Analysis 1B

Let D⊆R and c∈R. We say that D is a punctured neighbourhood of c if…

… ∃δ₀ > 0 s.t. (c-δ₀, c)∪(c, c+δ₀) ⊆ D.

Let f: D→R where D is a punctured neighbourhood of c. We say that lim_x→c f(x) = L if…

… ∀ε>0 ∃δ>0 s.t. ∀x∈D 0<|x-c|<δ => |f(x)-L| < ε

(< 2ε gives an equivalent statement)

Let f: D→R where D is a punctured neighbourhood of c.

We say that lim_x→c f(x) exists if…

… ∃L∈R s.t. lim_x→c f(x) = L

Let f: D→R with D a punctured neighbourhood of c.

If L₁, L₂∈R, lim_x→c f(x) = L₁ and lim_x→c f(x) = L₂, then…

… L₁ = L₂

Let f: D→R where D is a punctured neighbourhood of c.

Then the following are equivalent…

… a) lim_x→c f(x) = L

b) for any sequence(x_n)_n∈N, if
∀n x_n∈D, x_n→c as n→∞, ∀n x_n≠c
then f(x_n)→L as n→∞

Let f:D→R, with D, a punctured neighbourhood of c. Let L,M∈R. If lim_x→c f(x) = L and L>M then…

… ∃δ>0 ∀x∈D 0<|x-c|<δ => f(x) > M

Let f:D→R, with D, a punctured neighbourhood of c.

Assume ∀x∈D, f(x)≤M. If lim_x→c f(x) = L exists then…

… L≤M.

Let f,g:D→R with a punctured neighbourhood of c.
Assume that lim_x→c f(x) = L_f and lim_x→c g(x) = L_g.

Then lim_x→c (f+g)(x) = ?

L_f + L_g

Let f,g:D→R with a punctured neighbourhood of c.
Assume that lim_x→c f(x) = L_f and lim_x→c g(x) = L_g.

If L∈R, then lim_x→c (αf)(x) = ?

αL_f

Let f,g:D→R with a punctured neighbourhood of c.
Assume that lim_x→c f(x) = L_f and lim_x→c g(x) = L_g.

The lim_x→c (fg)(x) = ?

L_fL_g

Let f,g:D→R with a punctured neighbourhood of c.
Assume that lim_x→c f(x) = L_f and lim_x→c g(x) = L_g.

If L_g ≠ 0, then lim_x→c (f/g)(x) = ?

L_f / L_g

Let p(x) be a real polynomial, p:R→R of the form p(x) = a₀ + a₁x + … + aₙxⁿ where a₀,…,aₙ∈R.

Then for any c∈R, lim_x→c p(x) = ?

p(c)

Let f:D→R, D⊆R, D≠0. Let c,L∈R.

Assume that ∃δ₀ > 0 s.t. (c, c+δ₀) ⊆ D.

Then we say lim_x→c⁺ f(x) = L iff…

… ∀ε>0 ∃δ>0 ∀x∈D c<x<c+δ => |f(x) - L| < ε

Let f:D→R, D⊆R, D≠0. Let c,L∈R.

Assume that ∃δ₀ > 0 s.t. (c-δ₀, c) ⊆ D.

Then we say lim_x→c^- f(x) = L iff…

… ∀ε>0 ∃δ>0 ∀x∈D c-δ<x<c => |f(x) - L| < ε

Let f:D→R with D a punctured neighbourhood of c∈R.

Let L∈R. Then lim_x→c f(x) = L iff…

… lim_x→c⁺ f(x) = L and lim_x→c^- f(x) = L

Let f:D→R and c∈D. We say f is continuous at c if…

… ∀ε>0 ∃δ>0 ∀x∈D |x-c|<δ => |f(x) - f(c)| < ε

We say f is continuous on D if…

… it is continuous at all points on D.

