Math 8
Name _KEY_
Quiz 6
All numbers are assumed to be integers.
1. Prove or give a counterexample: a | b
a | c a | b+c
a | b aj = b
a | c ak = c
So assuming a | b and a | c, we have a(j+k) = aj + ak
John Anthony Leon Sanchez
Professor Drogalis
PHIL 11A
Your Self
Looking now into who weve become, we see how we came to be who we are, but we
can never pin point what exactly makes this I, I. It can b
John Anthony Sanchez
Dr. Drogalis
PHIL 11A
Descartes first three principles are the basis for what he knows. The first principles are
claims that must be true in order for other claims to be true. His
John Anthony Sanchez
Dr. Kuczenski
Friday 2:15-3:20
Application of Design
Develop a theft proof bicycle lock.
I should first ask the customer Did you have a lock at all? because there are some introdu
John Anthony Leon Sanchez
Interviewee: Kelsey Stone (Campisi RD)
[email protected]
Kelsey first wanted to major in marine biology, but quickly turned to anthropology. At home, she
began to have a passion
John Anthony Sanchez
Dr. Kuczenski
1/23/15
Economist Article Reflection
The article The World Turned Upside Down published by The Economist on April 17,
2010 talks about why Western multinationals are
John Anthony Leon Sanchez
Interviewee: Kelsey Stone (Campisi RD)
[email protected]
Kelsey first wanted to major in marine biology, but quickly turned to anthropology. At home, she
began to have a passion
John Anthony Sanchez
Dr. Drogalis
PHIL 11A
The definition of what the self is, that is, what makes us who we are, varies from
person to person. Platos theory of what it may be is one to note. Plato hi
Product and Quotient Rules
1 The Product Rule:
If a function Is the product oftwo differentiable functions then the derivative' Is thd rst
times the derivative of the second plus the second times the
DEFlNlTlON:
The line 3! = b is a horizontal asymptote of the graph of a function y = x)
it ith r.
e 8 ign;f(x)=b or xErme(x):b 3~(WK 93,17
+0~M
These uldelineeonl a l tolimlteatinnit so be careful.
mix will 3 NM mm Xa/
Basie Differentiation Rules
/\
l . The Constant Rule: 7) x)
The derivative of a constant function is O.
For any real number, 5: i [c] = 0
A.) y = -4 a.) s0) = 45 a) f(x) = 0 DJ
A Continuityand OneSided Limits
Definition of Continuity
Continuity atla Point: Fro erties of Continuit :
A function fis continuous at cif the following Given functions fand goontnuous at x = c,
t
RELATIONSHIP OF f ' TO lNCREASING/DECREASING A
r l
When f>0,ther1fis MM;-
When f'<0,thenfisI cfw_Cm-5&5]
WHERE TO FIND EXTREMA
Relative (or Local Extrema (Maximums and Mnhnums) are values of the funct
The Fundamental Theorem ofCalculus
We have now seen the two major branches of calculus:
l) differential (tangent line problem)
2) integral (area problem)
Leibniz and Newton independently discovered a
THE GENERAL POWER RULE FOR INTEGRATION:
R
if g is a differentiable Function 0H: then
n+1
f 9001" g'(x)dx= 93311 + Cm. a 1
Equivaiently, 'rF u = g (x), then
un+1 .
71d = _
fu u n+1+C,n
1St DERIVATIVE TEST FOR EXTREMA
\ c
Using the Criticai Numbers (values of x where MACMM )
create a Sign chart and look to the left and right of these critical numbers
0R
Look at a graph of the derivat