Queen Mary University of London ECN115 Lecture 2 PDF

ECN115 Mathematical Methods in Economics and Finance Evgenii Safonov 2024-2025. LECTURE 2 1 Quiz I ◁ Online Quiz I will be on Monday, 14 October ◁ Based on the material of first 2 lectures and homeworks (3rd week’s material is...

ECN115 Mathematical Methods in Economics and Finance Evgenii Safonov 2024-2025. LECTURE 2 1 Quiz I ◁ Online Quiz I will be on Monday, 14 October ◁ Based on the material of first 2 lectures and homeworks (3rd week’s material is not included) ◁ 7.5% of the total grade ◁ 1 hour since you start the quiz; you can start at any time on 14 October ◁ Work alone; there are several variants of each question and each student gets a random variant of each question ◁ Between 5 and 10 questions in total (to be decided) ◁ 3 types of questions: either write down an answer (number), or choose one of the options, or choose all answers that are correct (potentially, the number of correct answers can be one or even zero) 2 Outline ◁ ”There exists” and ”For all” ◁ Absolute value and triangle inequality ◁ Mathematical induction ◁ Binomial formula ◁ Functions: definitions ◁ Common numerical functions 3 “There exists” ◁ Given set A and property (proposition) P, the notation “∃x ∈ A : P” or “∃x ∈ A such that P” means “there exists at least one element x in the set A such that property P holds for x.” ◁ The negation of this statement is denoted by “ ̸ ∃x ∈ A : P ′′ or “̸ ∃x ∈ A such that P.” It means “there does not exist an element x in the set A such that property P holds for x.” ✓ Let A = {2, 4, 6, 8, 10, 12}, and property P be x ≥ 10. Then ∃x ∈ A : x ≥ 10. Indeed, both numbers 10 and 12 are elements of the set A and they are greater or equal to 10 ✓ Let A = {2, 4, 6, 8, 10, 12}, and property Q be x ≥ 20. Then ̸ ∃x ∈ A : x ≥ 20. Indeed, 2 < 20, 4 < 20, 6 < 20, 8 < 20, 10 < 20, 12 < 20 ✓ Let B = {students enrolled in ECN115}, R be “a student is attending today’s lecture.” Is it true that ∃x ∈ B : R? 4 “For all” ◁ Given set A and property (proposition) P, the notation “∀x ∈ A, P” means “for all elements x of the set A the property P is true.” ✓ Let A = {2, 4, 6, 8, 10, 12}, and property T be x ≥ 0. Then ∀x ∈ A, x ≥ 0. Indeed, 2 ≥ 0, 4 ≥ 0, 6 ≥ 0, 8 ≥ 0, 10 ≥ 0, 12 ≥ 0 ✓ Let A = {2, 4, 6, 8, 10, 12}, and property P be x ≥ 10. Then not ∀x ∈ A, x ≥ 10. Indeed, 2 < 10 ✓ Let A = {2, 4, 6, 8, 10, 12}, and property Q be x ≥ 20. Then ̸ ∀x ∈ A, x ≥ 20. Indeed, 2 < 20 ✓ Let B = {students enrolled in ECN115}, R be “a student is attending today’s lecture.” Is it true that ∀x ∈ B, R? 5 “There exists” and “For all” ◁ A = {bicycle, car , boat}. Vehicles do not use rails ✓ ̸ ∃x ∈ A : x uses rails ✓ not ∀x ∈ A, x uses rails ◁ B = {bus, tram, underground}. Some vehicles use rails and some don’t ✓ ∃x ∈ B : x uses rails ✓ not ∀x ∈ B, x uses rails ◁ C = {tram, overground, train}. All vehicles use rails ✓ ∃x ∈ C : x uses rails ✓ ∀x ∈ C , x uses rails 6 “There exists” and “For all” ◁ The statements “∃x ∈ A : P” and “not ∀x ∈ A, not P” are equivalent ✓ “∃x ∈ {bus, tram, underground} : x uses rails” is equivalent to “Not all vehicles in the set {bus, tram, underground} do not use rails” ◁ Similarly, “∃x ∈ A : not P” and “not ∀x ∈ A, P” are equivalent ✓ “There exist a non-negative real number” is equivalent to “not all real numbers are negative” ◁ The statements “∀x ∈ A, P” and “̸ ∃x ∈ A : not P” are equivalent ✓ “∀x ∈ {tram, overground, train}, x uses rails” is equivalent to “there is no vehicle in the set {tram, overground, train} that does not use rails” ◁ Similarly, “∀x ∈ A, not P” and “̸ ∃x ∈ A : P” are equivalent ✓ “All natural numbers are non-negative” is equivalent to “there is no negative natural number” 7 Outline ◁ ”There exists” and ”For all” ◁ Absolute value and triangle inequality ◁ Mathematical induction ◁ Binomial