Definitions Flashcards
The absolute value of a ∈ R is defined by
|a| = max{a,−a} = { a, a > 0 , -a, a =>0
Inequalities -
For all a,b,c ∈ R:
(i) Eithera<aora=b.
Triangle inequality
|a+b| ≤ |a|+|b| and furthermore |a−b| ≥ ||a|−|b||
Least Upper Bound Axiom - A non–empty subset S ⊂ R of the real numbers is said to be…
(i) bounded above if there exists M ∈ R so that x ≤ M for all x ∈ S; in this case M is
called an upper bound of S;
(ii) boundedbelow ifthereexistsm∈Rsothatm≤xforallx∈S;inthiscasemis
called a lower bound of S;
(iii) bounded if S is both bounded above and below.
LUB Axiom - M ∈ R is called the least upper bound (or, also frequently used, supremum) of S,
written as lub(S) (or sup(S)), if…
(a) M is an upper bound of S, and
(b) ifM ̃ isanupperboundofSthenM≤M ̃.
m ∈ R is called the greatest lower bound (or, also frequently used, infimum) of S,
written as glb(S) (or inf(S)), if
(a) m is a lower bound of S, and
(b) if m ̃ is a lower bound of S then m ̃ ≤ m.
LUB Axiom - non empty subsets
Every non–empty subset of the real numbers which has an upper bound has a least upper bound.
If M = lub S and ε > 0 then there exists s ∈ S with…
M−ε<s≤M
Axiom of Induction!!
Let S ⊂ N. If (i) 1 ∈ S
(ii) k ∈ S =⇒ k + 1 ∈ S then S = N .
Epsilon/Delta definition of a limit
Let f : D → R be a function with (c−p, c)∪(c, c+p) ⊂ D forsomep>0. Thenwesaythatthelimit off atcisL,thatis
lim f (x) = L , x→c
if for all ε > 0 there exists δ > 0 such that :
0<ε.
Method for Epsilon/Delta argument
”Find δ” scratch work to find δ
“δ works” verify that your candidate for δ indeed satisfies the conditions.
Limits - Equivalent Formulations
The following are equivalent:
(i) limx→c f(x) = L
(ii) limx→c(f(x) − L) = 0
(iii) limh→0 f(c + h) = L (iv) limx→c|f(x)−L|=0.
Limits - Uniqueness
If limx→c f(x) = L and limx→c f(x) = M then L = M.
Limit Laws - If limx→c f(x) = L and limx→c g(x) = M then…
(i) limx→c[f(x) + g(x)] = L + M (ii) limx→c[αf(x)] = αL for α ∈ R
(iii) limx→c[f(x)g(x)] = LM
Limits of Rational Functions
If limx→c g(x) = M, M ̸= 0, limx→c f(x) = L, then
lim f(x) = L . x→c g(x) M
Criterion for a limit not to exist:
If lim f(x) = L, L ̸= 0, and lim g(x) = 0 then lim f(x) does not x→c x→c x→c g(x)
exist.
Pinching Theorem
Let f,g,h : D → R. Let p > 0 with (c−p,c)∪(c,c+ p)⊂D. Suppose that 0<p implies
h(x) ≤ f(x) ≤ g(x). If limx→c h(x) = L and limx→c g(x) = L then
lim f (x) = L . x→c
Definition of Continuity!!
Let f : D → R a function and c ∈ R such that there exists p > 0 such that(c−p,c+p)∈D. Thenf iscalledcontinuous atcif
lim f(x) = f(c)
Continuity - Using ε, δ–criterion
f is continuous at c if and only if ∀ε > 0∃δ > 0: |x−c|<ε.
Continuous on an interval
A function f is said to be continuous on an interval if it is continuous for all interior points of the interval, and one–sidedely continuous at the endpoints.
Intermediate Value Theorem!!
Let f be continuous on [a, b] and K any number between f (a) and f (b). Then there exists at least one c ∈ (a, b) with
f(c) = K .
