precalc Flashcards
Conjuction
Compound Statement joined by AND - true when both are true (^)
Disjunction
Compound Statement joined by OR - true when 1 is true (V)
Intersection
a SET of elements formed where 2 sets OVERLAP (∩)
Union
a SET of elements formed where 2 sets combine (U)
Conditional
If p, then q (implies, only if, is sufficient for)
Logically Equivalent
statements with the same final values in a truth table
Biconditional
“if and only if” - “equivalences” F-F and T-T
Converse
switch p and q - If q, then p
Principal of Inference
__→T = T then __ is T
Principal of Contraposition
__ →F = T then __ is F
Principal of Syllogism
p→q = T and q→r =T then p→r is T (p→q→r)
Principal of Substitution of Statements
- Math Teachers are cool.
- danny is a Math Teacher.
Therefore, Danny is cool.
Disjunctive Inference
F V __ = T then __ = T
Equivalence Inference
T ←→ __ = T then __ = T
Empty (Null) Set
set which has NO elements {}, ø - proper subset of ALL sets
Tautology
True no matter what values are assigned to p and q
Contrapositive
p → q becomes q’ → p’
Vertex Theorem
Corners of a shaded region - max or min value of f(x,y) = ax+by+c
Determinant
in a SQUARE matrix ( detA or |A| ) is -/
Multiplying Matricies
must have same inside dimensions when put next to each other
axb x bxc - b have to be the same and axc is product
Adding/Subtracting Matricies
Must have same dimensions then add/subtract from same position in both
Inverse of a Matrix
[ : : ] - change sign of \
- switch /
Dependent system of equations
infinite solutions - same line
Independent system of equations
one solution - lines intersect
Consistent system of equations
lines that touch - not parallel
Inconsistant system of equations
parallel lines
Horizontal Asymptote
HA
Vertical Asymptote
occurs when denominator has an exponent that doesn’t cancel
Slant Asymptote
divide numerator by denominator?
Hole
occurs when a factor of the denominator cancels
Inflection Points
where a curve in a graph changes direction - f”(x)
Critical Points
where nature of graph changes - (max / min / inflection) f’(x)
Symmetry
Point, origin, line
First derivative of a graph
tells the max/min
Second derivative of a graph
tells the inflection points
- concave up or down
GOAL
Greater than - OR
Less than - AND
Used with absolute value inequalities
Even Function
Symmetric to y-axis
Odd Function
Symmetric to origin
YEOO
Y-axis - even
Origin - odd
Types of Graph Discontinuity
Infinite (Asymptotes)
Jump
Point
Fundamental Theorem of Algebra
Every polynomial with n degrees has n roots
Corollary
describes the relationship between roots and factors - if 2 is a root, (x-2) is a factor
Discriminant
“b^2 - 4ac” tells the # and type of roots
= positive - 2 real roots
= 1 - 1 real rood
= neg - 2 imaginary roots
Complex Conjugates Theorem
If (a + bi) is a root, then (a - bi) is also a root
Remainder Theorem
If a polynomial is divided by (x-r), the remainder is a constant, P(r)
Factor Theorem
A binomial (x-r) is a factor of a polynomial if there is NO remainder
Location Principal
If you plug 3 and 4 into a function and they are opposite in sign, there is a real ZERO between them
Rational Root Theorem
If you set a polynomial equal to 0
8x^3 … + 15 = 0 , then the rational roots are the last constant over the first constant. L/F
Present Value of an Annuity
(Loans) Pn = …
n = total # payments
i = interest / # payments per year
Future Value of an Annuity
(Savings) Fn = …
Interest Compounded Continuously
A = Pe^rt
Inverse of Log Function
Exponential Function
Log PRODUCT property
logmn = logm+logn
Log QUOTIENT property
logm/n = logm - logn
Log POWER property
logm^p = p logm
Log EQUALITY property
logm = logn, then n = m
Common Log
Log base 10
Natural log
Ln
Log base e
Arithmetic / Geometric
Add / Multiply
Convergent Series
infinite series that has a sum / limit
Geometric when |r|< 1 (decimal)
Divergent Series
infinite series that doesn’t have a sum / limit
All Arithmetic
Geometric when |r|>1
Binomial Theorem
choose statement
Derivative SUM rule
x’ + y’ (derive each piece separately)
Derivative PRODUCT rule
1st x D2nd + 2nd x D1st
Derivative CHAIN rule
D(outer) + D(inner)
Derivative QUOTIENT rule
LoDhi minus HiDLo all over the square of whats below
Definite Integral
area under f(x) from vertical lines x=a to x=b bound to the x-axis
(S with the #s)
Indefinite Integral
the general antiderivative of a function
(S without the #s)
Constant of Integration
+C
Fundamental Theorem of Calculus
Integration is the inverse of Differentiation (derivation)
Integration CONSTANT Rules
Skdx = kSdx + c
(move constant out front)
Integration x^-1 rule
Ln |x| + c
