0.1 Real Number System - - Hwk: Std (1, 6, 9, 14, 17,…)
Reading asgn: It’s mostly review.
“rational numbers” fractions, terminating decimals or repeating dec.
Ex: 2.37 versus 2.373737… versus 2.1010010001… Order: -3<x<7 is OK but 7<x<-3 is not.
Order of Ops & Distrib Law:
The only ordinary op/function that distrib across +/- is * (except “linear operators")
Ex: 5*(2-3) = 5*2 – 5*3
Non-Ex: sqrt(4+1), (4+2)^3, log(10+1), sin(pi/3-pi/6), abs(5-2)¹abs(5)-abs(2)
Parentheses on Calcr - - Biggest prob for most students
All grouping is done with ( ) not [ ] or { } etc.
Ex: 7+{3-[(6-1)4]} = -10
Use ( ) for complete num, denom, radicands (& other func args), & fractions in probs.
(Unless the entry is a single character)
Ex: 
sqrt(8^(1/3)+23) / (4+1) = 1
Negatives on Calcr - - Negation key (-) vs Minus key -
Use (-) as first entry in a calculation (or 1st entry after an operation) is a neg nr.
Use - if your intention is to subtract from the prev entry.
Ex: 10-4 vs 10-4 Note type style on calcr
Refer to prev ans:
Start line with an op key (^ / * - +) or
2nd ANS
Ex: 4+3 ENTER *2 ENTER sqrt 2nd ANS ENTER
Refer to prev entry:
2nd ENTRY
Ex: (10+1)^2 ENTER; 2nd ENTRY change exp to 3 ENTER; chg exp to 4, etc.
Deep recall: Hit 2nd ENTRY several times. Does calcr takes you back several steps?
"Abs value"
Def -- Text
Ex: Write abs (sqrt 5 - 4) without abs value brackets
If variable is involved, work problem twice:
Ex: abs(x-3) < x+1 Ans: {x|x>1} Check, test critical vals (1,3) and test intervals.
Notes: Shortcuts fail on this one. Changing just middle sign (x+3) is wrong.
On calculator: 2nd Math Num menu or first entry in catalog.
abs(c) = sqrt(c)
Triangle inequality: abs(a+b) £ abs(a) + abs(b) How do you get < option?
Abs value and distance
Dist calls for subtraction.
On nr line, dist = high value - low value (high & low relative to nr line; e.g. depth gauge)
Typical app: dist btwn fixed point and variable: x - -3 if x > -3, -3 - x if x < -3, or..
abs(x+3) if x could be either above or below -3.
So: abs(c) = abs (c-0) = dist btwn c and 0 [implying c could be either above or below 0]
abs(c-d) = dist btwn c and d [either c or d could be higher] = abs(d-c)
Ex: Solve abs(x+5)>7 on nr line
We want dist btwn x and -5; Locate -5 on nr line, move both R and L 7 units;
Solution is everything beyond those points (hence two separate intervals)
Ex: Solve earlier ex with nr line/table: abs(x-3) < x+1. We want LHS < RHS. Ans: {x|x>1}
|
|
Nr Line |
-1 |
0 |
1 |
2 |
3 |
4 |
|
|
LHS |
4 |
3 |
2 |
1 |
0 |
1 |
|
|
RHS |
0 |
1 |
2 |
3 |
4 |
5 |
Have students do (on their own) the calculator investigation for 0-1. Discuss tomorrow.
Replay
Abs value
Greatest integer (follow "int" with both pos and neg decimal fractions)
STO button
To get B2-4AC use mult symbol in A*C
Try to figure out why you get 1's and 0's for inequalities (2<4 or 6<3, etc.
0.2 Integral Exponents Hwk asgn: Std (1, 6, 9, 14, 17, etc.)
Read/review rules for
Mult/div with exponents
Power of a power
Product to a power (Note - distrib exponents & radicals across mult/div)
Quotient to a power
Zero exponent
Neg exponents
Goal for all these: simplify expressions w exponents
Ex: Work all 4 examples in Ex6 and Ex7, except include some constants
Suggested sequence of actions:
1. Distrib exp to all factors in ( ), whether in num or denom
[Remember: (1)not across sums; (2)coeffs are factors]
2. Move factors with neg exponents [you're taking recip when crossing fraction
bar]
3. Combine factors with like bases [same vs opp
side of frac bar]
4. Expand coeffs with exponents if reasonable.
5. Add/subt fractions if necessary
Note the 5 CAUTIONs in the text
For caution on pg 19, note that you may combine if either the base or the exp is the same
(what about both? go either way, but not both ways)
Scientific Notation
Read (i.e. work out) Note you combine the coeffs one way and the exp's another way.
On calcr, (1) use EE key; (2) be able to read answer
0.3 Roots, Radicals and Rational Exponents Hwk Std (1, 6, 9, 14, 17, …)
Even roots vs odd
roots for "nth root of c"
Even: c must be nonneg; result is the pos root (use
the - symbol for the neg root
Odd: c may be any real nr; result has the same sign
"products and quotients of roots" radicals (just like exponents) distrib across * and / (not ±)
"nth roots of nth powers" Even vs odd
Even: Since cn is pos, the base c may be pos or neg; Since result is the pos root, use |c|
Odd: both cn and the result have the same sign as c, result is just c
"reciprocal exponents" c^(1/n) is nth root of c (provided it exists)
Applic: general roots on calculator.
"rational exponents" c^(t/k) is the k root of c^t (provided it exists)
Exp laws apply to rational exponents as well as integer exponents --
they actually apply to any real nr (irrational exp's as well)
Examples to look at from exercises:
10. sqrt 75 + sqrt 192
18. cubert(40x6*y-5)
25. 4throot( 4throot( a^3))
32. (x1/2y3)(x0y7)-2
41. Note that base can be an entire alg expr; so: (x+y)1/2*(1+x+y) = …
0.4 Polynomials Hwk: Std (1, 6, 9, 14, 17, …)
Read defs and know bold-print terms on first two pages.
Ops:
+ (Combine like terms)
- (Clear paren and add)
* (Mult every term in first paren times every term in 2nd paren)
Applic to squaring, cubing
Do (5x-3)3 = 125x^3-225x^2+135x-27
show setup of (5x-3)2
factoring
greatest common factor
count terms
4+ Grouping
4: New: perfect cubes: for a^3±…±b^3, try long div by a±b:
Do you get (a±b)^2 ? [or check for pattern]
2: special patterns
(sum) or diff of squares
sum or diff of cubes For prob above, div by (5x-3) -- Why this factor?
3: trial and error
perfect sq (get by trial and error
X66 Box method of factoring, when one factor is proposed: back into other boxes
Try with finding factorization of 27x^3+8
0.5 Fractional Expressions Hwk: Std(1, 6, 9, 14, 17, …)
Simplify (factor and reduce)
Mult and Div (rearrange if div, then simplify)
Add and Subt (use LCD)
Ex: Do example 6, but w 1st - 2nd + 3rd
Compound (complex) fractions
Ex 10-11; do #51 but w neg expo
Ex 12 Rationalize denom 7/sqrt5; 2/(3+sqrt 6)
Ex 13 Rationalize num (sqrt(x-h) - sqrt x ) / h
Do #59
Return to: Merced College; Don Power Updated 08/16/05 by Don Power