X10^-6  ?

well on paper or with calculator mult
1.5 and 2.09

but don't do the 10s on the calculator!  just add 3 and -6!

so it's about 3X10^-3

dividing is opposite:

4.5X10^8 divided by 1.5X10^2 is

4.5/1.5 and 10^8/10^2 we subtract exponents right?

3X10^6

adding is trickier:

1.5X10^4 +3X10^2  well it's a big number plus a smaller number... best to expand them

1.5 move the dec point 4 places: X10 X10 X10 X10..
15000
and 3X10^2 is just 3 hundreds right:

15,000+300=15,300

adding and subtracting is a pain in the ass.

but you can see that
3.45X10^5+9.887X10^-4

well that's a giant number plus a smidgen, maybe don't even bother?  depends on how many decimal places of accuracy you are using.

(oops we forget to talk about significant figures.  arghh)

hell, you can even do sqare roots!

sqareroot of 4X10^6 is

sqrrt(4) times sqrrt(10^6)
that's 2X sqrrt (10X10X10 X10X10X10)
or 2X sqrrt( 1,000,000)
oh, that's 2X 1000!
2X10^3

doing sqrt of 4X10^5 is trickier:

sqrt(4) X sqrt(10^4) X sqrt(10)
2 X 10^2 X well, sqrt(10) you got to look up or approximately 3.

basically, you can do any maner of complex math with these things with pencil and paper or even in your head if you just need it approximately!  which you shold do anyway before you plug things into your calculator


I.5)
significant figures
should go after measurements


i forgot to ask problems like:

write the number of seconds in a year in scientific notation

write the number of km in an inch in scientific notation...


II)
graphing basic functions

ok, lets get to this:

1) LINES
graph y=3x-2:

set up a table:

x......................y
-2..y=3(-2)-2=-8
-1...y=3(-1)-2=-5
0...y=.. (don't forget negatives and zero)

do a few more points, x=1, 2, 3

now you know how to plot points on a graph?  (-2,-8), (0,-2) etc...?  x comes first (horizontal axis) y comes second (vertical axis)

so review y=mx+b, is a streight line with slope m, crosses y axis at b (y intercept)

review slopes between two points (x1, y1) and (x2, y2)

m=(y2-y1)/(x2-x1)

draw some lines with slopes of 0, 1/10, 1/2, 1, 2, 10, -10, -2 etc..

***you should be able to visualize these slopes instantly and be able to approxximate the slope of a line (or tangent line) that you see***

**
**
more on slope.  slope is rise over run, that is if you plot the point (3,2) and draw a line from the point down to the x axis, and a line fromacross  (0,0) to (3,0) and draw the hypotenuse from the origin to the point you got a triangle.  the slope is also the oposite length over the adjacent bottom length.

if i give you a point (3,2) and i ask you to make a line through that point with the slope of m=4, which is 4/1, you can plot (3,2), then go by 4 and across by 1 to the next point (4,6), do it again and get (5,10).  now you can draw a line through them.

you can make an equation for it too.

first way:

use y=mx+b.  we said that m=4 so we got

y=4x+b.  we also said that it goes through the point (3,2), so x=3 when y=2, so plug those in:

2=4*3+b

now you gotta use algebra:

2=12+b  oh well no you don't obviously b=-10.

so your equation is y=4x+-10

or y=4x-10.

now check it and see if the points (3,2) and (4,6) work in it:

6=4*4-10?  yes.

another concept is the y intercept.  a line hits the y axis when x=0.  if we plug x=0 in the equation we get:
y=4*0-10 or
y=-10.  so b is the y intercept.

see if the line you drew hits the y axis way down there.

II)
next kind of problem: if i give you two points (3,2) and (9, 5) can we make a line?  yes draw it.

next we can get an equation for it.  fist find the slope: (y1-y2)/(x1-x2)
(5-2)/(9-3)=3/6=1/2

now we use the slope formula to make a new kind of equation for a line called the point slope formula:

m= (y1-y2)/(x1-x2)

using algebra mult by sides by (x1-x2) and cancell out and get:

m(x1-x2)=y1-y2

now rearrange
y1-y2=m(x1-x2)

now we let one of the points be variable and the other fixed:

y-y1=m(x-x1)

now we can plug our point and slope into the formula:

y-2=(1/2)*(x-3)

move things around:

y-2=(1/2)x - 3/2

add 2

y=(1/2)x -3/2 +2

2=4/2 so 4/2-3/2 =1/2 and we get

y=(1/2)x +1/2

plug the other point into this equation and see if it fits.  see if your line has a shallow slope of 1/2.  see if the line you draw looks like it hits the y axis at 1/2.