Let f:D→R and c∈D. Then “f is continuous at c” is equivalent to…

… for any sequence (x_n)_n∈N, if ∀n x_n∈D and lim_n→∞ x_n = c, then lim_n→∞ f(x_n) = f(c)

Let x₀∈R. Then ∃ a sequence (g_n)_n∈N s.t. ∀n g_n∈Q, …

… lim_n→∞ g_n = x₀

Let x₀∈R. Then ∃ a sequence (i_n)_n∈N s.t. ∀n i_n∈R\Q …

… lim_n→∞ i_n = x₀

Let f:D→R and c∈D. Assume that c∈D and D is a punctured neighbourhood of c.

f is continuous at c is equivalent to…

… lim_x→c f(x) = f(c)

Let a<b and f:[a,b]→R.

Then f is continuous at a iff…

… lim_x→a⁺ f(x) = f(a)

Let a<b and f:[a,b]→R.

Then f is continuous at b iff…

… lim_x→b^- f(x) = f(b)

Let f,g: D→R and c∈D. Assume f and g are continuous at c. Then…

…
f+g is continuous at c
if α∈R, then αf is continuous at c
f-g is continuous at c
if g(c) ≠ 0, then f/g is continuous at c

Polynomials are…

… continuous on R.

For x∈R, we define exp(x) = ?

∑^∞_n=0 xⁿ/n! = 1 + x + x²/2 +…

(or exp(x) = lim_n→∞ (1+x/n)ⁿ)

For x∈R, we define sin(x) = ?

∑^∞_n=0 (-1)ⁿ[x²ⁿ⁺¹/(2n+1)!] = x - x³/3! + x⁵/5! -…

For x∈R, we define cos(x) = ?

∑^∞_n=0 (-1)ⁿ[x²ⁿ/(2n)!] = 1 - x²/2! + x⁴/4! -…

The functions exp, cos, sin: R→R are…

… continuous

Let g:D→R, f:E→R, where D,E⊆R, be s.t. g(x) ∈E ∀x∈D.

Then f_^og is defined as…

… the function f_^og: D→R, (f_^og)(x) = f(g(x)).

Let g:D→R, f:E→R, where D,E⊆R, be s.t. g(x) ∈E ∀x∈D. Let c∈D.

Assume that g is continuous at c and f is continuous at g(c). Then f_^og…

… is continuous at c.

Let a < b and f:[a,b]→R be continuous on [a,b].

If f(a) ≤ f(b), then ∀ f(a) ≤ y ≤ f(b)…

… ∃c∈[a,b] s.t. f(c) = y.

(Same if you flip the inequalities)

Let f:D→R where D⊆R, D≠∅.

f is bounded above if…

… ∃M∈R s.t. ∀x∈D f(x) ≤ M

Let f:D→R where D⊆R, D≠∅.

f is bounded below if…

… ∃m∈R s.t. ∀x∈D f(x) ≥ m

Let f:D→R where D⊆R, D≠∅.

f is bounded if…

… if it is bounded above and below.

Let f:D→R.

f is not bounded above iff…

… ∃ a sequence (x_n)_n∈N with ∀x∈D and f(x_n)→∞ as n→∞.

Let f:D→R (D⊆R, D≠∅).

If f is bounded above we define supf = ?

sup{f(x)|x∈D}

Let f:D→R (D⊆R, D≠∅).

If f is bounded above we define inff = ?

inf{f(x)|x∈D}

Let f:D→R (D⊆R, D≠∅) be bounded. We say f attains its supremum if…

… ∃p∈D s.t. supf = f(p), i.e. iff ∃p∈D s.t. ∀x∈D f(x) ≤ f(p)

If f attains its supremum, we write supf = maxf

Let f:D→R (D⊆R, D≠∅) be bounded. We say f attains its infimum if…

… ∃q∈D s.t. inff = f(q) iff ∃q∈D s.t. ∀x∈D f(x) ≥ f(q)

If f attains its infimum, we write inff = minf

The Weierstrass Extremal Value Theorem

Let a<b and let f:[a,b]→R be continuous on [a,b].

Then f attains its bounds on [a,b], that is, there exists p,q∈[a,b] s.t. ∀x∈[a,b] f(q) ≤ f(x) ≤ f(p).

An interval is…

… a set of the form: [a,b], (a,b), [a,b), (a,b], [a,∞), (a,∞), (-∞,b], (-∞,b), (-∞,∞)

Let I⊆R be an interval and f:I→R.

