1 | Sums, logarithms, derivatives Flashcards

1
Q

Terminology - name all parts of a summation

eg

Σ1 ≤ k ≤ n a

A

Σ = Sigma
1 = lower bound
k = index
n = upper bound
a = formula
a_k = term / summand

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2
Q

Summation for odd numbers

A

Σ0 ≤ k ≤ n (2k +1)

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3
Q

Summation for even numbers

A

Σ0 ≤ k ≤ n (2k)

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4
Q

True or false:

Σ1 ≤ k ≤ n ak = Σ0 ≤ k ≤ n-1 ak+1

A

True
(when in doubt — expand!)

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5
Q

True or false:

Σ1 ≤ k ≤ n ak1 ≤ k ≤ n ak + 1

A

False
(when in doubt — expand!)

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6
Q

True or false:

Σ2≤k≤n-1 k(k-1)(n-k) = Σ0≤k≤n k(k-1)(n-k)

A

True
(when in doubt — expand!)

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7
Q

True or false:

Σ0 ≤ k ≤ n-1 ak+1 = Σ1 ≤ k ≤ n ak + 1

A

False
(when in doubt — expand!)

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8
Q

True or false:

Σ1≤k≤n 2(k+1) = 2Σ1≤k≤n (k+1)

A

True
(Distributive law)

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9
Q

True or false:

Σ1≤k≤n (k+1) = Σ1≤k≤n k + Σ1≤k≤n 1

A

True
(Associative law)

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10
Q

Commutative law?

A

Σk∈K ak = Σp(k)∈K ak

𝑎1 + 𝑎2 + 𝑎3 = 𝑎2 + 𝑎1 + 𝑎3 = 𝑎3 + 𝑎1 + 𝑎2

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11
Q

Arithmetic progression - generic formula?

A

S = Σ0≤k≤n (a + bk)

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12
Q

Arithmetic progression - describe with words and a formula

A

an - an-1 = b

Whenever subtracting 2 adjacent terms, the solutions is a constant, b

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13
Q

Arithmetic progression - closed form solution

S = Σ0≤k≤n (a + bk)

A

(2a + bn)(n + 1)/2

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14
Q

∩ symbol?

A

Intercept - all elements that are common between two sets

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15
Q

∪ symbol?

A

Union - two sets combined

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16
Q

Manipulation of sums
Σk∈K ak + Σk∈K’ ak = ?

A

Σk∈K∩K’ ak + Σk∈K∪K’ ak

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17
Q

Example of perturbation method?

A

Σ0≤k≤n ak = a0 + Σ1≤k≤n ak

= operation of splitting off a term

18
Q

Geometric progression - generic formula

A

Sn = Σ0≤k≤n axk

19
Q

Geometric progression - describe with formula and words

A

an / an-1 = q

Whenever dividing 2 adjacent terms, the resulting ratio is always a constant, q

20
Q

Geometric progression

Sn = Σ0≤k≤n axk

closed form solution?

A

(𝑎 − 𝑎𝑥𝑛+1)/(1 − x)

21
Q

Σ1≤k≤nc = ?

22
Q

Σ1≤k≤nk

23
Q

Σ1≤k≤nk2

A

n(n+1)(2n+1)/6

24
Q

What is the rule for multiple sums?

Which sum is calculated first?

A

ΣP(j,k)ajbk

= Σj,kajbk I[P(j,k)]
= ΣjΣkajbkI[P(j,k)]

innermost sum first

25
Logarithms logbx = y x = ?
x = by
26
Logarithms Product rule
logbxy = logbx + logby
27
Logarithms logbxy = ?
= logbx + logby
28
Logarithms Quotient rule?
logbx/y = logbx - logby
29
Logarithms logbx - logby = ?
logbx/y
30
Logarithms Power rule?
logbxp = p*logbx
31
Logarithms logbxp
= p*logbx
32
Logarithms 1/p*logbx
logb(pth root of x)
33
Logarithms Change of base rule?
logbx = logkx / logkb
34
Derivatives f(x) = ex f'(x) = ?
ex
35
Derivatives f(x) = ax
axln(a)
36
Derivatives f(x) = ln(x) f'(x) = ?
1/x
37
Derivatives f(x) = logax f'(x) = ?
1/xln(a)
38
Derivatives f(x) = ag(x) + bh(x) f'(x) = ?
ag'(x) + bh'(x)
39
Derivatives f(x) = g(x)*h(x) f'(x) = ?
g'(x)h(x) - g(x)h'(x)
40
Derivatives f(x) = g(x)/h(x) f'(x) = ?
[ g'(x)h(x) - g(x)h'(x) ] / (h(x))2
41
Derivatives f(x) = h(g(x)) f'(x) = ?
h'(g(x))g'(x)