now do these:


p.s.

y=3 is a horizontal line, all the points have y value of 3.  what is it's slope?

x=4 is a vertical line all the pints have x value of 4.  so it has the points (4,0) and (4,2)  what's the slope between em?  2/0?  that's undefined, vertical lines have undefined slope.  and they aren't techically functions.  (advanced topic)

okl, go

BASIC MATH I    38103   EXAM #1

COPY DOWN THE PROBLEM, AND THE PROBLEM NUMBER. SHOW ALL WORK. CIRCLE YOUR
ANSWER.  YOU DON'T HAVE TO CIRCLE THE GRAPHS.


GRAPH EACH OF THE GIVEN EQUATIONS ON ITS OWN GRAPH AT LEAST 3 INCHES
   BY 3 INCHES. NEATLY!  LABEL EVERYTHING.  MAKE YOUR SCALE MARKS WELL

1) Y=X+2

2) 3X-2Y=-5

3) X=4

4) Y=-3

5) Y=-1/2X+2

6) Y=3/2X-3

7) Y=2X

8) WRITE AN EQUATION FOR A LINE WITH A NEGATIVE SLOPE

9) WRITE AN EQUATION FOR A HORIZONTAL LINE


CONVERT EACH EQUATION INTO THE FORM Y=mX+B BY SOLVING FOR Y:



10) 3X-2Y=8

do you remember the algebra?  add 2y to both sides

3x=8+2y

sub 8

3x-8 =2y

switch

2y=3x-8

divide by 2

y=(3/2)x -4

11) 2X+3Y=4


FIND THE SLOPE OF THE LINE BETWEEN EACH PAIR OF POINTS:

12) (2,-4) AND (6,8)

13) (10, 5) AND (10, -5)

14) (-1,1) AND (0,0)

15) (4,4) AND (-6, 2)

16) (6,3) AND (-2,3)

17) (-1,-4) AND (-3,-1)


18) WRITE THE EQUATION OF THE LINE WITH SLOPE OF -3/2 THAT PASSES THROUGH
    THE POINT (-4,3).

19) WRITE THE EQUATION OF THE LINE PASSING THOUGH THE POINTS (-1,-5) AND
    (3,3)

20) IS THE ORDERED PAIR (-5, 4) A SOLUTION TO THE EQUATION 3/2Y-X=4?  SHOW
    THE WORK TO JUSTIFY YOUR ANSWER

21) WRITE THE EQUATION OF THE LINE PASSING THOUGH THE POINTS (4,-5) AND
    (4,3)


FIND THE SLOPE AND Y INTERCEPT OF EACH OF THE FOLLOWING LINES:

22) Y=-3X+2

23) 4X-3Y=8

24) Y=-3X

FIND THE X AND Y INTERCEPTS OF EACH OF THE FOLOWING LINES:

x intercept is when y is set to 0, y intercept is when x is set to 0.  (so much to learn!)

25) 5X+2Y=4

26) Y=-X-1


27) WRITE THE EQUATION OF A LINE THAT DOES NOT HAVE A Y INTERCEPT

28) WRITE THE FORMULA FOR THE SLOPE BETWEEN THE POINTS (X1,Y1) AND (X2,Y2).


29) GRAPH THE PAIR OF EQUATIONS, THIS MUST BE DONE ON A VERY CAREFULLY
    DONE GRAPH.  MAKE SURE ALL SCALE MARKS ARE EQUALLY SPACED!  IF THEY
    INTERSECT, AT WHAT ORDERED PAIR DO THEY INTERSECT?  VERIFY THAT
    THE ORDERED PAIR IS A SOLUTION TO BOTH EQUATIONS. IF THEY DO NOT
    INTERSECT, VERIFY THAT BOTH EQUATIONS HAVE THE SAME SLOPE.

    Y=2X-3  AND Y=-X+3



30) you should be able to look at a line drawn on a graph and find the y intercept and approximate the slope by dividing rise over run between two points.  then you can write an equation for it.


