1. Limiting Behaviour of Functions Flashcards
List the three levels of approach to determining the limiting behaviour of a function
A1) minimally (as a single value) - what is the value or limit of the function
A2) qualitatively (single term in x) how does the function approach its limit
A3) quantitatively (e.g. several terms in decreasing powers of x or 1/x) - how can we usefully approximate the function locally
Binomial Series
(1+t)^n
= 1 + nt + n(n-1)/2! t² +
n(n-1)(n-2)/3! t³ +…
for |t|<1
When is the binomial series valid?
-for small t
(1+t)^n = …
Taylor’s Theorem
-suppose f is n+1 times differentiable and f^(n+1) is continuous
-then Taylor’s theorem centred at xo∈ℝ is:
f(x) = f(xo) + f’(x0)(x-xo) +
f’‘(xo)/2! (x-xo)² + … +
f’n(xo)/n! (x-xo)^n + Rn(x)
-where Rn(x) is the remainder term
Taylor’s Theorem
Remainder in Lagrange Form
Rn(x) =
f’n+1(c)/(n+1)! [x-xo]^(n+1)
-with c being an unknown number between xo and x
-the remainder can’t be evaluated but it can be bounded which is useful in various contexts
Convergent Taylor Series
-if f is infinitely differentiable and if the remainder term vanishes as N->∞, then we obtain a convergent Taylor Series representation of f centred at xo: f(x) = f(xo) + f'(x0)(x-xo) + f''(xo)/2! (x-xo)² + ... = Σ f'n(xo)/n! (x-xo)^n -where the sum is from n=0 to n=∞
Radius of Convergence
-convergence of a Taylor series typically only happens for |x-xo|
Taylor Series for sin(x)
sin(x) = x - x³/6 + x^5/120 +…
= Σ[(-1)^n x^(2n+1)]/(2n+1)!
-sum from n=0 to n=∞
-infinite radius of convergence, valid for x∈ℝ
Taylor Series for cos(x)
cos(x) = 1 - x²/2 + x^4/24 + …
= Σ[(-1)^n x^2n]/(2n)!
-sum from n=0 to n=∞
-infinite radius of convergence, valid for x∈ℝ
Taylor Series for exp(x)
exp(x) = x - x²/2 + x³/3 + …
= Σ x^n/n!
-sum from n=0 to n=∞
-valid for x∈ℝ
Taylor Series for log
log(1+x) = x - x²/2 + x³/3 + …
= Σ[(-1)^(n+1) x^n] / n
-sum from n=1 to n=∞
-valid for |x|<1
L’Hopital’s Rule
if lim f(xo) = 0 & lim g(xo) = 0
OR
lim |f(xo)| = ∞ & lim |g(xo)| = ∞
THEN
lim f(x)/g(x) = lim f’(x)/g(x)
-where limits are taken as x->xo
When is l’hopital’s rule not the best choice?
- even if it is possible to apply l’hopital’s rule, it is not always the best choice
- sometimes a loop results
- inserting the relevant Taylor series expansions can be much more efficient
Implicit Representation
-functions are often defined implicitly:
f(x) + log(f(x)) = x² + 1
rather than f(x) = …
-we can still approximate these functions locally
-suggest a form for the function
-balance coefficients
Dominant Balance
-a technique that can be used when the form of an expansion is unknown
-assume the function takes the form:
δ1(ε) + δ2(ε) + δ3(ε) + …, ε«1
-precise dependence of δ on ε is unspecified, we only assume that the terms are decreasing:
1»_space; δ1»_space; δ2»_space; δ3»_space; …. when ε«1
