Math Tutorial

Lecture 2

Some of these slides are courtesy of D. Plaisted, UNC and M. Nicolescu, UNR

Exponentials – What are they?

• Simply: a number (base) raised to a power (exponent).

• Simply: a number (base) raised to a power (exponent).

• Easy enough to calculate:

• ab = a × a × a … × a • So there are ‘b’ lots of ‘a’• What is 43?

• 4 × 4 × 4 = 64

Exponentials – A quick explanation

b

Exponentials in Computer Science

• Have you ever noticed the following numbers popping up in your computer science studies?

• 1, 2, 4, 8, 16, 32, 64, 128…1024…• What do they all have in common?• They are all powers of 2!!

Exponentials in Computer Science

• The powers of 2 should be (very) familiar to you by now.

• All the computers you have been using work in binary bits (at the lowest level)…

• 2 values: 0/1, true/false etc• all ‘data sizes’ must be expressed in powers of

2.

Exponentials in Computer Science

• 1 kilobyte ≠ 1000 bytes (‘standard’ use of kilo)• 1 kilobyte = 1024 bytes• Because…• 29 = 512• 210 = 1024• 211 = 2048• So 210 is the closest we can get to 1000

Working with Exponentials

• First, some easy exponentials to remember:

• a0 = 1

• a1 = a

Exponentials

• Useful Identities:

nmnm

mnnm

aaa

aaa

a

)(

11

Logarithms

• This is because log(arithm)s are just ‘reversed’ exponentials

• E.g. if 24 = 16• log216 = 4

– base is 2• Logarithms ‘map’ large numbers onto smaller

numbers

Logarithms - Bases

• There are several common bases:– 10: very common base, Richter scale etc.– e: used by a lot of scientists– 2: very common in computer science. WHY?

Logarithms – An example

• Imagine US open: 8 players left.• It is a knockout tournament, so every time that 2 players

play the losing player is eliminated from the tournament and the winning plays goes on to the next round.

• How many rounds must be played to determine an overall winner?

Logarithms – An example

• Imagine US open: 8 players left.• It is a knockout tournament, so every time that 2 players

play the losing player is eliminated from the tournament and the winning plays goes on to the next round.

• How many rounds must be played to determine an overall winner?

• What stage: semi-final? quarterfinal? 1/8? 1/16?

Logarithms – An example

Round 0: 8 players

Round 1: 4 players

Round 2: 2 players

Round 3: 1 player

3 rounds needed!

Logarithms – Example explained

• 3 rounds are needed to determine the winner of 8 teams, competing 2 at a time (i.e. one-on-one)

• This can be easily calculated using logs.• 2 teams play at a time, so the base is 2. (i.e. 2x = 8, so

we need to use log2)• So, log2 8 = ….• 3

Logarithms• In algorithm analysis we often use the notation

“log n” without specifying the base

nnnn

eloglnloglg 2

yxlog

Binary logarithm

Natural logarithm )log(logloglog)(loglog

nnnn kk

xy logxylog yx loglog

yxlog yx loglog

xalog xb ba loglog

abx logxba log

Logarithms and exponentials – Bases

• If the base of a logarithm is changed from one constant to another, the value is altered by a constant factor.– Ex: log10 n * log210 = log2 n.– Base of logarithm is not an issue in asymptotic notation.

• Exponentials with different bases differ by a exponential factor (not a constant factor).– Ex: 2n = (2/3)n*3n.

Some Simple Summation Formulas• Arithmetic series:

• Geometric series:

– Special case: x < 1:

• Harmonic series:

• Other important formulas:

2)1( nn

n

k

nk1

...21

1111

xxxn

nn

k

k xxxx ...1 2

0

x11

0k

kx

nln

n

k nk1

1...2111

n

k

k1

lg nn lg

1

11

pn

p

n

k

pppp nk1

...21

Limit• Def (informal): a limit is the intended height of a function

x

f(x)

3

x

g(x)4

Parabola• Different x values have different height (f(x) value)• Function changes its height for x within the x domain

f(x) = x2

f(1) = 1

f(2) = 4

4)(2lim

xfx

f(x)

When x is getting close to the value of 2the function f(x) getting close to 4

x

f(x)

2

-2 2)(lim00

2282*62)2(

286)(

2

2

2

xf

f

xxxxf

x

f(2) is not defined but limit for f(2) exists

Limit: Formal Definition

Karl Weierstrass formally defined a limit as follows (epsilon-delta definition):

Let f be a real-valued function defined on an open interval of real numbers containing c (except possibly at c) and let L be a real number. Then

means that

for each real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x − c| < δ, we have |f(x) − L| < ε.

Lxfcx

)(lim

c

xc

Whenever a point x is within δ units of c, f(x) is within ε units of L

When does a Limit Exist?

)(lim)(lim)(lim222

xfxfifexistsxfxxx

x

f(x)

2

Right-hand limit(limit from above)

Left-hand limit(limit from below)

5.1)(lim

3)(lim

2

2

xf

xf

x

x

x

f(x)

2

x

f(x)

2

Limit does not exist for x=2, but exists for other x’s (for example, for x=1)

Limit exists for x=2 even if f(x) is not defined for x=2

How to Find/Evaluate a Limit. 1

• Substitution

31

30*210)(lim

321)(

2

0

2

xf

xxxf

x

How to Find/Evaluate a Limit. 2

• Factoring

242)(lim

42

)2)(4(2

86)(

2

2

xf

xxxx

xxxxf

x

How to Find/Evaluate a Limit. 3

• Conjugate method

81

41lim)(lim

)4(*)16(16

44*

164)(

?)(lim16

4)(

1616

16

xxf

xxx

xx

xxxf

xfxxxf

xx

x

Limits and Infinity (lim and ∞)

)(lim

)(lim

4

4

xf

xf

x

x

Non-zero number divided by 0 vertical asymptote

Asymptote cannot be reached

Limits and Infinity (lim and ∞)

)(lim

)(lim

4

4

xf

xf

x

x

Non-zero number divided by 0 vertical asymptote

Asymptote cannot be reached

x=-4 is a vertical asymptote

Limits and Infinity (lim and ∞)

3)(lim

3)(lim

xf

xf

x

x

y=3 is a horizontal asymptote

Limit at ∞• x is getting infinitely large limit at ∞• To calculate a limit at infinity, compare the degrees in the top and

bottom of the fraction (in general case, the top and the bottom of the fraction)

277534)(

276542)(

276544)(

)()()(

3

23

35

23

2

23

xxxxxf

xxxxxxf

xxxxxf

xbxaxf

74)(lim

0)(lim

)(lim

xf

xf

xf

x

x

x

Degree a(x) > degree b(x) There is no horizontal asymptote

Degree a(x) < degree b(x) Horizontal asymptote: y=0

Compare the degree of the nominator a(x) and the degree of the denominator b(x)

Degree a(x) = degree b(x).The ratio of the coefficiencies of the highest terms

Limit and ∞• Limit equals ∞ function increases infinitely (∞ is not

number)• Vertical asymptote• Horizontal asymptote• L’Hopital’s Rule

32

Monotonicity

• f(n) is – monotonically increasing if m n f(m) f(n).– monotonically decreasing if m n f(m) f(n).– strictly increasing if m < n f(m) < f(n).– strictly decreasing if m > n f(m) > f(n).

33

Exponentials

• Useful Identities:

• Exponentials and polynomials

nmnm

mnnm

aaa

aaa

a

)(

11

)(

0lim

nb

n

b

n

aonan