Suppose you want lim x-> a   [f(x) - f(a)](x-a) where a = 2


get [f(x) - f(a)](x-a) with a = 2, then take the limit as x -> 2
f(x) - f(a)/(x-a) =( 4.5x^2 -3x + 2 - (4.5(2)^2 - 3(2) + 2)]/(x- 2)

= (4.5x^2 - 3x + 2 - 14)/(x-2)
= (4.5x^2 -3x -12)/(x-2)
= (4.5x + 6)(x-2)/(x-2)
= 4.5x + 6

Now we take lim x-> 2 (4.5x + 6) = 4.5(2) + 6 = 15

Suppose you have an angle and you want to know where its terminal side lies. How do you go about figuring this out?  It's simply seeing where the angle ends. For example, if you have the 4 quadrants, first ending at 90 degrees, 2nd at 180, 3rd at 270 and 4th at 360, then a 225 degree angle would have its terminal side in the third quadrant since 225 falls between 180 and 270.

If the angle is negative in value, simply add 360 each time until you get a number between 0 and 360 and follow the same guidelines as above.

For example, an angle of -556 degrees would have it's terminal side in quadrant 2. Add 360 to -556 and you get -196, then add  360 again to get 164, which is a second quadrant angle

Here's an easy way to simplify higher powers of i.

First know the first 3 powers of i

i^1 = sqrt(-1)
i^2 = -1
i^3 = -i


To get any power of i greater than 4, simply divide by 4 and the integer value remainder gives the answer. It will be one of the above.

Example:

i^45....   45 divided by 4 is 11 with a remainder of 1, so the answer is sqrt(-1)

i^202..... 202 divided by 4 is 50 with a remainder of 2, so the answer is -1

i^79.... 79 divide by 4 is 19 with a remainder of 3, so the answer is -i

Chi square test

Suppose  the frequency tablef or the counties and votes for a particular candidate are as follows
Miami Dade 6
Broward 8
Palm Beach 34
Pinellas 10
Hillsbourough 8
Total is 66

Relative frequencies are each frequency divided by the total
Miami Dade = 6/66 = .09
Broward = 8/66 = .12
Palm Beach = 34/66 = .52
Pinellas = 10/56 = .18
Hillsborough = 8/66 = .12

b. If the votes were equal across each county, you'd expect 66/5 = 13.2 for each. So we get a X^2 statistic
(6 - 13.2)^2/13.2 + (8-13.2)^2/13.2 + (34-13.2)^2/13.2 + (10 - 13.2)^2/13.2 + (8-13.2)^2/13.2 = 41.59
The critical value for the test for X^2, 4df at alpha = .05 = 9.488
Since the 41.59 > 9.488 critical value, reject Ho

c. The conclusion is that there is significant evidence say that the proportion of voters for Buchanon is not the same across all the counties.

remember key trigonometric identities that can help you in intergration

sin^2 + cos^2 = 1
1 + tan^2 = sec^2
1 + cot^2 + csc^2

Also you need to know the derivatives of trig functions

sinx ..... derivative    cosx
cosx.....  derivative  -sinx
tanx ......derivative   sec^2 x
cotx ....  derivative  -csc^2 x
secx....   derivative   tanxsecx
cscx ....  derivative   -cotxcscx

the trig identities come in handy to do substitutions to obtain antiderivatives.