We say that f is strictly increasing if…

… ∀x₁,x₂∈I x₁<x₂ => f(x₁) < f(x₂)

Let I⊆R be an interval and f:I→R.

We say that f is strictly decreasing if…

… ∀x₁,x₂∈I x₁<x₂ => f(x₁) > f(x₂)

Let I⊆R be an interval and f:I→R.

We say that f is monotone…

… if it is strictly increasing or decreasing.

Let f:I→J be bijective.

We define f^-1:J→I, y∈J, as…

…f^-1(y) = x where x∈I is the unique element s.t. f(x) = y.

Let I≠∅ be an interval and f:I→R be continuous on I and monotone.

Then what three things are true?

J = f(I) is an interval

f: I→J is bijective

f^-1: J→I is strictly monotone and continuous on J

Let I,J⊆R be intervals and f:I→J be bijective.

If f is strictly increasing…

… then f^-1: J→I is strictly increasing.

(and similarly for strictly decreasing)

Let (a_n)_n∈N be a sequence and L∈R. Assume a_n does not tend to L as n→∞.

Then…

… ∃c>0 and a subsequence (aₙ_k)_k∈N s.t. ∀k∈N |aₙ_k - L| ≥ ε.

Let a<b and f:(a,b)→R and let c∈(a,b). We say that f is differentiable at c if…

… lim_h→0 [f(c+h)-f(c)]/h exists.
When this limits, we write f'(c) = lim_h→0 [f(c+h)-f(c)]/h = the derivate of f at c.

Let a<b and f:(a,b)→R. We say f is differentiable in (a,b) if…

… f is differentiable for all c∈(a,b).

By change of variable, lim_h→0 [f(c+h)-f(c)]/h can be written as…

… lim_x→c [f(x)-f(c)]/x-c if you let x = c+h

Let f:(a,b)→R be differentiable at c∈(a,b). Then f is…

… continuous at c.

(This is a one-way implication)

Let f,g:(a,b)→R be differentiable at c∈(a,b).

Then f+g is…

… differentiable at c and (f+g)'(c) = f'(c)+g'(c)

Let f,g:(a,b)→R be differentiable at c∈(a,b).

Then for α∈R, αf is…

… differentiable at c and (αf)'(c) = αf'(c)

Let f,g:(a,b)→R be differentiable at c∈(a,b).

Then fg is…

… differentiable at c and (fg)'(c) = f'(c)g(c) + f(c)g'(c)

Let f,g:(a,b)→R be differentiable at c∈(a,b).

If g(c) ≠ 0, then 1/g is…

… differentiable at c and (1/g)'(c) = - g'(c)/g²(c)

If f,g:(a,b)→R are differentiable at c∈(a,b) and g(c)≠0, then f/g is…

… differentiable at c and (f/g)'(c) = [f'(c)g(c) - f(c)g'(c)]/g²(c)

If p(x) = a₀ + a₁x + … + aₙxⁿ is a polynomial, then for any c∈R p(x) is…

… differentiable at c and p'(c) = a₁ + 2a₂x + … + naₙx^n-1

exp(x), sin(x) and cos(x) are…

… differentiable at all c∈R and exp'(c) = exp(c), sin'(c) = cos(c) and cos'(c) = -sin(c)

Let f:(a,b)→R and y₀∈(a,b).

Then f is differentiable at y₀ and f'(y₀) = L is equivalent to…

… ∃g:(a,b)→R continuous at y₀ s.t. ∀y∈(a,b) f(y)-f(y₀) = g(y)(y-y₀) where g(y₀) = L.

If c∈(a,b), g is differentiable at c and f is differentiable at c, then f_^og…

… is differentiable at c and (f_^og)'(c) = f'(g(c))g'(c)

Let f:(a,b)→R be differentiable at x₀∈(a,b).

We define the tangent line to the graph of f at (x₀, f(x₀)) by…

… y = f'(x₀)(x-x₀) + f(x₀)

Let f:(a,b)→(α,β) be bijective. Let c∈(a,b).
Assume f is differentiable at c, f'(c)≠0 and f^-1 is continuous at f(c).

Then f^-1(c) is…

… differentiable at f(c) and (f^-1(c))'(f(c)) = 1/f'(c)

Let I be an interval. Let p∈I. Let f:I→R.