formula ◁ Functions: definitions ◁ Common numerical functions 8 Absolute value and Triangle inequality ◁ This is the Triangle Inequality: |x − z| is a distance between x and z on the number line, and similarly for |x − y | and |y − z| ◁ Going from x to z should not be longer than going first from x to y and then from y to z (Whiteboard analysis) ◁ The absolute value |x| of the real number x is define as follows: |x| = x if x ≥ 0 and |x| = −x if x < 0 ◁ Problem B.1 from HW-1. Show that for any real numbers x, y , z, |x − z| ≤ |x − y | + |y − z| 9 Outline ◁ ”There exists” and ”For all” ◁ Absolute value and triangle inequality ◁ Mathematical induction ◁ Binomial formula ◁ Functions: definitions ◁ Common numerical functions 10 Mathematical Induction ◁ A proof technique applied when there are several similar propositions P1 , P2 ,...,Pn ,... that depend on the parameter n ∈N 11 Mathematical Induction ◁ A proof technique applied when there are several similar propositions P1 , P2 ,...,Pn ,... that depend on the parameter n ∈ N ◁ The idea is: instead of proving each of the propositions P1 , P2 , P3 to prove only two propositions: ✓ The induction base is the first proposition P1 ✓ The induction step is the proposition that for any index n, Pn =⇒ Pn+1 11 Mathematical Induction ◁ A proof technique applied when there are several similar propositions P1 , P2 ,...,Pn ,... that depend on the parameter n ∈ N ◁ The idea is: instead of proving each of the propositions P1 , P2 , P3 to prove only two propositions: ✓ The induction base is the first proposition P1 ✓ The induction step is the proposition that for any index n, Pn =⇒ Pn+1 ◁ When both these the induction step and the induction base are proven, all propositions P1 ,...,Pn ,... are proven P1 , P2 , P3 , P4 , P5 ,... 11 Mathematical Induction ◁ A proof technique applied when there are several similar propositions P1 , P2 ,...,Pn ,... that depend on the parameter n ∈ N ◁ The idea is: instead of proving each of the propositions P1 , P2 , P3 to prove only two propositions: ✓ The induction base is the first proposition P1 ✓ The induction step is the proposition that for any index n, Pn =⇒ Pn+1 ◁ When both these the induction step and the induction base are proven, all propositions P1 ,...,Pn ,... are proven P1 ; P2 , P3 , P4 , P5 ,... 11 Mathematical Induction ◁ A proof technique applied when there are several similar propositions P1 , P2 ,...,Pn ,... that depend on the parameter n ∈ N ◁ The idea is: instead of proving each of the propositions P1 , P2 , P3 to prove only two propositions: ✓ The induction base is the first proposition P1 ✓ The induction step is the proposition that for any index n, Pn =⇒ Pn+1 ◁ When both these the induction step and the induction base are proven, all propositions P1 ,...,Pn ,... are proven P1 =⇒ P2 ; P3 , P4 , P5 ,... 11 Mathematical Induction ◁ A proof technique applied when there are several similar propositions P1 , P2 ,...,Pn ,... that depend on the parameter n ∈ N ◁ The idea is: instead of proving each of the propositions P1 , P2 , P3 to prove only two propositions: ✓ The induction base is the first proposition P1 ✓ The induction step is the proposition that for any index n, Pn =⇒ Pn+1 ◁ When both these the induction step and the induction base are proven, all propositions P1 ,...,Pn ,... are proven P1 =⇒ P2 =⇒ P3 ; P4 , P5 ,... 11 Mathematical Induction ◁ A proof technique applied when there are several similar propositions P1 , P2 ,...,Pn ,... that depend on the parameter n ∈ N ◁ The idea is: instead of proving each of the propositions P1 , P2 , P3 to prove only two propositions: ✓ The induction base is the first proposition P1 ✓ The induction step is the proposition that for any index n, Pn =⇒ Pn+1 ◁ When both these the induction step and the induction base are proven, all propositions P1 ,...,Pn ,... are proven P1 =⇒ P2 =⇒ P3 =⇒ P4 ; P5 ,... 