Extreme Value Theorem
Let f be continuous on a bounded closed in- terval [a, b]. Then f takes on both its maximum and its minimum on [a, b].
Differentiable/Derivative
A function f : (a, b) → R is said to be differentiable at x ∈ (a, b) if lim f(x + h) − f(x)
h→0 h
exists. In this case, the limit is called the derivative of f at x, and is denoted by
f′(x) = lim f(x + h) − f(x) . h→0 h
Differentiable/Continuous
If f is differentiable at x then f is continuous at x.
Product Rule!!
If f, g are differentiable at x then so is f · g and (fg)′(x) = f′(x)g(x) + f(x)g′(x) .
Reciprocal Rule!!
Theorem 4.10 (Reciprocal rule). Let g be differentiable at x and g(x) ̸= 0. Then 1 is g
differentiable at x and
Quotient Rule!!
Theorem 4.12 (Quotient rule). If f,g are differentiable at x and g(x) ̸= 0 then f is g
differentiable at x and
Chain Rule!!
If F = f ◦ g and g is differentiable at x and f differentiable at g(x) then F is differentiable at x and
F′(x) = (f ◦ g)′(x) = f′(g(x))g′(x).
Increasing/Decreasing Functions
(strictly) increasing on an interval I if for all x1,x2 ∈ I, x1 < x2: f(x1) < f(x2). (strictly) decreasing on an interval I if for all x1,x2 ∈ I, x1 < x2: f(x1) > f(x2).
Injective Functions
A function f is said to be one–to–one (or injective) if for all x1,x2 ∈ dom(f ):
f(x1)=f(x2) =⇒ x1 =x2.
Inverse Function
If f is one–to–one, then there exists a unique function g : range(f) → dom(f) such that
f(g(x)) = x
for all x ∈ range(f). denoted by g = f−1.
Inverse/One-to-One Function
Let f be one–to–one and f−1 its inverse function. Then f−1(f(x)) = x
for all x ∈ dom(f). In particular, f−1 is one–to–one and the inverse function of f−1 is (f−1)−1 = f. Furthermore If f is one–to–one and continuous on (a,b) so is f−1.
Derivative of Inverse Function!!
Let f : I → R be one–to–one and differentiable on an open interval I. Let a∈I and f(a)=b. If f′(a)̸=0 then f−1 is differentiable at b and
(f−1)′(b) = 1 = 1 . f′(a) f′(f−1(b))
Newton’s Method - IVT
Let xn ∈ [a,b]. The tangent line at xn is given by y = f(xn) + f′(xn)(x − xn). The intersection xn+1 with the x–axis is given by 0 = f(xn) + f′(xn)(xn+1 − xn). Solving for xn+1 we obtain
xn+1 = xn − f(xn) . f′(xn)
Mean Value Theorem!!!
Let f be differentiable on (a,b), continuous on [a, b]. Then there exists at least one number c ∈ (a, b) for which
f′(c) = f(b) − f(a) . b−a
Mean Value Theorem 2
Let f be differentiable at x0.
(i) If f′(x0)>0 then f(x0 −h)0. (ii) If f′(x0)0.
Rolle’s Theorem!!
Suppose f is differentiable on (a, b) and continuous on [a,b]. If f(a) = f(b) = 0 then there exists c ∈ (a,b) such that f′(c) = 0.
Increasing/Decreasing Theorem!!
Suppose f : I → R is differentiable on the interior I0 of an interval I and continuous on I. Then
• If f′(x)>0 for all x∈I0 then f is increasing on I. • If f′(x)<0 for all x∈I0 then f is decreasing on I. • If f′(x)=0 for all x∈I0 then f is constant on I.
Local Max/Min
Suppose f is a function on an interval I and c an interior point of I. Then f is said to have a
local maximum at c if f(c) ≥ f(x) for all x sufficiently close to c. local minimum at c if f(c) ≤ f(x) for all x sufficiently close to c.