We call p a maximum point of f if…

… ∀x∈I f(x) ≤ f(p)

(equivalently f(p) = supf)

Let I be an interval. Let q∈I. Let f:I→R.

We call q a minimum point of f if…

… ∀x∈I f(x) ≥ f(q)

(equivalently f(q) = inff)

Let I⊆R be an interval, f:I→R and p∈I.

We call p a local maximum of f if…

… ∃δ>0 s.t. ∀x∈I∩(p-δ, p+δ) f(x) ≤ f(p)

Let I⊆R be an interval, f:I→R and q∈I.

We call q a local minimum of f if…

… ∃δ>0 s.t. ∀x∈I∩(q-δ, q+δ) f(x) ≥ f(q)

Let f:(a,b)→R have a local max or local min at c∈(a,b).

If f is differentiable at c, then f'(c) = ?

0

Let I⊆R be an interval and f:I→R.

We say that c∈I is a critical point of f if…

… c is in the interior of I, f is differentiable at c and f'(c) = 0.

Let f:[a,b]→R, f be continuous on [a,b] and differentiable in (a,b).

Then…

… ∃p,q∈[a,b], a max and a min point of f and any local max or min of f in (a,b) is a critical point of f.

Rolle’s Theorem

Let a,b∈R and a<b.
Let f:[a,b]→R be continuous on [a,b] and differentiable in (a,b).

If f(a) = f(b) then ∃c∈(a,b) s.t. f'(c) = 0.

Mean Value Theorem (MVT)

Let a<b and f:[a,b]→R be continuous on [a,b] and differentiable in (a,b).

Then ∃c∈(a,b) s.t. f'(c) = f(b)-f(a) / b-a

Cauchy Mean Value Theorem (CMVT)

Let a<b and let f,g:[a,b]→R and assume f,g are continuous on [a,b] and differentiable in (a,b).

Then ∃c∈(a,b) s.t. (f(b)-f(a))g'(c) = (g(b)-g(a))f'(c)

Let a<b and f:[a,b]→R be continuous on [a,b] and differentiable in (a,b).

If ∀x∈(a,b) f'(x) = 0, then…

… f is constant in [a,b].

Let I⊆R be an interval.

f is increasing on I if…

… ∀x₁,x₂∈I x₁ ≤ x₂ => f(x₁) ≤ f(x₂)

(f is strictly increasing if you lose the equality)

Let I⊆R be an interval.

f is decreasing on I if…

… ∀x₁,x₂∈I x₁ ≤ x₂ => f(x₁) ≥ f(x₂)

(f is strictly decreasing if you lose the equality)

Let a<b and f:[a,b]→R be continuous on [a,b] and differentiable on (a,b).

Then f is increasing on [a,b] iff…

… ∀x∈(a,b) f'(x) ≥ 0

Let a<b and f:[a,b]→R be continuous on [a,b] and differentiable on (a,b).

If ∀x∈(a,b) f'(x) > 0, then…

… f is strictly increasing on [a,b].

Let a<b and f:[a,b]→R be continuous on [a,b] and differentiable on (a,b).

Then f is decreasing on [a,b] iff…

… ∀x∈(a,b) f'(x) ≤ 0.

Let a<b and f:[a,b]→R be continuous on [a,b] and differentiable on (a,b).

If ∀x∈(a,b) f'(x) < 0 then…

… f is strictly decreasing on [a,b].

De l’Hopital’s Rule (Version 1)

Let a<b and f,g:(a,b)→R. Let c∈(a,b).

Assume f, g are differentiable at c, f(c)=g(c)=0 and g'(c)≠0.

Then lim_x→c f(x)/g(x) = f'(c)/g'(c)

De l’Hopital’s Rule (Version 2)

Let a<b and f,g:(a,b)→R.

Assume lim_x→a⁺ f(x) = 0, lim_x→a⁺ g(x) = 0, ∀x∈(a,b) g'(x)≠0 and lim_x→a⁺ f'(x)/g'(x) exists.

Then lim_x→a⁺ f(x)/g(x) = lim_x→a⁺ f'(x)/g'(x)

Let a<b and f:(a,b)→R. Let c∈(a,b).