11 Mathematical Induction ◁ A proof technique applied when there are several similar propositions P1 , P2 ,...,Pn ,... that depend on the parameter n ∈ N ◁ The idea is: instead of proving each of the propositions P1 , P2 , P3 to prove only two propositions: ✓ The induction base is the first proposition P1 ✓ The induction step is the proposition that for any index n, Pn =⇒ Pn+1 ◁ When both these the induction step and the induction base are proven, all propositions P1 , P2 , P3 ,... are proven P1 =⇒ P2 =⇒ P3 =⇒ P4 =⇒ P5... 11 Mathematical Induction ◁ A proof technique applied when there are several similar propositions P1 , P2 ,...,Pn ,... that depend on the parameter n ∈ N ◁ The idea is: instead of proving each of the propositions P1 , P2 , P3 to prove only two propositions: ✓ The induction base is the first proposition P1 ✓ The induction step is the proposition that for any index n, Pn =⇒ Pn+1 ◁ When both these the induction step and the induction base are proven, all propositions P1 , P2 , P3 ,... are proven P1 =⇒ P2 =⇒ P3 =⇒ P4 =⇒ P5... ◁ When proving the induction step Pn =⇒ Pn+1 , the statement that proposition Pn is true is called the induction hypothesis. 11 Example: sum of first n natural numbers ◁ For example, let the propositions Pn for n = 1, 2, 3,... be n ∑︁ n(n + 1) j = j=1 2 12 Example: sum of first n natural numbers ◁ For example, let the propositions Pn for n = 1, 2, 3,... be n ∑︁ n(n + 1) j = j=1 2 Put it differently, the proposition Pn states that the sum of all natural n(n + 1) numbers from 1 to n is equal to 2 12 Example: sum of first n natural numbers ◁ For example, let the propositions Pn for n = 1, 2, 3,... be n ∑︁ n(n + 1) j = j=1 2 Put it differently, the proposition Pn states that the sum of all natural n(n + 1) numbers from 1 to n is equal to 2 ◁ The induction base is the first proposition, P1 ; thus, n = 1. In our example, it is 1 ∑︁ 1 · (1 + 1) j = j=1 2 Since the left hand side is 1 and the right hand side is 1, then P1 is true 12 Example: sum of first n natural numbers The induction step is the proposition that for any natural n, Pn =⇒ Pn+1 In our example, the proposition Pn+1 is n+1 ∑︁ (n + 1)((n + 1) + 1) j = j=1 2 13 Example: sum of first n natural numbers The induction step is the proposition that for any natural n, Pn =⇒ Pn+1 In our example, the proposition Pn+1 is n+1 ∑︁ (n + 1)((n + 1) + 1) j = j=1 2 To prove the induction step in our example: n+1 n ∑︁ ∑︁ n(n + 1) j = j + (n + 1) = + (n + 1) = j=1 j=1 2 (︁ n )︁ (n + 2)(n + 1) (n + 1)((n + 1) + 1) = + 1 (n + 1) = = 2 2 2 which we wanted to prove. Note that we used proposition Pn (the induction hypothesis) in the proof. 13 Alternative way to prove the induction step in our example Equivalently, the induction step is the proposition that for any natural n ≥ 2, Pn−1 =⇒ Pn. In this formulation, the induction hypothesis is the proposition Pn−1 which is n−1 ∑︁ (n − 1)((n − 1) + 1) (n − 1)n j = = j=1 2 2 14 Alternative way to prove the induction step in our example Equivalently, the induction step is the proposition that for any natural n ≥ 2, Pn−1 =⇒ Pn. In this formulation, the induction hypothesis is the proposition Pn−1 which is n−1 ∑︁ (n − 1)((n − 1) + 1) (n − 1)n j = = j=1 2 2 To prove the induction step in our example: n n−1 ∑︁ ∑︁ (n − 1)n j = j +n = +n = j=1 j=1 2 (︂ )︂ n−1 (n + 1)n = +1 n = 2 2 hence Pn holds, and since this argument works for all n ≥ 2, Pn−1 =⇒ Pn , proving the induction step. 14 Alternative formulation of the principle ◁ Want to prove propositions P1 , P2 ,...,Pn ,... ✓ The induction base is the first proposition P1 ✓ The induction step is the proposition that for any index n, {P1 ,..., Pn } =⇒ Pn+1 ✓ Remark: we’ve formulated the induction step originally as Pn =⇒ Pn+1 for any n. The two formulations are, in fact, equivalent ◁ When proving the induction step Pn =⇒ Pn+1 , the statement that propositions P1 ,..., Pn are true is called the induction hypothesis. 