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ISYE 2028 A and B
Notation and Prerequisites
Dr. Kobi Abayomi
January 8, 2009
1 Introduction
Just a brief review of some notation you should know and some short questions.
We often use roman letters {X, Y, x, y...} for things we hope to measure or model; greek letters {α, β, θ} for quantities we’ll infer from directly measured quantities.
2 Summation Notation
We work with indexed vectors alot, like:
X = (X 1 , X 2 , ..., Xn)
or
x = (x 1 , x 2 , ..., xn)
or any
stuf f = (stuf f 1 , stuf f 2 , ..., stuf fn)
∑^ n
i=
stuf fi = stuf f 1 + stuf f 2 + · · · + stuf fn
is translated as: Start with stuf f 1 and add it to stuf f 2 and keep on adding until stuf fn. The stuf f to do (n times) can be as simple to do as taking a bunch of numbers {x 1 , x 2 , ...x 3 } and dividing it by the total number, i.e.
∑^ n
i=
xi n
n
∑^ n
i=
xi =
x 1 + x 2 + · · · + xn n
or something more complicated, like taking each of those numbers, subtracting some other number, squaring the result, and dividing that by n − 1
∑^ n
i=
(xi − μ)^2 n − 1
2.1 Product notation
Just like summation, but the indexed items are separated by multiplication in place of addition
∏^ n
i=
xi = x 1 · x 2 · · · xn
3 Integration, differentiation, exp, log...
...of simple functions should all be familiar from calculus. Principally, remember that inte- gration and differentiation are linear operators meaning you can integrate and differentiate — usually – across sums.
3.1 exp, log
You might have forgotten this important result about the derivative of the inverse of a function f , (f −^1 )′:
(f −^1 (x))′^ =
f ′(f −^1 )(x)
3.2 integrate, differentiate
We will look at multiple integration some time in the future. Remember that you can usually switch the order of integration without too much trouble.
∫ (^) b
a
∫ (^) d
c
f (x)g(y)dydx = · · ·
∫ (^) b
a
(f (x)
∫ (^) d
c
g(y)dy)dx = · · · ∫ (^) d
c
g(y)dy
∫ (^) b
a
f (x)dx
You should also remember the very important substitution method for integration, with u = g(x).
∫ f (u)du =
f (u)
du dx
dx
And also integrate by parts
∫ (^) b
a
f g′^ = f g|ba −
∫ (^) b
a
f ′g
which might look more familiar to you if you let u = f and v = g.
4 Suggested Exercises
∑n i=1 c^ ·^ i^ =?, when^ c^ is a constant.
i=0 r i (^) =?, when |r| < 1.
∏n i=1 c^ ·^ e
xi (^) , when x = (x 1 , ..., xn) and c is a constant.
- If f (x) = aλx^ then f ′(x) =?, if λ is a constant.
- If g(x) = loga(λx) then g′(x) =?, if λ is a constant.
- The so-called Taylor series for a function f at a, is:
f (x) =
∑^ ∞
i=
f (i)(a) i!
(x − a)i
where f (i)^ indicates the ith derivative of f and i! = i · (i − 1) · (i − 2) · · · 1. Find the taylor series for f (x) = ex^ at a = 0.
- (i) Use integration by parts to evaluate
0 e
−ttk− (^1) dt. (ii) Use substitution to evaluate
0 e
−ste−λtdt and express as a function of s, if λ is a constant.
