16-SEP

Some basic algebra:

(a-b)#card=math&code=a%5E2-b%5E2%3D%28a%2Bb%29%28a-b%29&id=F0Oje)

(a%5E2%2Bab%2Bb%5E2)#card=math&code=a%5E3%20-%20b%5E3%20%3D%20%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29&id=P49d0)

%5E3#card=math&code=-b%5E3%20%3D%20%28-b%29%5E3&id=Uhdea)

(a%5E2%20-ac%20%2Bc%5E2)#card=math&code=c%20%3D%20-b%20%5CRightarrow%20a%5E3%20%2B%20c%5E3%20%3D%20%28a%2Bc%29%28a%5E2%20-ac%20%2Bc%5E2%29&id=Kz7Ny)

Example: rationalizing the denominator.

%20%3D%20a%2Bb%5Csqrt%7B5%7D#card=math&code=12%2F%283-%5Csqrt%7B5%7D%29%20%3D%20a%2Bb%5Csqrt%7B5%7D&id=Jkrgo) , where a, b are integers.

%7D%7B3%5E2%20-%20%5Csqrt%7B5%7D%5E2%7D%20%3D%203(3%2B%5Csqrt%7B5%7D)#card=math&code=%5Cfrac%7B12%7D%7B3-%5Csqrt%7B5%7D%7D%20%5Cfrac%7B3%2B%5Csqrt%7B5%7D%7D%7B3%2B%5Csqrt%7B5%7D%7D%20%3D%20%5Cfrac%7B12%283%2B%5Csqrt%7B5%7D%29%7D%7B3%5E2%20-%20%5Csqrt%7B5%7D%5E2%7D%20%3D%203%283%2B%5Csqrt%7B5%7D%29&id=hrOJ0)

Quadratic functions:

is nonzero.

%2C%20aB%20%3D%20b%2C%20aC%20%3D%20c#card=math&code=y%20%3D%20a%28x%5E2%20%2B%20Bx%20%2B%20C%29%2C%20aB%20%3D%20b%2C%20aC%20%3D%20c&id=QkTyF).

If we only want to solve , it is equivalent to solve .

monic, which means the leading coefficient of the highest degree term is 1.

. , where is the discriminant .

%5E2%20-%20(B%2F2)%5E2%20%2B%20C%20%3D%20(x%2BB%2F2)%5E2%20%2B%20C%20-%20B%5E2%2F4%20%3D%200#card=math&code=x%5E2%20%2B%20Bx%20%2B%20%28B%2F2%29%5E2%20-%20%28B%2F2%29%5E2%20%2B%20C%20%3D%20%28x%2BB%2F2%29%5E2%20%2B%20C%20-%20B%5E2%2F4%20%3D%200&id=oGAcL) .

%5E2%20%3D%20B%5E2%2F4%20-%20C%20%3D%20%5Cfrac%7BB%5E2%20-%204C%7D%7B4%7D#card=math&code=%28x%2BB%2F2%29%5E2%20%3D%20B%5E2%2F4%20-%20C%20%3D%20%5Cfrac%7BB%5E2%20-%204C%7D%7B4%7D&id=wstPn) .

Solution exists if and only if (abbreviated as “iff”) .

.

This is related to the mock test problem:

Write into the form %5E2%2Bq#card=math&code=2%28x%2Bp%29%5E2%2Bq&id=POQcm) , —> completing the square.

%20%3D%202(x%5E2%2B%205x%2F2%20%2B%20(5%2F4)%5E2%20-%20(5%2F4)%5E2)%20%3D%202(x%2B5%2F4)%5E2%20-%2025%2F8%20%5Cgeq%20-25%2F8#card=math&code=2%28x%5E2%20%2B%205x%2F2%29%20%3D%202%28x%5E2%2B%205x%2F2%20%2B%20%285%2F4%29%5E2%20-%20%285%2F4%29%5E2%29%20%3D%202%28x%2B5%2F4%29%5E2%20-%2025%2F8%20%5Cgeq%20-25%2F8&id=HhpDo).