15 Outline ◁ ”There exists” and ”For all” ◁ Absolute value and triangle inequality ◁ Mathematical induction ◁ Binomial formula ◁ Functions: definitions ◁ Common numerical functions 16 The Binomial formula Consider (x + y )n where x, y ∈ R, n ∈ {0} ∪ N. (x + y )0 = 1 (x + y )1 = x + y (x + y )2 = x2 + 2xy + y2 (x + y )3 = x3 + 3x 2 y + 3xy 2 + y3 (x + y )4 = x4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y4 (x + y )5 = x 5 + 5x 4 y + 10x 3 y 2 + 10x 2 y 3 + 5xy 4 + y5 17 The Binomial formula Consider (x + y )n where x, y ∈ R, n ∈ {0} ∪ N. (x + y )0 = 1 (x + y )1 = x + y (x + y )2 = x2 + 2xy + y2 (x + y )3 = x3 + 3x 2 y + 3xy 2 + y3 (x + y )4 = x4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y4 (x + y )5 = x 5 + 5x 4 y + 10x 3 y 2 + 10x 2 y 3 + 5xy 4 + y5 The general formula is n ∑︁ n! (x + y )n = x n−j y j j=0 (n − j)!j! 17 Binomial coefficients The term (︃ )︃ n n! = Cnj = j (n − j)!j! is called a Binomial coefficient. It is defined for j, n ∈ {0} ∪ N such that n ≥ j. 18 Binomial coefficients The term (︃ )︃ n n! = Cnj = j (n − j)!j! is called a Binomial coefficient. It is defined for j, n ∈ {0} ∪ N such that n ≥ j. N Given k ∈ {0} ∪ , the notation k! denotes the factorial of the number k, defined recursively by 0! = 1, (k)! = (k − 1)! · k 18 Binomial coefficients The term (︃ )︃ n n! = Cnj = j (n − j)!j! is called a Binomial coefficient. It is defined for j, n ∈ {0} ∪ N such that n ≥ j. N Given k ∈ {0} ∪ , the notation k! denotes the factorial of the number k, defined recursively by 0! = 1, (k)! = (k − 1)! · k For instance, 0! = 1, 1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120 (︃ )︃ 5 5! 5! 5·4·3·2 = C52 = = = = 10 2 (5 − 2)! · 2! 3! · 2! (3 · 2) · 2 18 Examples from an exam (whiteboard) ◁ Write down the Binomial formula for (1 + x)n ◁ Using the Binomial formula, show that for any x > 0, (1 + x)10 − 10 · x · (1 + 4x) > 1 19 Outline ◁ ”There exists” and ”For all” ◁ Absolute value and triangle inequality ◁ Mathematical induction ◁ Binomial formula ◁ Functions: definitions ◁ Common numerical functions 20 Functions The concept of a function Definition (informal). A function f : D → R is a rule that associates each element of a (domain) set D with an element of a (range) set R. 21 The concept of a function Definition (informal). A function f : D → R is a rule that associates each element of a (domain) set D with an element of a (range) set R. Definition (formal). For a given sets D and R, a function f is a set of ordered pairs of elements (x, y ), x ∈ D, y ∈ R such that for every x ∈ D there is a unique y ∈ R such that (x, y ) ∈ f. In this case we write y = f (x). 21 The concept of a function Definition (informal). A function f : D → R is a rule that associates each element of a (domain) set D with an element of a (range) set R. Definition (formal). For a given sets D and R, a function f is a set of ordered pairs of elements (x, y ), x ∈ D, y ∈ R such that for every x ∈ D there is a unique y ∈ R such that (x, y ) ∈ f. In this case we write y = f (x). Examples ◁ D = objects, R = masses, and g (x) is the mass of object x. For instance, g (Jupiter) ≈ 1.898 × 1027 kg. 21 The concept of a function Definition (informal). A function f : D → R is a rule that associates each element of a (domain) set D with an element of a (range) set R. Definition (formal). For a given sets D and R, a function f is a set of ordered pairs of elements (x, y ), x ∈ D, y ∈ R such that for every x ∈ D there is a unique y ∈ R such that (x, y ) ∈ f. In this case we write y = f (x). Examples ◁ D = objects, R = masses, and g (x) is the mass of object x. For instance, g (Jupiter) ≈ 1.898 × 1027 kg. R ◁ D = R = and h(x) = x 2. For instance, h(10) = 102 = 100 and h(−1) = (−1)2 = 1. 