Calculus: Applications of the Derivative

Applications of the Derivative

• The lecture covers applications of derivatives through examples, focusing on finding absolute maximums and minimums, and critical points of functions.

Example 1: Finding Absolute Maxima and Minima

• Given function: ( f(x) = x^3 - 3x + 5 ) on interval ([-3, 2]).
• Compute the derivative, ( f'(x) = 3x^2 - 3 ), and factor to find critical points.
• Set ( f'(x) = 0 ) to find critical points at ( x = 1 ) and ( x = -1 ).
• Evaluate ( f(x) ) at critical points and endpoints: ( f(-3) = -13 ), ( f(-1) = 7 ), ( f(1) = 3 ), ( f(2) = 7 ).
• The absolute maximum is ( 7 ) at ( x = -1 ) and ( x = 2 ), and the absolute minimum is ( -13 ) at ( x = -3 ).

Example 2: Finding Critical Points

• Given function: ( f(x) = x^{3/5}(4-x) ).

• Apply the product rule and simplify:

  ( f'(x) = \frac{3}{5}x^{-2/5}(4-x) + x^{3/5}(-1) = \frac{12-8x}{5x^{2/5}} )

• Find where ( f'(x) = 0 ), which occurs at ( x = \frac{3}{2} ).

• Find where ( f'(x) ) is undefined, which occurs at ( x = 0 ).

• Critical points are ( x = 0 ) and ( x = \frac{3}{2} ).

Example 3: Finding Critical Points

• Given function: ( f(x) = \sqrt{x}(x-3) ).
• Rewrite as ( f(x) = x^{1/2}(x-3) ).
• Apply the product rule and simplify: ( f'(x) = \frac{1}{2}x^{-1/2}(x-3) + x^{1/2}(1) = \frac{3x-3}{2\sqrt{x}} ).
• ( f'(x) = 0 ) when ( x = 1 ).
• ( f'(x) ) is undefined at ( x = 0 ), but ( x = 0 ) is not in the domain of ( f ).
• The only critical point is ( x = 1 ).

Calculus Cheat Sheet: Derivatives

• Basic Derivative formulas for constants, sums, products and quotients.
• Chain Rule for composite functions: $(f(g(x)))' = f'(g(x))g'(x)$
• Common Derivatives include:
  • Power rule: ((x^n)' = nx^{n-1})
  • Exponential functions:((e^x)' = e^x)
  • Logarithmic functions:((\ln(x))' = \frac{1}{x})
  • Trigonometric functions: (\sin(x)' = \cos(x)), (\cos(x)' = -\sin(x)), (\tan(x)' = \sec^2(x)), etc.
  • Inverse Trigonometric functions: ((\sin^{-1}(x))' = \frac{1}{\sqrt{1-x^2}}), ((\tan^{-1}(x))' = \frac{1}{1+x^2}), etc.

Calculus Cheat Sheet: Integrals

• Basic Integral formulas for constants, sums and integration by parts.
• Common Integrals include:
  • Power rule: (\int x^n,dx = \frac{x^{n+1}}{n+1} + C, n\neq -1)
  • Reciprocal Rule: (\int \frac{1}{x},dx = \ln|x| + C)
  • Exponential functions: (\int e^x,dx = e^x + C)
  • Trigonometric functions: (\int \cos(x),dx = \sin(x) + C), (\int \sin(x),dx = -\cos(x) + C), etc.
  • Reciprocal Polynomials: (\int \frac{1}{x^2+1},dx = \tan^{-1}(x) + C), (\int \frac{1}{\sqrt{1-x^2}},dx = \sin^{-1}(x) + C)
• U-Substitution:
  • Method: Substitute (u = g(x)) and (du = g'(x)dx) into (\int f(g(x))g'(x) , dx) to get (\int f(u) , du).

Calculus Cheat Sheet: Trigonometry

• Common identities:
  • Pythagorean: (\cos^2(x) + \sin^2(x) = 1), (\tan^2(x) + 1 = \sec^2(x)), (1 + \cot^2(x) = \csc^2(x))
  • Double angle: (\sin(2x) = 2\sin(x)\cos(x)), (\cos(2x) = \cos^2(x) - \sin^2(x))
  • Power reducing: (\sin^2(x) = \frac{1}{2}(1 - \cos(2x))), (\cos^2(x) = \frac{1}{2}(1 + \cos(2x)))
• Right Triangle definitions for (0 < \theta < \frac{\pi}{2}):
  • (\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}), (\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}), (\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}})
  • (\csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}}), (\sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}}), (\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}})
• Unit Circle Definitions:
  • (\sin(\theta) = \frac{y}{1}) and (\cos(\theta) = \frac{x}{1}) for any angle (\theta)

