Podcast
Questions and Answers
What is the relationship between log(x) and ln(x)?
What is the relationship between log(x) and ln(x)?
The function f(x) = x^a is decreasing for any positive value of a.
The function f(x) = x^a is decreasing for any positive value of a.
False
What is the condition for the function f(x) = b^ax to be strictly increasing?
What is the condition for the function f(x) = b^ax to be strictly increasing?
b^a > 1
The expression log_b(xy) is equal to ______.
The expression log_b(xy) is equal to ______.
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Match the following logarithmic expressions with their corresponding properties:
Match the following logarithmic expressions with their corresponding properties:
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If a = -2 in f(x) = x^a, what is the nature of the function?
If a = -2 in f(x) = x^a, what is the nature of the function?
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The logarithmic function log_b(x) is defined for negative values of x.
The logarithmic function log_b(x) is defined for negative values of x.
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What is the value of log_e(e^5)?
What is the value of log_e(e^5)?
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What is the correct expression for tangent in trigonometry?
What is the correct expression for tangent in trigonometry?
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Tangent is defined for all values of x.
Tangent is defined for all values of x.
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What is the value of sin(π/2)?
What is the value of sin(π/2)?
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The equation sin²(x) + cos²(x) = _____ must always hold true.
The equation sin²(x) + cos²(x) = _____ must always hold true.
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Match the trigonometric function with its corresponding output value.
Match the trigonometric function with its corresponding output value.
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The function cotangent is defined as cotan(x) = sin(x) / cos(x).
The function cotangent is defined as cotan(x) = sin(x) / cos(x).
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What is the periodic property of the tangent function?
What is the periodic property of the tangent function?
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What characterizes the limit of a sequence as n approaches infinity?
What characterizes the limit of a sequence as n approaches infinity?
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The sequence (an) = 2^n diverges as n approaches infinity.
The sequence (an) = 2^n diverges as n approaches infinity.
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What is the limiting behavior of the sequence (an) = 2^(-n) as n approaches infinity?
What is the limiting behavior of the sequence (an) = 2^(-n) as n approaches infinity?
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A sequence (an) is said to converge if there exists a natural number N such that for all n > N, |an - x| < ___ for any real number 𝜖 > 0.
A sequence (an) is said to converge if there exists a natural number N such that for all n > N, |an - x| < ___ for any real number 𝜖 > 0.
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For the sequence (an) = 2^(-n), what value best describes its limit as n approaches infinity?
For the sequence (an) = 2^(-n), what value best describes its limit as n approaches infinity?
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Match the following sequences with their limits as n approaches infinity:
Match the following sequences with their limits as n approaches infinity:
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State the definition of a limit for a sequence.
State the definition of a limit for a sequence.
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If a sequence does not converge, it means its limit is necessarily infinity.
If a sequence does not converge, it means its limit is necessarily infinity.
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Which of the following sequences tends to infinity?
Which of the following sequences tends to infinity?
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The sequence an = ln(n) tends to minus infinity.
The sequence an = ln(n) tends to minus infinity.
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What does it mean for a sequence to tend to minus infinity?
What does it mean for a sequence to tend to minus infinity?
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The limit of the sequence an = -n is ____.
The limit of the sequence an = -n is ____.
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Match the following sequences with their limits:
Match the following sequences with their limits:
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If lim an = ∞ and lim bn = ∞, what is lim (an + bn)?
If lim an = ∞ and lim bn = ∞, what is lim (an + bn)?
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The sequence an = (1, 0, 2, 0, 3, 0, 4,...) has a finite limit.
The sequence an = (1, 0, 2, 0, 3, 0, 4,...) has a finite limit.
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What is the relationship between the limits of two sequences?
What is the relationship between the limits of two sequences?
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What does the triangle inequality help establish in the proof of the limit of the sum of sequences?
What does the triangle inequality help establish in the proof of the limit of the sum of sequences?
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If lim $b_n$ = 0, the limit of the ratio $ rac{a_n}{b_n}$ does not exist.
If lim $b_n$ = 0, the limit of the ratio $ rac{a_n}{b_n}$ does not exist.
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What is the limit of the product of two sequences if both sequences have finite limits?
What is the limit of the product of two sequences if both sequences have finite limits?
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If $a_n o a$ and $b_n o b$, then the limit of $a_n + b_n$ is ______.
If $a_n o a$ and $b_n o b$, then the limit of $a_n + b_n$ is ______.
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Match the following limits with their results:
Match the following limits with their results:
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Which condition needs to be satisfied for the limit of the ratio $ rac{a_n}{b_n}$ to exist?
Which condition needs to be satisfied for the limit of the ratio $ rac{a_n}{b_n}$ to exist?
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For any arbitrary $ heta > 0$, it can be concluded that |$a_n - a$| < $ heta$ for all n.
For any arbitrary $ heta > 0$, it can be concluded that |$a_n - a$| < $ heta$ for all n.
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The limit of $ rac{1}{n^k}$ as n approaches infinity is ______.
The limit of $ rac{1}{n^k}$ as n approaches infinity is ______.
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What is the limit of the sequence $b_n = 3 + \frac{4}{n} + \frac{2}{n^2}$ as $n$ approaches infinity?
What is the limit of the sequence $b_n = 3 + \frac{4}{n} + \frac{2}{n^2}$ as $n$ approaches infinity?