21 The concept of a function Definition (informal). A function f : D → R is a rule that associates each element of a (domain) set D with an element of a (range) set R. Definition (formal). For a given sets D and R, a function f is a set of ordered pairs of elements (x, y ), x ∈ D, y ∈ R such that for every x ∈ D there is a unique y ∈ R such that (x, y ) ∈ f. In this case we write y = f (x). Examples ◁ D = objects, R = masses, and g (x) is the mass of object x. For instance, g (Jupiter) ≈ 1.898 × 1027 kg. R ◁ D = R = and h(x) = x 2. For instance, h(10) = 102 = 100 and h(−1) = (−1)2 = 1. Definition. We say that two functions f and g are equal if they have the same domain D and range R, and for any x ∈ D, f (x) = g (x). 21 The concept of a function Definition (informal). A function f : D → R is a rule that associates each element of a (domain) set D with an element of a (range) set R. Definition (formal). For a given sets D and R, a function f is a set of ordered pairs of elements (x, y ), x ∈ D, y ∈ R such that for every x ∈ D there is a unique y ∈ R such that (x, y ) ∈ f. In this case we write y = f (x). Examples ◁ D = objects, R = masses, and g (x) is the mass of object x. For instance, g (Jupiter) ≈ 1.898 × 1027 kg. R ◁ D = R = and h(x) = x 2. For instance, h(10) = 102 = 100 and h(−1) = (−1)2 = 1. Definition. We say that two functions f and g are equal if they have the same domain D and range R, and for any x ∈ D, f (x) = g (x). Definition. When D, R ⊆ R, we call f : D → R a numerical function. 21 Monotone functions Definition. Consider a numerical function f : X → R with X ⊆ R. If for all x, z ∈ X such that x > z, 1. f (x) ≥ f (z), then function f is called increasing (or non-decreasing) 22 Monotone functions Definition. Consider a numerical function f : X → R with X ⊆ R. If for all x, z ∈ X such that x > z, 1. f (x) ≥ f (z), then function f is called increasing (or non-decreasing) 2. f (x) > f (z), then function f is called strictly increasing 22 Monotone functions Definition. Consider a numerical function f : X → R with X ⊆ R. If for all x, z ∈ X such that x > z, 1. f (x) ≥ f (z), then function f is called increasing (or non-decreasing) 2. f (x) > f (z), then function f is called strictly increasing 3. f (x) ≤ f (z), then function f is called decreasing (or non-increasing) 22 Monotone functions Definition. Consider a numerical function f : X → R with X ⊆ R. If for all x, z ∈ X such that x > z, 1. f (x) ≥ f (z), then function f is called increasing (or non-decreasing) 2. f (x) > f (z), then function f is called strictly increasing 3. f (x) ≤ f (z), then function f is called decreasing (or non-increasing) 4. f (x) < f (z), then function f is called strictly decreasing 22 Monotone functions Definition. Consider a numerical function f : X → R with X ⊆ R. If for all x, z ∈ X such that x > z, 1. f (x) ≥ f (z), then function f is called increasing (or non-decreasing) 2. f (x) > f (z), then function f is called strictly increasing 3. f (x) ≤ f (z), then function f is called decreasing (or non-increasing) 4. f (x) < f (z), then function f is called strictly decreasing ◁ If a function is either increasing, or decreasing, we call it monotone (or monotonic) function ◁ If a function is either strictly increasing, or strictly decreasing, we call it strictly monotone function ◁ The definitions are extended for mononotonicity properties on an interval of real numbers Y ⊆ X. It this case, we consider x > z such that x, z ∈ Y 22 Outline ◁ ”There exists” and ”For all” ◁ Absolute value and triangle inequality ◁ Mathematical induction ◁ Binomial formula ◁ Functions: definitions ◁ Common numerical functions 23 Integer Powers Positive integer powers: k ∈ N (i.e. k is natural). f : R → R