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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