**
**

2) PARABOLA
Do the same for y=x^2, make a table and plot points, don't forget to use x=-2, x=0 etc...  also plot x=1/2, y=(1/2)^2

1/2 X 1/2 =1/4 right?

plot x=1/4, y=1/16 etc..

note that x^2 is always >=0, positive!

note that the slope between the points keeps changing sometimes negative on the left, positive on the right and getting steeper

draw on the same graph with y=x^2, the streight line y=x.  they should intersect at the point (1,1).  notice the graph of y=x^2 falls below the line for 0<x<1
and y=x^2 is above the line for x>1

3) CUBIC

graph y=x^3-4x

note if yo set y=x^3-4x =0, you can factor:
x(x^2-4)=0 and set each =0

x=0,
x^2-4=0
so x^2=4 (add 4 to both sides)
and so x can be both 2 and -2

so the graph should cross the x axis (where y=0) 3 times at -2, 0, and 2.  those are the zeros of the function

4) SQUARE ROOT
y=sqrt(x)

this time from right to left:

start with x=16, y=4
x=9, y=3
don't forget:
x=1, y=1
and even if x=1/4, then y=1/2!

onward
x=0, what's sqrt(0)?  well 0X0=0 so sqrt(0)=0

now x=-1 what's sqrt(-1)?  can any number times itself be negative?  no!  -2X2=-4, 2X-2=-4, -2X-2 is defined as +4!  so the function is undefined for all x less than 0!  so we only get half a graph.  it should look like a sideways parabola.

again, draw on the same graph the line y=x and note where they cross and where the sqrt graph is above and below.

these facts about numbers are useful:

small number (<1) squared gets smaller
big number (>1) squared gets bigger
square root of a small number (<1) is bigger
square root of a big number (>1) is smaller

5) HYPERBOLA
y=1/x

again start from right to left, graph
x=4, y=1/4
x=2, y=1/2
x=1, y=1/1
x=1/2 y=1/(1/2)  oh how do we do that?

1/(1/2) = 1 divided by 1/2 so it's
1/1 divided by 1/2, or
1/1 X 2/1 =2!

in general 1/(a/b) is = b/a

x=1/10, y=10!

x=0? y=1/0 ?  looks like it's infinity?

now from left side,

x=-4, y=-1/4
...

x=-1/10, y=-10

oh, the graph doesn't come together in the middle it looks like on the negative side 1/0 should be negative infinity!  damm it's broken!  so we say that 1/0 is undefined.  for real physical process, we almost never get to 0 so the universe doesn't usually blow up

6) EXPONENTIAL
Y=2^X

this is exponential growth, i.e. a bacteria, 20minutes later 2 bacteria, 40 minutes later 4 bacteria, an hour later 8 bacteria!  so 2 hours later its 64 bacteria, keep multiplying by 2, how many bacteria after 1 day?  this is why the human body does not tolerate rampant bacterial growth!  of course it slows down because diffusion of food and waste doesn't grow exponentially as things get bigger...

ok, graph it:

start from right:

x=4, y=2^4=16
x=3
x=2
x=1, y=2^1=2
x=0, y=1 (remember?  from way above, we discussed this)

we can even do 4^(1/2)  becuase

4^(1/2) X 4^(1/2) we add exponents =4^1!=4

so a number times itself = 4, so the number is =sqrt(4)=2, so
4^(1/2) =sqrt(4)=2

and 8^(1/3) * 8^(1/3) * 8^(1/3) =8 so it's
2X2X2=8
so 8^(1/3) =2 the cube root!  and so on

hell if i had to calculate 7.4^1.3 i could do it on paper!

=7.4^1 X 7.4^.3
and that's 7.4 X the cube root of 7.4 and the cube root of 8 is 2 so cube root of 7.4 is a little less, maybe 1.8 and my answer is
7.4X 1.8

we dont have to RELY on calculators

onward to plot
x=-1 remember 2^-1 is 1/2
x=-2 and so on.

note that 2^something is never negative it's never even =0

so the graph is always >0

on the right side is exponential growth.

if i graphed y=2^(-x) then on the right side the positive xs would turn negaitve and the graph would get smaller on the right, that's exponential decay you find in chemistry (drugs get used up with half lives etc..) and in radioactive decay

we can graph y=2.7^x also

there is a special number in mathematics called 'e' which is equal to approx 2.7 and the function y=e^x is very important.  you can graph it the same way:

x=2, y=2.7^2 etc...
x=0, y=2.7^0 anything ^0=1
x=-1, y=2.7^-1=1/2.7 etc... it looks the same

we'll do log and sin and cos and tan next time

7) logs

when we are graphing functions like y=2^x or y=4*x^10  the points go way up the y axis quickly.  also when we deal with data that looks like this:

mass of organism vs life span:

0.0001 grams   20 min

0.1 gram     2days

1 gram   10weeks

10 grams  etc...