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The limit of the sequence $c_n = 5 - \frac{1}{n^2}$ as $n$ approaches infinity is 6.
The limit of the sequence $c_n = 5 - \frac{1}{n^2}$ as $n$ approaches infinity is 6.
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What does the definition of an infinite sum state about the existence of the sum?
What does the definition of an infinite sum state about the existence of the sum?
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For the infinite sum $\sum_{j=1}^{\infty} a_j$, it is defined as the limit of the sequence $b_n = \sum_{j=1}^{n} a_j$ as $n$ approaches ______.
For the infinite sum $\sum_{j=1}^{\infty} a_j$, it is defined as the limit of the sequence $b_n = \sum_{j=1}^{n} a_j$ as $n$ approaches ______.
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What is the behavior of the sum $\sum_{j=1}^{\infty} q^j$, if $0 < |q| < 1$?
What is the behavior of the sum $\sum_{j=1}^{\infty} q^j$, if $0 < |q| < 1$?
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The limit of the sum of a sequence can be calculated by interchanging summation and multiplication.
The limit of the sum of a sequence can be calculated by interchanging summation and multiplication.
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State the limit of the sequence $\frac{3n + 4n + 2}{5n^2 - 1}$ as $n$ approaches infinity.
State the limit of the sequence $\frac{3n + 4n + 2}{5n^2 - 1}$ as $n$ approaches infinity.
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Study Notes
Lecture 3: Mathematical Methods in Economics & Finance
- Course: ECN115 Mathematical Methods in Economics and Finance
- Lecturer: Evgenii Safonov
- Academic Year: 2024-2025
Quiz Information
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Quiz I: Online quiz on Monday, October 14th
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Coverage: Material from the first two lectures and associated homework assignments (excluding the third week's material)
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Weighting: 7.5% of the total grade
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Duration: 1 hour, starting anytime on October 14th (recommended to start before 23:00)
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Instructions: Work independently; each question has multiple variations, and each student receives a different variant
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Question Types:
- Write down an answer (a numerical value)
- Choose one correct option
- Choose all correct options
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Number of Questions: 10 questions, with 3 more challenging questions
Topic Outline:
- Exponential and logarithmic functions
- Trigonometric functions
- Sequences and their limits
- Arithmetic properties of limits
Exponential Function Definition (Informal)
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For given values a, b (where b > 0) ∈ R, a function f : R → R is defined as:
- If ax ∈ Q, then f(x) = bax
- Otherwise if ax ∉ Q, f(x) is defined to render the graph of f on the (x, y) plane a continuous line.
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Special Case: When b = 2.71828… (Euler's number 'e'), the function is represented as f(x) = eax or exp(ax)
Logarithmic Function Definition
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Given b (where b > 0 and b ≠ 1) ∈ (0, 1) ∪ (1, ∞), the function logb : (0, ∞) → R is defined by logb(x) = y such that by = x.
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Remarks:
- The existence and uniqueness of y is assumed.
- The domain comprises strictly positive real numbers since by > 0 for all y ∈ R
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Special Case: When b = 2.71828… = e, the function loge(x) is expressed as ln(x) or loge(x).
Algebraic Properties of Exponential and Logarithmic Functions
- Properties are listed in the respective images.
Monotonicity of Common Functions
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Functions of the form f(x) = xa:
- Strictly increasing when a > 0
- Constant when a = 0
- Strictly decreasing when a < 0
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Functions of the form f(x) = bax (where b > 0):
- Strictly increasing when ba > 1
- Constant when ba = 1
- Strictly decreasing when ba < 1
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Functions of the form f(x) = logb(x) (where b > 0, b ≠ 1):
- Strictly increasing when b > 1
- Strictly decreasing when 0 < b < 1
Trigonometric functions
- sin(x): Defined as the vertical coordinate on a unit circle corresponding to x radians of rotation.
- cos(x): Defined as the horizontal coordinate on a unit circle corresponding to x radians of rotation.
- tan(x): Defined as sin(x)/cos(x). Note that it's undefined when cos(x) = 0.
Properties of Trigonometric functions
- sin2(x) + cos2(x) = 1
- Basic trigonometric identities are listed.
Sequences and Their Limits
- A sequence is an ordered list of numbers, indexed by natural numbers (N). It's a function from N to R.
- Numerical sequences examples are given.
Limits of Sequences
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Definition: The number x is the limit of sequence (an) if, for any є > 0, there is a natural number N such that for all n > N, |an - x| < є. In this case, the sequence converges to x.
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Illustration: This is further illustrated with a worked example involving the sequence an = 2−n.
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Proofs examples and illustrations are also given for both of the above categories.
Arithmetic properties of limits
- Theorem on calculating limits of sums, products, and quotients of sequences. Detailed proofs included.
Calculating Limits
- Examples demonstrate use of the theorems.
Infinite Sums
- Definition for infinite sums of sequences of numbers.
Squeeze Theorem for Sequences
- Statement of the squeeze theorem as it applies to sequences.
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Description
Test your knowledge on logarithmic and trigonometric functions with this comprehensive quiz. Explore concepts such as relationships between logarithms, properties of functions, and trigonometric identities. Perfect for students wanting to deepen their understanding of these critical mathematical areas.
