1 | Sums, logarithms, derivatives

Question 1

Terminology - name all parts of a summation

eg

Σ_{1 ≤ k ≤ n} a

Answer 1

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

Question 2

Summation for odd numbers

Answer 2

Σ_{0 ≤ k ≤ n} (2k +1)

Question 3

Summation for even numbers

Answer 3

Σ_{0 ≤ k ≤ n} (2k)

Question 4

True or false:

Σ_{1 ≤ k ≤ n} a_k = Σ_{0 ≤ k ≤ n-1} a_k+1

Answer 4

True
(when in doubt — expand!)

Question 5

True or false:

Σ_{1 ≤ k ≤ n} a_k =Σ_{1 ≤ k ≤ n} a_k + 1

Answer 5

False
(when in doubt — expand!)

Question 6

True or false:

Σ_2≤k≤n-1 k(k-1)(n-k) = Σ_0≤k≤n k(k-1)(n-k)

Answer 6

True
(when in doubt — expand!)

Question 7

True or false:

Σ_{0 ≤ k ≤ n-1} a_k+1 = Σ_{1 ≤ k ≤ n} a_k + 1

Answer 7

False
(when in doubt — expand!)

Question 8

True or false:

Σ_1≤k≤n 2(k+1) = 2Σ_1≤k≤n (k+1)

Answer 8

True
(Distributive law)

Question 9

True or false:

Σ_1≤k≤n (k+1) = Σ_1≤k≤n k + Σ_1≤k≤n 1

Answer 9

True
(Associative law)

Question 10

Commutative law?

Answer 10

Σ_k∈K a_k = Σ_p(k)∈K a_k

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

Question 11

Arithmetic progression - generic formula?

Answer 11

S = Σ_0≤k≤n (a + bk)

Question 12

Arithmetic progression - describe with words and a formula

Answer 12

a_n - a_n-1 = b

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

Question 13

Arithmetic progression - closed form solution

S = Σ_0≤k≤n (a + bk)

Answer 13

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

Question 14

∩ symbol?

Answer 14

Intercept - all elements that are common between two sets

Question 15

∪ symbol?

Answer 15

Union - two sets combined

Question 16

Manipulation of sums
Σ_k∈K a_k + Σ_k∈K’ a_k = ?

Answer 16

Σ_k∈K∩K’ a_k + Σ_k∈K∪K’ a_k

Question 17

Example of perturbation method?

Answer 17

Σ_0≤k≤n a_k = a₀ + Σ_1≤k≤n a_k

= operation of splitting off a term

Question 18

Geometric progression - generic formula

Answer 18

S_n = Σ_0≤k≤n ax^k

Question 19

Geometric progression - describe with formula and words

Answer 19

a_n / a_n-1 = q

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

Question 20

Geometric progression

S_n = Σ_0≤k≤n ax^k

closed form solution?

Answer 20

(𝑎 − 𝑎𝑥^𝑛+1)/(1 − x)

Question 21

Σ_1≤k≤nc = ?

Answer 21

nc

Question 22

Σ_1≤k≤nk

Answer 22

n(n+1)/2

Question 23

Σ_1≤k≤nk²

Answer 23

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

Question 24

What is the rule for multiple sums?

Which sum is calculated first?

Answer 24

Σ_P(j,k)a_jb_k

= Σ_j,ka_jb_k I[P(j,k)]
= Σ_jΣ_ka_jb_kI[P(j,k)]

innermost sum first

Question 25

Logarithms

log_bx = y

x = ?

Answer 25

x = b^y

Question 26

Logarithms

Product rule

Answer 26

log_bxy

= log_bx
+ log_by

Question 27

Logarithms

log_bxy
= ?

Answer 27

= log_bx
+ log_by

Question 28

Logarithms

Quotient rule?

Answer 28

log_bx/y

= log_bx
- log_by

Question 29

Logarithms

log_bx
- log_by = ?

Answer 29

log_bx/y

Question 30

Logarithms

Power rule?

Answer 30

log_bx^p

= p*log_bx

Question 31

Logarithms

log_bx^{p</sup =?}

Answer 31

= p*log_bx

Question 32

Logarithms

1/p*log_bx

Answer 32

log_b(pth root of x)

Question 33

Logarithms

Change of base rule?

Answer 33

log_bx

= log_kx / log_kb

Question 34

Derivatives

f(x) = e^x

f’(x) = ?

Answer 34

e^x

Question 35

Derivatives

f(x) = a^x

Answer 35

a^xln(a)

Question 36

Derivatives

f(x) = ln(x)

f’(x) = ?

Answer 36

1/x

Question 37

Derivatives

f(x) = log_ax

f’(x) = ?

Answer 37

1/xln(a)

Question 38

Derivatives

f(x) = ag(x) + bh(x)

f’(x) = ?

Answer 38

ag’(x) + bh’(x)

Question 39

Derivatives

f(x) = g(x)*h(x)

f’(x) = ?

Answer 39

g’(x)h(x) - g(x)h’(x)

Question 40

Derivatives

f(x) = g(x)/h(x)

f’(x) = ?

Answer 40

[ g’(x)h(x) - g(x)h’(x) ] / (h(x))²

Question 41

Derivatives

f(x) = h(g(x))

f’(x) = ?

Answer 41

h’(g(x))g’(x)