100 grams

100 kilograms

etc...  (think from bacteria to whale)

it would be hard to graph on x-y axes!  if you made it go from 0 to 100,000 grams all the points for the tiny critters would get squashed together near the origin.

notice i got them as powers of ten.  notice scientific notation uses power of 10.

i could make a weird x axis and label the x axis this way:  make marks one inch apart.  the origin is x=0, then the next mark is 10, next is 100, next is 1000, and so on  and backwards the mark to the left of 0 is 0.1, next is 0.01 etc...

so our graph is really showing the number 0f zeros to the left or ritght of the dec point.

or the exp0nent of the scientific notation number, i.e if its'  10^4 then our mark is really at 4.

the exponent of a number is called the logarithm.

formally:

if 1000=10^3 then we say that

log base 10 of a 1000 is =3.  or

log(1000)=3.

in science and engineering log stands for log base 10.  and ln stands for log base e which is a number about 2.718... more on that later.

so log(10^7)=7

log(1/10) = log (10^-1)=-1.

notice that log(10)=log(10^1)=1
and log(1)=log(10^0)=0

what is log(0)?

well if 1000=10^3 converts to

log(1000)=3  then

log(0) = x converts back to:

0 = 10^x.

but remember our graphing exponential functions?  10 to the something never gets to 0.

there is no such x and log(0) is undefined.

you can graph y=log(x), the points are (10, 1), (100, 2) (1,0) (1/10, -1) (1/100, -2) right?  see how it goes infinitely down but doesnt touch the y axis.

the log function grows slowly.

now if you had to graph the data about mass and life expectancy with real data you'd have to do things like take the log(345)  that's the same as

arggh  i don't have time.  i'll do more later.

III)
algebra

IV)
geometry, triangles, trig

so we start with:
PYTHAGOREAN THEOREM
draw a square 4" on each side. (approx)  now make a point on the top side on inch in from the right.  make a point on the left side one inch from the bottom.  join these two points.  draw a point on the bottom, 1 inch from the right side.  join with the point on the side.  draw a point on the right side 1 inch from the top, join with last point.  now join the poinits on the left and top.  label the line segments 'a' for the one inch ones and 'b' for the 3 inch ones.  and label the 'diagonals', 'c'.   'c' is the HYPOTENUSE of each of the right triangles.  Note they make a square in the middle.  is it a square?  at each corner there are three angles.  the center one surrounded by two different ones.  these differnt ones are however just the small angles in one of the triangles.  since all angles of a triangle add to 180degrees, the angle that's left is the same as the right angle of the triangle, so the thing in the center is a square.

how does 'c' relate to 'a' and 'b'?

well the area c^2 + 4 triangles is the area of the whole square.  lets join the triangles by rearranging:  make a square 4" on each side. draw a horizontal line through the square one inch from the top.  draw a vertical line throught he square on inch from the right.  the sqare at the top left is aXa, the one on the bottom is bXb and the two rectangles are made of the 4 triangles in the first square, right?

each big square is the same area.  take the 4 triangles (2 rectangles) out of each one.  so the areas that are left are the same.  a^2 +b^2 = c^2

2)
now that we have that we can make some right triangles.  make one with 2 45degree angles and a 90degree angle.  so the two sides are the same.  so if the hypotenuse is =1, we have: s^2+s^2=1,
2*(s^2)=1
4*(s^2)=2
(square root both sides:)
2*s=sqrt(2)
s=sqrt(2)/2  

that goes with 45deg triangle

make an equilateral triangle with sides =1.  since it's an equlateral triangle, each angle is the same.  3A=180, or each angle =60degrees.  drop a vertical line fromt he top vertex to the base.  call it 's'. it cuts the base in half. it also cuts the top angle in half (30degrees)  look at one of the small triangles: it has 30degrees, 60degrees and right angle.  and the base =1/2, the huypotenuse=1, what's the vertical side?

s^2 + (1/2)^2 = 1^2
s^2 + 1/4=1
s^2=3/4
take square roots
s=sqrt(3)/2

3) sin, cos and tangent

draw x-y axes.  make points on the x axis 1 inch on either side of the origin.  two points 1" from the origin on the y axis .  use em to draw a circle with center at origin and radius =1"

now draw a ray from origin to perimeter of circle with angle 45degrees from the horizontal.  and vertical line from the intersection point on the circle to the x axis.  (45 45 90 triangle on it's side).  the ray is the hypotenuse.  lable the angle at the origin alpha or theta or t or whatever.

