Definitions

Question 0

The absolute value of a ∈ R is defined by

Answer 0

|a| = max{a,−a} = { a, a > 0 , -a, a =>0

Question 1

Inequalities -

Answer 1

For all a,b,c ∈ R:
(i) Eithera<aora=b.

Question 2

Triangle inequality

Answer 2

|a+b| ≤ |a|+|b| and furthermore |a−b| ≥ ||a|−|b||

Question 3

Least Upper Bound Axiom - A non–empty subset S ⊂ R of the real numbers is said to be…

Answer 3

(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.

Question 4

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…

Answer 4

(a) M is an upper bound of S, and
(b) ifM ̃ isanupperboundofSthenM≤M ̃.

Question 5

m ∈ R is called the greatest lower bound (or, also frequently used, infimum) of S,
written as glb(S) (or inf(S)), if

Answer 5

(a) m is a lower bound of S, and
(b) if m ̃ is a lower bound of S then m ̃ ≤ m.

Question 6

LUB Axiom - non empty subsets

Answer 6

Every non–empty subset of the real numbers which has an upper bound has a least upper bound.

Question 7

If M = lub S and ε > 0 then there exists s ∈ S with…

Answer 7

M−ε<s≤M

Question 8

Axiom of Induction!!

Answer 8

Let S ⊂ N. If (i) 1 ∈ S
(ii) k ∈ S =⇒ k + 1 ∈ S then S = N .

Question 9

Epsilon/Delta definition of a limit

Answer 9

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.

Question 10

Method for Epsilon/Delta argument

Answer 10

”Find δ” scratch work to find δ
“δ works” verify that your candidate for δ indeed satisfies the conditions.

Question 11

Limits - Equivalent Formulations

Answer 11

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.

Question 12

Limits - Uniqueness

Answer 12

If limx→c f(x) = L and limx→c f(x) = M then L = M.

Question 13

Limit Laws - If limx→c f(x) = L and limx→c g(x) = M then…

Answer 13

(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

Question 14

Limits of Rational Functions

Answer 14

If limx→c g(x) = M, M ̸= 0, limx→c f(x) = L, then
lim f(x) = L . x→c g(x) M

Question 15

Criterion for a limit not to exist:

Answer 15

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.

Question 16

Pinching Theorem

Answer 16

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

Question 17

Definition of Continuity!!

Answer 17

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)

Question 18

Continuity - Using ε, δ–criterion

Answer 18

f is continuous at c if and only if ∀ε > 0∃δ > 0: |x−c|<ε.

Question 19

Continuous on an interval

Answer 19

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.

Question 20

Intermediate Value Theorem!!

Answer 20

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 .

Question 21

Extreme Value Theorem

Answer 21

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].

Question 22

Differentiable/Derivative

Answer 22

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

Question 23

Differentiable/Continuous

Answer 23

If f is differentiable at x then f is continuous at x.

Question 24

Product Rule!!

Answer 24

If f, g are differentiable at x then so is f · g and (fg)′(x) = f′(x)g(x) + f(x)g′(x) .

Question 25

Reciprocal Rule!!

Answer 25

Theorem 4.10 (Reciprocal rule). Let g be differentiable at x and g(x) ̸= 0. Then 1 is g
differentiable at x and

Question 26

Quotient Rule!!

Answer 26

Theorem 4.12 (Quotient rule). If f,g are differentiable at x and g(x) ̸= 0 then f is g
differentiable at x and

Question 27

Chain Rule!!

Answer 27

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).

Question 28

Increasing/Decreasing Functions

Answer 28

(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).

Question 29

Injective Functions

Answer 29

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.

Question 30

Inverse Function

Answer 30

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.

Question 31

Inverse/One-to-One Function

Answer 31

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.

Question 32

Derivative of Inverse Function!!

Answer 32

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))

Question 33

Newton’s Method - IVT

Answer 33

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)

Question 34

Mean Value Theorem!!!

Answer 34

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

Question 35

Mean Value Theorem 2

Answer 35

Let f be differentiable at x0.
(i) If f′(x0)>0 then f(x0 −h)0. (ii) If f′(x0)0.

Question 36

Rolle’s Theorem!!

Answer 36

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.

Question 37

Increasing/Decreasing Theorem!!

Answer 37

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.

Question 38

Local Max/Min

Answer 38

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.

Question 39

Critical Point

Answer 39

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.

Question 40

First Derivative Test!!

Answer 40

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.

Question 41

Second Derivative Test!!

Answer 41

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.

Question 42

Absolute Max/Min

Answer 42

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).

Question 43

Find Absolute Extrema?! - continuous function f on a closed bounded interval [a, b]:

Answer 43

(i) Find critical points of f.
(ii) Compare f(a),f(b) and f(c) for all critical points c of f.

Question 44

Concavity

Answer 44

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.

Question 45

Points of Inflection

Answer 45

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).

Question 46

Find the concavity type by examining the second derivative:

Answer 46

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.

Question 47

Cauchy Mean-Value Theorem!!!

Answer 47

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)

Question 48

L’Hopital’s Rule!!

Answer 48

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)

Question 49

Definite Integral!!

Answer 49

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

Question 50

Integral-Derivative

Answer 50

Theorem 6.8. Let f be continuous on [a, b], c ∈ [a, b]. Then the function

Question 51

Antiderivative!!

Answer 51

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).

Question 52

Fundamental Theorem of Calculus!!!

Answer 52

Let f be continuous on [a, b]. If G is any antiderivative for f on [a, b] then

Question 53

Linearity Definition

Answer 53

Lemma 6.11 (Linearity). For f, g continuous on [a, b], α ∈ R:

Question 54

Substitution Law!!

Answer 54

If f is continuous with antiderivative F , then

Question 55

Change of Variable Formula!!

Answer 55

Theorem 6.14 (Change of Variable Formula). If f is continuous on [a, b] with antideriva- tive F, then

Question 56

MVT for Integrals

Answer 56

If f is continuous on [a,b] then there exists c ∈ (a, b) such that

Question 57

Exponential Growth/Decay

Answer 57

If f′(t) = kf(t) for some k ∈ R then there exists a constant c ∈ R such that
f(t)=cekt, t∈R.

Question 58

Integration by parts!

Answer 58

If u, v are differentiable with continuous derivatives then