Critical Point
If c is an interior point of the domain of a function f and f′(c) = 0 or f′(c) does not exist
then c is called a critical point of f.
First Derivative Test!!
Suppose c is a critical point of f and f is contin- uous at c. If there exists δ > 0 such that
(i) f′(x)>0 for all x∈(c−δ,c) and f′(x)0 for all x∈(c,c+δ) then f has a local minimum at c.
(iii) If f′ keeps constant sign on (c−δ,c)∪(c,c+δ) then f(c) is not a local extreme value.
Second Derivative Test!!
Suppose f′(c) = 0 and f′′(c) exists.
(i) If f′′(c) < 0 then f has a local maximum at c. (ii) If f′′(c) > 0 then f has a local minimum at c.
Absolute Max/Min
The function f is said to have an
absolute maximum at d ∈ dom(f) if f(d) ≥ f(x) for all x ∈ dom(f); absolute minimum at d ∈ dom(f) if f(d) ≤ f(x) for all x ∈ dom(f).
Find Absolute Extrema?! - continuous function f on a closed bounded interval [a, b]:
(i) Find critical points of f.
(ii) Compare f(a),f(b) and f(c) for all critical points c of f.
Concavity
Let f be a function differentiable on an open interval I. The graph of f is said to be
concave up if f′ increases on I, and concave down if f′ decreases on I.
Points of Inflection
Let f be continuous at c, differentiable near c. Then (c,f(c)) is called a point of inflection if the graph of f changes its type of concavity at c, that is, if there exists δ > 0 such that graph(f ) is concave up (or concave down) on (c − δ, c) and the opposite concave down (or concave up) on (c, c + δ).
Find the concavity type by examining the second derivative:
Let f be twice differentiable on an open interval I.
(i) If f′′(x)>0 for all x∈I then the graph of f is concave up. (ii) If f′′(x) < 0 for all x ∈ I then the graph of f is concave down.
Cauchy Mean-Value Theorem!!!
Let f, g be differentiable on (a, b), con- tinuous on [a,b]. If g′(x) ̸= 0 for all x ∈ (a,b) then there exists r ∈ (a,b) with
f′(r) = f(b) − f(a) . g′(r) g(b) − g(a)
L’Hopital’s Rule!!
Suppose that limx→c f(x) = limx→c g(x) = 0 with g(x)̸=0 and g′(x)̸=0 for all x near c. Then if
lim f′(x) = L x→c g′(x)
exists, then lim f(x) exists and equals L. x→c g(x)
Definite Integral!!
Let f be a continuous function on [a,b]. The unique number I with Lf(P) ≤ I ≤ Uf(P) for all partitions P of [a,b] is called the definite integral of f from a to b and is denoted by
Integral-Derivative
Theorem 6.8. Let f be continuous on [a, b], c ∈ [a, b]. Then the function
Antiderivative!!
Let f be continuous on [a, b]. A function G is called an antiderivative for f on [a,b] if
(i) G is continuous on [a, b], and (ii) G′(x) = f(x) for all x ∈ (a,b).
Fundamental Theorem of Calculus!!!
Let f be continuous on [a, b]. If G is any antiderivative for f on [a, b] then
Linearity Definition
Lemma 6.11 (Linearity). For f, g continuous on [a, b], α ∈ R:
Substitution Law!!
If f is continuous with antiderivative F , then
Change of Variable Formula!!
Theorem 6.14 (Change of Variable Formula). If f is continuous on [a, b] with antideriva- tive F, then
MVT for Integrals
If f is continuous on [a,b] then there exists c ∈ (a, b) such that
Exponential Growth/Decay
If f′(t) = kf(t) for some k ∈ R then there exists a constant c ∈ R such that
f(t)=cekt, t∈R.
Integration by parts!
If u, v are differentiable with continuous derivatives then