given by f (x) = x k. 2 1 x0 x1 x2 -2 -1 1 2 x3 x6 -1 x9 -2 24 Integer Powers Negative integer powers: k ∈ {..., −3, −2, −1, 0} (i.e. k is either an opposite of a natural number, or k is zero). f : R∖{0} → R given by f (x) = x k. In this case, f (x) = 1 x −k , where −k ∈ N. 2 1 x0 x-1 x-2 -2 -1 1 2 x-3 x-6 -1 x-9 -2 25 Inverse natural powers Definition. For a given n ∈ 1 N and x ∈ [0, ∞), define the number y = x n ∈ [0, ∞) to be the unique non-negative number that satisfies y n = x. 26 Inverse natural powers Definition. For a given n ∈ 1 N and x ∈ [0, ∞), define the number y = x n ∈ [0, ∞) to be the unique non-negative number that satisfies y n = x. Remarks ◁ We assume that such number exists. We might actually prove it (later). ◁ We assume that such number is unique. We might actually prove it. 1 ◁ The requirement that y = x n is non-negative is essential: say, 22 = 4, but also (−2)2 = 4. 1 √ ◁ When n = 2, the other notation for x 2 is x. Less frequently, people use √ n 1 1 √ x for x n ; for example, x 3 = 3 x, etc. 26 Rational powers Z Definition. For a given p ∈ , q ∈ p N, and x ∈ (0, ∞), define the number y = x q by the following formula: p (︁ 1 )︁p xq = xq 2.0 1.5 x0 x1/4 1.0 x1/2 x1 x3/2 0.5 x10/3 0.0 0.0 0.5 1.0 1.5 2.0 27 Rational powers Z Definition. For a given p ∈ , q ∈ p N, and x ∈ (0, ∞), define the number y = x q by the following formula: p (︁ 1 )︁p xq = xq 2.0 1.5 x0 x-1/4 1.0 x-1/2 x-1 x-3/2 0.5 x-10/3 0.0 0.0 0.5 1.0 1.5 2.0 28 Polynomials N R Definition. Given n ∈ {0} ∪ , and numbers a0 , a1 ,..., an ∈ , define a R R function f : → as follows: n ∑︁ f (x) = a0 + a1 x + a2 x 2 + · · · + an x n = ak x k k=0 The function f is called a polynomial. The numbers a0 , a1 ,..., an are called the coefficients of the polynomial f. The largest number k such that ak ̸= 0 is called the degree of the polynomial f. The functions given by a0 , a1 x, a2 x 2 ,...,an x n are called the monomials (terms) of the polynomial f. 29 Polynomials N R Definition. Given n ∈ {0} ∪ , and numbers a0 , a1 ,..., an ∈ , define a R R function f : → as follows: n ∑︁ f (x) = a0 + a1 x + a2 x 2 + · · · + an x n = ak x k k=0 The function f is called a polynomial. The numbers a0 , a1 ,..., an are called the coefficients of the polynomial f. The largest number k such that ak ̸= 0 is called the degree of the polynomial f. The functions given by a0 , a1 x, a2 x 2 ,...,an x n are called the monomials (terms) of the polynomial f. Examples ◁ f (x) = 1 + 5x 2. Coefficients: a0 = 1, a1 = 0, a2 = 5. The degree is 2 (quadratic). The monomials (terms) are 1 and 5x 2. ◁ f (x) = − 13 15 x. Coefficients: a0 = 0, a1 = −13/15. The degree is 1 (linear). ◁ f (x) = −7. Coefficient: a0 = −7. The degree is 0 (constant). ◁ f (x) = −x 3 + 0.9x − 0.3 + 2x 2. Coefficients: a0 = −0.3, a1 = 0.9, a2 = 2, a3 = −1. The degree is 3 (cubic). 29 Polynomials—linear ◁ A generic linear polynomial f (x) = a1 x + a0 (it is convenient to include also the case when a1 = 0, i.e constant polynomial). ◁ Graph is a straight line. Intersects the vertical Axis at point (0, a0 ); if a1 ̸= 0, intersects the horizontal Axis at point (−a0 /a1 , 0). 10 5 5 -4 3x+2 -2 -1 1 2 -5 x + 10 4x-8 -5 -14 x + 10 -10 30 Polynomials—general ◁ Even-degree polynomials (an ̸= 0, n = 2, 4,... is even). For very large x and very small x, f (x) > 0 if an > 0, and f (x) < 0 if an < 0. ◁ Odd-degree polynomials (an ̸= 0, n = 1, 3, 5,... is even). For very large x, f (x) > 0 if an > 0, and f (x) < 0 if an < 0. For very small x, f (x) < 0 if an > 0, and f (x) > 0 if x < 0. 