Mathematical Review — Full Definitions for Teaching
Contents
1 Laws of Exponents
1.1 Exponent Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Radical (Root) Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
2
2 Logarithms
3
3 Series
3.1 Arithmetic Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Geometric Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
3
4 Limits
4.1 Definition of a Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Left-Hand and Right-Hand Limits . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Limit Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Limits at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Infinite Series & Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
4
4
4
4
4
5 Modular Arithmetic
5
6 Proof Methods
6.1 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Proof by Contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
5
5
7 Intro to Calculus
7.1 Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Power Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Antiderivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . .
7.5 Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
6
6
6
6
1
1
Laws of Exponents
1.1
Exponent Rules
Definition (Exponent): For a real number a and integer n > 0,
an = |a · a ·{z· · · · a}
n times
For n = 0, a0 = 1 (if a ̸= 0). For negative n,
a−n =
Exponent Laws:
1. Zero Exponent Rule: a0 = 1
2. Product of Powers: an · am = an+m
3. Quotient of Powers:
an
= an−m
am
4. Power of a Power: (an )m = anm
5. Power of a Product: (ab)n = an bn
a n an
6. Power of a Quotient:
= n
b
b
√
7. Rational Exponents: am/n = n am
8. Negative Exponents: a−n =
1.2
1
an
Radical (Root) Rules
√
n
1. ( an )n = a
√
√ √
2. n ab = n a · n b
r
√
n
a
a
n
3.
= √
n
b
b
√
√
4. n am = ( n a)m
p√
p√
√
5. m n a = mn a = n m a
2
1
.
an
2
Logarithms
Definition:
loga b = x means ax = b
Properties:
loga (bc) = loga b + loga c
b
loga
= loga b − loga c
c
loga (bk ) = k loga b
Special logarithms: ln x = loge x (base e).
3
Series
3.1
Arithmetic Sequences and Series
Arithmetic Sequence: A sequence with constant difference d:
an = a1 + (n − 1)d
Arithmetic Series: Sum of first n terms:
n
Sn = (a1 + an )
2
3.2
Geometric Sequences and Series
Geometric Sequence: Each term is multiplied by r:
an = a1 rn−1
Geometric Series: Sum of first n terms:
1 − rn
,
Sn = a1
1−r
r ̸= 1
Infinite Geometric Series: If |r| < 1,
S∞ =
4
Limits
4.1
Definition of a Limit
a1
1−r
lim f (x) = L
x→a
If we can make f (x) as close to L as we want by choosing x sufficiently close to a.
Epsilon-delta definition: For every ε > 0, there exists δ > 0 such that
0 < |x − a| < δ =⇒ |f (x) − L| < ε
3
4.2
Left-Hand and Right-Hand Limits
lim f (x),
lim f (x)
x→a−
x→a+
Limit exists if:
lim f (x) = lim+ f (x)
x→a−
4.3
x→a
Continuity
f (x) continuous at a if:
1. f (a) exists
2. limx→a f (x) exists
3. limx→a f (x) = f (a)
4.4
Limit Laws
If lim f (x) and lim g(x) exist:
lim(f + g) = lim f + lim g
lim(f g) = (lim f )(lim g)
lim f
f
, lim g ̸= 0
=
lim
g
lim g
lim(cf ) = c · lim f
4.5
Limits at Infinity
lim f (x) = L
x→∞
Example:
1
=0
x→∞ x
lim
4.6
Infinite Series & Limits
An infinite series
P∞
n=1 an converges if
Sn =
n
X
ak
k=1
has a finite limit:
lim Sn = S
n→∞
Otherwise, the series diverges.
4
5
Modular Arithmetic
Two integers a and b are congruent modulo n if
a≡b
(mod n) ⇐⇒ n | (a − b)
Meaning a and b have the same remainder when divided by n.
Properties:
1. If a ≡ b (mod n) and c ≡ d (mod n), then
a+c≡b+d
(mod n),
ac ≡ bd
(mod n)
2. Modular arithmetic is compatible with addition, subtraction, and multiplication.
3. Division requires care: a ≡ b (mod n) does not always imply a/c ≡ b/c (mod n) unless
c and n are coprime.
6
Proof Methods
6.1
Mathematical Induction
Prove statement for all n ≥ n0 :
1. Base case: prove for n = n0
2. Inductive hypothesis: assume true for n = k
3. Inductive step: prove true for n = k + 1
6.2
Counterexample
To disprove “for all” statements, provide a single counterexample.
6.3
Proof by Contradiction
Assume the negation of the statement. If it leads to a contradiction, the original statement
is true.
7
Intro to Calculus
7.1
Derivative
f (x + h) − f (x)
h→0
h
Instantaneous rate of change / slope of tangent.
f ′ (x) = lim
5
7.2
Power Rule
If f (x) = xn , then
f ′ (x) = nxn−1
7.3
Antiderivative
F (x) is an antiderivative of f (x) if F ′ (x) = f (x):
Z
f (x) dx = F (x) + C
7.4
Fundamental Theorem of Calculus
Part 1:
d
dx
Part 2:
Z x
f (t) dt = f (x)
a
Z b
f (x) dx = F (b) − F (a)
a
7.5
Definite Integral
Z b
f (x) dx = lim
a
n→∞
6
n
X
k=1
f (x∗k )∆x