we define sin(t)=opposite/hypotenuse.  since hypotenuse is =1, then sin(t) is the vertical side or even sin(t)=the y coordinate of the point.  (these are three ways to remember what sin is)

so sin(45deg)=sqrt(2)/2

do the same thing with a 30 60 degree triangle with smaller (s=1/2) base on x axis and longer side vertical, hypotenuse from center to circle.  60degree angle will be at origin.

so sin(60deg) = sqrt(3)/2

now flip the triangle over with long side on x axis and 30deg angle at origin.

sin (30deg)= smaller side = 1/2

if we made a triangle with 0deg, the vertical side would be even smaller, in fact nothing at all so

sin(0deg) =0

if we made a triangle with 90 degree angle at origin, the oposit side would go al the way up:

sin(90deg) = 1

arrange these in a series:

angle    sin(angle)
0        sqrt(0)/2    or     0
30        sqrt(1)/2    or     1/2
45        sqrt(2)/2     or    sqrt(2)/2
60        sqrt(3)/2     or    sqrt(3)/2
90        sqrt(4)/2     or    2/2=1

notice also that these weird things are really just numbers between 0 and 1:
sqrt(2)=?  well 1.5 X 1.5 =2.25, close!  so sqrt(2) is approx =1.4
sqrt(3)=?  1.6X1.6 =?  lets call it 1.7, so the pattern is:


angle    sin(angle)                in decimal
0        sqrt(0)/2    or     0            0.0
30        sqrt(1)/2     or     1/2            0.5
45        sqrt(2)/2     or    sqrt(2)/2    0.7
60        sqrt(3)/2     or    sqrt(3)/2    0.85
90        sqrt(4)/2     or    2/2=1        1.0

that's nice.


it's a very simple pattern in the second collumn, yes?  you should have NO problem memorizeing it!  do so.

COS
now for each of those triangles, we define cos(t) as the adjacent side (side touching the angle which is lying on the x axis) divided by the hypotenuse.  so cos(t) is the adjacent side, or the x coordinate of the point on the circle.

so looking at each triangle we find:

cos(30deg) = sqrt(3)/2, cos(45)=sqrt(2)/2, cos(60deg)=1/2.  convince yourself that the table now becomes:


angle    sin(angle)                cos(angle)
0        sqrt(0)/2    or     0            1
30        sqrt(1)/2 or     1/2            sqrt(3/2)
45        sqrt(2)/2 or    sqrt(2)/2        sqrt(2/2)
60        sqrt(3)/2 or    sqrt(3)/2        1/2
90        sqrt(4)/2 or    2/2=1        0

you see that the cosines go in the opposite direction of the sines!  so in fact whenever you have to do sines and cosines you can sketch out this table from memory in 2 seconds.

look at the way we defined sin and cosine:

sin=opp/hyp, cos=adj/hyp

so sin(t)*sin(t) + cos(t)*cos(t)=opp^2/hyp^2 +adj^2/hyp^2
and this =
[opp^2+adj^2]/hyp^2

but pythagoream theorm says that the opp^2+adj^2 is     EQUAL TO the hyp^2, so we have

sin(t)*sin(t) +cos(t)*cos(t) = hyp^2/hyp^2 =1

we write sin(t)*sin(t) more compactly as sin^2(t)  [THIS DOES NOT MEAN WE MULTIPLY SOME QUANTITY CALLED SIN, BY ITSELF!  only take sin of t and THEN multiply]

so sin^2(t) + cos^2(t) =1

this comes in handy.

go back to the 30 60 triangle you have drawn with the 30deg angle at  the origin.  recall that  the slope of the line (hypotenuse) from teh origin is the rise over run!  so it's m=(1/2)/(sqrt(3)/2).  to divide flip the second fraction and multiply: m=1/sqrt(3)  [oops we don't like sqrts on bottom, multiply both top and bottom by sqrt(3):

1/sqrt(3) * sqrt(3)/sqrt(3)
=sqrt(3)/[sqrt(3)*sqrt(3)]
=sqrt(3)/3

we define tan(t)=opposite/adjacent.  it's the same thing as teh slope of the hypotenuse when the triangle is drawn this way.