10 x2 - 4 5 -5 x2 + 2 3 x3 - 2 -2 -1 1 2 3 x3 - 10 x - 4 -0.5 x4 + 8 -5 x4 - 2 x3 - x2 + 2 x -10 31 Outline ◁ ”There exists” and ”For all” ◁ Absolute value and triangle inequality ◁ Mathematical induction ◁ Binomial formula ◁ Functions: definitions ◁ Common numerical functions - Extra material 32 Exponential functions Definition (informal). For a given a, b ∈ R, b > 0 define function f : R → R as follows: Q, then f (x) = b. 1. If ax ∈ ax 2. Otherwise if ax ̸∈ Q, f (x) is the value that makes the graph of the function f on the (x, y ) plane a continuous (!?) line. 33 Exponential functions Definition (informal). For a given a, b ∈ R, b > 0 define function f : R → R as follows: Q, then f (x) = b. 1. If ax ∈ ax 2. Otherwise if ax ̸∈ Q, f (x) is the value that makes the graph of the function f on the (x, y ) plane a continuous (!?) line. Special case. When b = 2.71828... = e (Euler’s number), we write f (x) = e ax = exp(ax) 33 Exponential functions 6 5 4x 4 exp(x) 2x 3 20.5 x 2 1x 2-x 1 -3 -2 -1 0 1 2 3 34 Logarithmic functions Definition. Given number b ∈ (0, 1) ∪ (1, ∞) (that is, b > 0, b ̸= 1), define R function logb : (0, ∞) → as follows: logb (x) = y such that b y = x. 35 Logarithmic functions Definition. Given number b ∈ (0, 1) ∪ (1, ∞) (that is, b > 0, b ̸= 1), define R function logb : (0, ∞) → as follows: logb (x) = y such that b y = x. Remarks ◁ We assume that such number exists. We might actually prove it (later). ◁ We assume that such number is unique. We might actually prove it. ◁ The domain consists of strictly positive real numbers, since b y > 0 for b > 0. Special case. When b = 2.71828... = e (Euler’s number), we write f (x) = log(x) = ln(x) So, the notation is ln(x) = log(x) = loge (x). 35 Logarithmic functions 6 4 log10 (x) 2 log(x) log2 (x) 0.5 1.0 1.5 2.0 2.5 log1.1 (x) -2 log0.9 (x) -4 log0.5 (x) -6 36 Algebraic properties of the exponential and logarithm functions b logb (x) = x logb (b x ) = x (b x )y = b xy logb (x y ) = y · logb (x) bx · by = b x+y logb (xy ) = logb (x) + logb (y ) b logb (c)·x = cx logb (x) = logb (c) · logc (x) 37 Trigonometric functions 38 Trigonometric functions—sinus Definition. Define function sin : R → R as follows: 1. If x ∈ [0, 2𝜋), then sin(x) is the vertical coordinate of the point corresponding to x radians of anticlockwise rotation from the positive horizontal axis in a unit circle centered at the origin. 2. If x ̸∈ [0, 2𝜋), then sin(x) = sin(x − 2𝜋n) for some n ∈ Z such that x − 2𝜋n ∈ [0, 2𝜋). where 𝜋 = 3.14159... is the ratio of circle’s circumference to its diameter. The function sin is called sinus. 39 Trigonometric functions—cosinus Definition. Define function cos : R → R as follows: 1. If x ∈ [0, 2𝜋), then cos(x) is the horizontal coordinate of the point corresponding to x radians of anticlockwise rotation from the positive horizontal axis in a unit circle centered at the origin. 2. If x ̸∈ [0, 2𝜋), then cos(x) = cos(x − 2𝜋n) for some n ∈ Z such that x − 2𝜋n ∈ [0, 2𝜋). where 𝜋 = 3.14159... is the ratio of circle’s circumference to its diameter. The function sin is called cosinus. 40 Trigonometric functions—tangent Definition. The function tan : R∖ {..., −5𝜋, −3𝜋, −𝜋, 𝜋, 3𝜋, 5𝜋,...} → R by sin(x) tan(x) = cos(x) where 𝜋 = 3.14159... is the ratio of circle’s circumference to its diameter. The function tan is called tangent. 41 Trigonometric functions—tangent Definition. The function tan : R∖ {..., −5𝜋, −3𝜋, −𝜋, 𝜋, 3𝜋, 5𝜋,...} → R by sin(x) tan(x) = cos(x) where 𝜋 = 3.14159... is the ratio of circle’s circumference to its diameter. The function tan is called tangent. Remarks Z ◁ Tangent is not defined for x = 𝜋 + 2𝜋n for n ∈ , since the denominator in the formula above assumes value zero at these points 41 Trigonometric functions—tangent Definition. The function tan : R∖ {..., −5𝜋, −3𝜋, −𝜋, 𝜋, 3𝜋, 5𝜋,...} → R by sin(x) tan(x) = cos(x) where 𝜋 = 3.14159... is the ratio of circle’s circumference to its diameter. The function tan is called tangent. Remarks Z ◁ Tangent is not defined for x = 𝜋 + 2𝜋n for n ∈ , since the denominator in the formula above assumes value zero at these points ◁ We have tan(x + 2𝜋n) = tan(x) for n ∈ Z. 41 Trigonometric functions—properties and graphs Basic properties: ◁ sin2 (x) + cos2 (x) = 1. ◁ cos(x) = sin(x + 𝜋/2). ◁ sin(x + 𝜋) = − sin(x), cos(x + 𝜋) = − cos(x). ◁ sin(0) = 0, sin(𝜋/2) = 1, sin(𝜋) = 0, sin(3𝜋/2) = −1. 