We say that f is twice differentiable at c if…

… ∃δ>0 s.t. f is differentiable in (c-δ, c+δ) ⊆ (a,b) and the function f':(c-δ, c+δ)→R is differentiable at c.

In this case we write (f')'(c) or f''(c) or f⁽²⁾(c).

Let n∈N, n≥2, a<b, f:(a,b)→R, c∈(a,b).

We say that f is n-times differentiable at c if…

…∃δ>0 s.t. f is (n-1) differentiable in (c-δ, c+δ) ⊆ (a,b) and f^(n-1): (c-δ, c+δ)→R is differentiable at c.

In this case, we write (f^(n-1))'(c) or f⁽ⁿ⁾(c).

Taylor’s Theorem

Let n∈N, a<b and f:(a,b)→R be (n+1) times differentiable in (a,b). Let x₀∈(a,b).

Then for x∈(a,b) ∃c∈(a,b) s.t. f(x) = f(x₀) + f'(x₀)(x-x₀) + … + 1/n!f⁽ⁿ⁾(x₀)(x-x₀)ⁿ + R_n(x) where R_n(x) = [f⁽ⁿ⁺¹⁾(c)/(n+1)!](x-x₀)ⁿ⁺¹

Let a<b and f:(a,b)→R be n-times differentiable at x∈(a,b).

The Taylor Polynomial of f at x₀ is…

… T_n,x₀(x) = f(x₀) + f'(x₀)(x-x₀) + … + (fⁿ(x₀)/n!)(x-x₀)ⁿ = ∑ⁿ_k=0 (f^k(x₀)/k!)(x-x₀)^k

T_n,x₀ is a polynomial of degree at most n, satisfying T^(k)_n,x₀(x₀) = f^k(x₀)

Let a<b and f:(a,b)→R be n-times differentiable in (a,b). Let x₀∈(a,b).

Then the Taylor polynomial of order n for f at x₀, is…

… the unique polynomial of degree at most n s.t.
lim_x→x₀ [f(x)-T_n,x₀(x₀)]/(x-x₀)ⁿ = 0

Let f:(a,b)→R be differentiable in (a,b) and twice differentiable at p∈(a,b).

If f'(p) = 0, f''(p) < 0, then…

… p is a local maximum of f.

Let f:(a,b)→R be differentiable in (a,b) and twice differentiable at p∈(a,b).

If f'(p) = 0, f''(p) > 0, then…

… p is a local minimum of f.

Let f:(a,b)→R be differentiable in (a,b) and twice differentiable at p∈(a,b).

If p is a local minimum of f then…

… f'(p) = 0 and f''(p) ≥ 0.

Let f:(a,b)→R be differentiable in (a,b) and twice differentiable at p∈(a,b).

If p is a local maximum of f then…

… f'(p) = 0 and f''(p) ≤ 0.

A power series is…

… a series of the form ∑^∞_n=0 aₙxⁿ where aₙ are real numbers.

Series of the form ∑^∞_n=0 aₙ(x-x₀)ⁿ where aₙ,x₀∈R are…

… also power series, just centred at x₀.

Consider a power series of the form ∑^∞_n=0 aₙxⁿ where aₙ∈R are given.

Then ∃R∈[0,∞]∪{∞} s.t. …

… a) |x|<R => ∑^∞_n=0 aₙxⁿ converges

b) |x|>R => ∑^∞_n=0 aₙxⁿ diverges

If R = 0, the series converges for…

… x = 0

If R = ∞, the series converges…

… ∀x∈R

If R = |x| …

… we can’t say what happens

The radius of convergence of ∑^∞_n=0 aₙxⁿ is…

… the number R∈[0,∞)∪{∞}.

It depends on (a_n)_n∈N and the earlier theorem.

Consider ∑^∞_n=0 aₙxⁿ. Assume that ∃n₀∈N ∀n≥n₀ aₙ≠0 and lim_n→∞ |a_n+1|/|a_n| = Q exists (Q = ∞ is allowed).

Then the radius of convergence, R, of ∑^∞_n=0 aₙxⁿ is given by…

… R = 1/Q if Q>0.

If Q=0, R=∞ and if Q=∞, P=0.