When , attains minimum.

Say, how to deal with ? Can we find out its minimum or maximum?

The general way to find out mimima or maxima is by using differentiation (derivatives).

Solution: .

%20%3D3(x-1)(x-3)#card=math&code=%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%203x%5E2%20-%2012x%2B9%20%3D%203%28x%5E2-4x%2B3%29%20%3D3%28x-1%29%28x-3%29&id=t4TrJ). Letting , we get .

Which is which? (More precisely, at which point attains a maximum/minimum? )

Methond one: %3D6(x-2)#card=math&code=%5Cfrac%7Bd%5E2%20y%7D%7Bdx%5E2%7D%3D3%282x-4%29%3D6%28x-2%29&id=vVzwc).

Inserting into #card=math&code=6%28x-2%29&id=Oe4is)—> ; Conclusion: is a max. point.

Inserting into #card=math&code=6%28x-2%29&id=jiBeR) —> 6. Conclusion: is a mim. point.

Method two: (very powerful if the second order derivatives does not exist)

When , , when , <0; when , .

At #card=math&code=%28-%5Cinfty%2C%201%29&id=JVFZO), increasing, at #card=math&code=%283%2C%20%5Cinfty%29&id=v4ZXX) also increasing; while at #card=math&code=%281%2C%203%29&id=RySHD), y decreasing;

Conclusion: , attains maxium ; attains mimimum .

Functions

What is a function? (three ingredients)

Different ways of giving a function

1. Listing
2. Number sequences
3. Analytic expressions
4. The Graph of a function (to be continued next lecture)

A little complicated example:

The electricity fee is charging in the following way:

less than 120 kwh, then 6cents per kwh;

from 121 kwh, up to 180 kwh, 13 cents per kwh;

from 181 kwh and so on, 15 cents per kwh;

electricity fee (cent): #card=math&code=f%28x%29&id=rW9JF), as a function of how much electricity (denoted by , unit kwh) one uses.

%3D%206x#card=math&code=f%28x%29%3D%206x&id=gMrw0), if ;

%3D6*120%2B%2013(x-120)#card=math&code=f%28x%29%3D6%2A120%2B%2013%28x-120%29&id=VFJfh); if ;

%3D6120%2B136%2B15(x-180)#card=math&code=f%28x%29%3D6%2A120%2B13%2A6%2B15%28x-180%29&id=CCYGa), if ;

Another example: Sequences of numbers,

n = 1, 2, 3, 4, 5…

;

;

;

, ; (The Fibonacci sequences).

1. Polynomials
   Suppose (sometimes ) is an indeterminant (variable). (for a non-negative integer) a monomial. (This is a convention.)
   A polynomial is a (linear) combination (a sum) of monomials: %20%3D%20%207x-%2087x%5E2%2B1.4x%5E3%2B4x%5E5%2B5x%5E6%0A#card=math&code=p%28x%29%20%3D%20%207x-%2087x%5E2%2B1.4x%5E3%2B4x%5E5%2B5x%5E6%0A&id=isXMy)
2. Exponential functions %3De%5Ex#card=math&code=y%28x%29%3De%5Ex&id=nKV1h).
   Do you know what is ? What is its approximated value?
   is a constant, introducing by L. Euler.
   (1) is not a rational number (it is irrational)
   (2) , so %3De%5Ex#card=math&code=y%28x%29%3De%5Ex&id=U76LU) is increasing as increases;
   (3) .
   (4) is defined via the limit: %5En#card=math&code=e%3D%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%281%2B%5Cfrac%7B1%7D%7Bn%7D%29%5En&id=jzwzj) .
3. Exponential functions in general , .
   (, , ).
   When , increases as increases;
   when , decreases as increases;
   #card=math&code=%5Cfrac%7Bd%7D%7Bdx%7Da%5Ex%20%3D%20a%5Ex%20%5Clog_e%28a%29&id=z1XBT).