4 1.0 0.8 2 0.6 sin(x) cos(x) -6 -4 -2 2 4 6 0.4 tan(x) -2 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 -4 42 Monotone functions—examples R R ◁ A linear polynomial f : → given by f (x) = a1 x + a0 is monotone: if a1 > 0, then f is strictly increasing, if a1 < 0, then f is strictly decreasing, and if a1 = 0, then f is constant, and hence, both increasing and decreasing, but not strictly. 43 Monotone functions—examples R R ◁ A linear polynomial f : → given by f (x) = a1 x + a0 is monotone: if a1 > 0, then f is strictly increasing, if a1 < 0, then f is strictly decreasing, and if a1 = 0, then f is constant, and hence, both increasing and decreasing, but not strictly. R R ◁ A quadratic polynomial f : → given by f (x) = x 2 is not monotone. For instance, f (−2) = 4 < 9 = f (−3) for −2 > −3, but f (3) = 9 > 4 = f (2) for 3 > 2. 43 Monotone functions—examples R R ◁ A linear polynomial f : → given by f (x) = a1 x + a0 is monotone: if a1 > 0, then f is strictly increasing, if a1 < 0, then f is strictly decreasing, and if a1 = 0, then f is constant, and hence, both increasing and decreasing, but not strictly. R R ◁ A quadratic polynomial f : → given by f (x) = x 2 is not monotone. For instance, f (−2) = 4 < 9 = f (−3) for −2 > −3, but f (3) = 9 > 4 = f (2) for 3 > 2. R ◁ Function g : [0, ∞) → given by the same formula g (x) = x 2 is monotone (strictly increasing). Indeed, if x > z ≥ 0, then g (x) = x 2 = x · x > x · z ≥ z · z = z 2 = g (z). 43 Monotone functions—examples R R ◁ A linear polynomial f : → given by f (x) = a1 x + a0 is monotone: if a1 > 0, then f is strictly increasing, if a1 < 0, then f is strictly decreasing, and if a1 = 0, then f is constant, and hence, both increasing and decreasing, but not strictly. R R ◁ A quadratic polynomial f : → given by f (x) = x 2 is not monotone. For instance, f (−2) = 4 < 9 = f (−3) for −2 > −3, but f (3) = 9 > 4 = f (2) for 3 > 2. R ◁ Function g : [0, ∞) → given by the same formula g (x) = x 2 is monotone (strictly increasing). Indeed, if x > z ≥ 0, then g (x) = x 2 = x · x > x · z ≥ z · z = z 2 = g (z). R ◁ Function h : (−∞, 0] → given by the same formula h(x) = x 2 is monotone (strictly decreasing). Indeed, if 0 ≥ x > z, then h(x) = x 2 = x · x ≤ x · z < z · z = z 2 = h(z). 43 Monotone properties of the common functions ◁ f : (0, ∞) → R given by f (x) = x a is: ✓ Strictly increasing when a > 0; ✓ Constant when a = 0 (in this case, f (x) = 1); 1 ✓ Strictly decreasing when a < 0, since f (x) = x −a. 44 Monotone properties of the common functions ◁ f : (0, ∞) → R given by f (x) = x a is: ✓ Strictly increasing when a > 0; ✓ Constant when a = 0 (in this case, f (x) = 1); 1 ✓ Strictly decreasing when a < 0, since f (x) = x −a. ◁ f : R → R given by f (x) = b ax with b > 0 is: ✓ Strictly increasing when b a > 1; ✓ Constant when b a = 1 (in this case, f (x) = 1); ✓ Strictly decreasing when b a < 1. It is convenient to denote c = b a , then f (x) = (b a )x = c x. 44 Monotone properties of the common functions ◁ f : (0, ∞) → R given by f (x) = x a is: ✓ Strictly increasing when a > 0; ✓ Constant when a = 0 (in this case, f (x) = 1); 1 ✓ Strictly decreasing when a < 0, since f (x) = x −a. ◁ f : R → R given by f (x) = b ax with b > 0 is: ✓ Strictly increasing when b a > 1; ✓ Constant when b a = 1 (in this case, f (x) = 1); ✓ Strictly decreasing when b a < 1. It is convenient to denote c = b a , then f (x) = (b a )x = c x. ◁ f : (0, ∞) → R given by f (x) = log (x) with b > 0, b ̸= 1 is: b ✓ Strictly increasing when b > 1; ✓ Strictly decreasing when 0 < b < 1. 44