but wait!  we defined sin as opposite and cos as adjacent, so
tan(t)=sin(t)/cos(t)  so now you can easily fill in the table:


angle    sin(angle)                cos(angle)        tan(angle)
0        sqrt(0)/2    or     0            1                0/1    =0
30        sqrt(1)/2 or     1/2            sqrt(3/2)            sqrt(3)/3
45        sqrt(2)/2 or    sqrt(2)/2        sqrt(2/2)            you fill it in...
60        sqrt(3)/2 or    sqrt(3)/2        1/2
90        sqrt(4)/2 or    2/2=1        0

4)
AT THIS POINT FIND A BUNCH OF PROBLEMS IN A BOOK!

angles bigger than 90degrees.  if i pick a point on the circle above the x axis and left of the y axis (x negative) and draw a ray from origin to it.  and i draw a line from that point to the point (1,0) right most point of the circle.  i now have a triangle with an angle bigger than 90deg at the origin.

no matter we can still define sin cos and tan.  the sides are funny, so we will use the third form of the definitions:

sin(t)= the y coordinate.  for angles between 90 and 180, these are the same as for the y coordinates for the smaller angles, so here sin(t)=sin(t) for the corresponding angle <90.  one way to say this is:
sin(180-t)=sin(t)  [see if that makes sense by trying t=45, 30, 60

cos(t)= the x coordinate.  this time the x coordinates are negative so cos(180-t)=-cos(t).

hell we can make angles even bigger so that the hypotenuse drops below the x axis, now both x and y are negative, and thus the corresponding sin and cos.  going further to the 4th quadrant, sin is still negative and cos is again positive.

why not keep going round and round?  sure.  in fact we should graph these buggers in a new way:

make x y coords.  label the x axis with 360 deg about 2" to the right of the origin.  cut it in half and we get 180deg mark.  again half and 90deg mark.  if we cut that in half we get 45, if we cut 90 in thirds we get 30.  2/3 along is 90.

this gets a little awkward so ... since all the way around is the circumference of our original circle, and the radius is one then the circumference is 2pi* r or 2pi, so:

360=2pi
180=pi
90=pi/2
45=pi/4
30=pi/6
60=180/3 or pi/3
0=0

place these (use the symbol for pi) right below the angles measures in degrees on your graph (along the x axis)

i don't know if you will use these units in your physics class.  if you were doing it with calculus, you would.  you might use them when you get to rotary motion.

onward.  from our chart you see that the values of sin and cos go between -1 and 1. make a mark 1" above the oritgin and call it 1, and -1 below.

now we graph sin:  it's 0 at 0.  use the chart to plot angles up to 90.  .5, .7 etc..  when we get to 90.. well at 135 (90+45) on our circle we see that the y value of the point on teh circle is getting smaller again and we said that sin(t) here is =sin(t) on the right side, so we plot .7 at x=135deg.

at this point it is easier to use the pi's.  if the x axis is marked off 2pi, half way is pi, half of that is pi/2, half way between pi and 2pi is pi/2 and pi/2 and pi/2 so it's 3pi/2.  between 0 and pi/2 is pi/4 and half way between pi/2 and pi is pi/4 +pi/4+pi/4 = 3pi/4, etc..

so our graph is .7 at 3pi/4.  continue towards 180deg or pi.  the graph is going down to sin(180)=0.  now sin goes negative.  at 3pi/2 that's 90deg, 90deg, 90deg, sin is the y value on the circle 1" down the y axis.  sin(3pi/2)=-1.  plot that.  finally we rise back up to sin(360)=0.  it makes a sine wave.  of course you can repeat the whole thing to make another 2pi!  for instance at 2pi+pi/2 = 5pi/2 it's one again.

even go backwards!  if you start at sin(0) on the circle, and go clockwise, y values going down and we call the angles negative, so our x-y graph has sin(-pi/2) = -1 and you can divide the x axis back there as -2pi, -pi etc... and get a sine wave going that way too.

YOU SHOULD DO THE SAME FOR COS.
notice that cos(t)=cos(-t)


GOOD RULE OF THUMB FOR WHEN YOU ARE USING SIN AND COSINES IN PHYSICS WITH TRIANGLE DIAGRAMS.  SOMETIMES HARD TO TELL WHEN TO USE WHICH.  JUST REMEMBER SIN OF SMALLER ANGLE IS CLOSE TO 0.  COS OF SMALLER ANGLE IS CLOSE TO 1.

tangent is weird.

these graphs will be useful for the chapters on harmonic motion.  (waves and springs bouncing up and down.

FIND MATH BOOK AND DO PROBLEMS!
V)
sin and cosine functions

VI)
vectors