Calculus Cheat Sheet: Limits

• Precise Definition: (\displaystyle \lim_{x\to a} f(x) = L) if for every (\varepsilon > 0) there is a (\delta > 0) such that whenever (0 < |x-a| < \delta) then (|f(x) - L| < \varepsilon).
• "Working" Definition: (\displaystyle \lim_{x\to a} f(x) = L) if we can make (f(x)) as close to (L) as we want by taking (x) sufficiently close to (a) (on either side of (a)) without letting (x = a).
• Limit Laws:
  • If (\displaystyle \lim_{x\to a} f(x)) and (\displaystyle \lim_{x\to a} g(x)) exist and (c) is a constant, (\displaystyle \lim_{x\to a} cf(x) = c \lim_{x\to a} f(x)), (\displaystyle \lim_{x\to a} f(x) \pm g(x) = \lim_{x\to a} f(x) \pm \lim_{x\to a} g(x)), etc.
  • (\displaystyle \lim_{x\to a} c = c)
  • (\displaystyle \lim_{x\to a} x = a)
  • (\displaystyle \lim_{x\to a} x^n = a^n)
  • (\displaystyle \lim_{x\to a} \sqrt[n]{x} = \sqrt[n]{a}) (if (a) is positive)
  • (\displaystyle \lim_{x\to a} \sqrt[n]{x} = 0) (if (a) is negative and (n) is odd)
• Basic evaluations at infinity: (\displaystyle \lim_{x\to \pm \infty} \frac{1}{x^n} = 0) where (n>0)
• Continuous Functions:
  • Polynomials, Rational Functions (except at zeros of the denominator), Exponential Functions, Logarithmic Functions, Trigonometric Functions, Inverse Trig Functions are continuous.
• Intermediate Value Theorem:
  • If (f(x)) is continuous on ([a, b]) and (M) is any number between (f(a)) and (f(b)), then there exists a number (c) in ((a, b)) such that (f(c) = M).

Plan de travail: Comment étudier efficacement?

• A guide to effective study techniques, focusing on organization, efficient study methods, and creating an optimal study environment.

1. Organisation et planification

• 1.1. Définir des objectifs clairs: Set general goals (succeed academically) and specific goals (excel in math, improve writing, master biology).
• 1.2. Établir un emploi du temps réaliste: Plan weekly schedules, alternating subjects, and including regular breaks.
• 1.3. Prioriser les tâches: Prioritize important or difficult subjects and start with urgent or complex tasks.

2. Méthodes d'étude efficaces

• 2.1. Techniques de lecture active: Taking notes to find the main ideas, highlighting essential information, and formulating questions.
• 2.2. Techniques de mémorisation: Using spaced repetition, creating associations, visualization, and preparing review sheets.
• 2.3. Techniques de résolution de problèmes: Identifying the problem, breaking it into sub-problems, solving each part, and verifying the solution.

3. Environnement d'étude optimal

• 3.1. Choisir un lieu calme et bien éclairé: Avoid distractions and ensure good lighting.
• 3.2. Préparer son matériel: Have all necessary tools within reach and organize the workspace.
• 3.3. Adopter une posture adéquate: Sit correctly and take regular breaks.

4. Conseils supplémentaires

• Working in groups, asking questions, taking care of one's health (sleep, balanced diet, exercise), and staying motivated.

Algorithmic Trading

• Algorithmic trading automates trade execution via computer programs based on predefined instructions considering price, timing, and volume.

Advantages

• Reduced Emotional Influence, Order Execution Speed, Simultaneous Monitoring.

Disadvantages

• Technical Issues, Over-Optimization, Unforeseen Events.

Common Algorithmic Trading Strategies

• Trend Following: Capitalizes on trend persistence, using moving averages to identify and align with trends: Buy when short-term crosses above long-term moving average, sell vice versa.
• Mean Reversion: Bets on price reversion to average by identifying significant asset price deviations: Buy when price drops below threshold from average, sell when price rises above.
• Arbitrage: Exploits asset price discrepancies in different markets by automating identification and execution: Buy low, sell high on different exchanges.
• Market Making: Provides market liquidity by placing buy/sell orders and profiting from the spread.

Important Considerations

• Backtesting: Simulating algorithm performance on historical data.
• Risk Management: Implementing controls to limit losses.
• Regulatory Compliance: Adhering to legal and financial standards.

Estadística Descriptiva

• Descriptive statistics involves gathering, organizing, analyzing, and interpreting data to describe dataset characteristics.

Methods of Descriptive Statistics

• Involve measures of central tendency, dispersion, frequency tables, and various graphs.

Applications of Descriptive Statistics

• Market and Social Studies. In Health, Engineering and Finance.

Example

• Age of 10 people: 25, 30, 28, 22, 24, 32, 26, 29, 27, 31.
  • Mean is 27.4.
  • Median is 27.5.
  • Standard deviation is 2.91.

Fonction Convexe

• Definition: A function ( f: I \rightarrow \mathbb{R} ) is convex if for all ( x, y \in I ) and ( t \in [0,1] ), ( f(tx + (1-t)y) \leq tf(x) + (1-t)f(y) ). It is concave if the inequality is reversed.
• $f$ is convexe if sa courbe représentative est située en dessous de chacune de ses cordes et $f$ est concave si sa courbe représentative est située au dessue de chacune de ses cordes. (Geometric Interpretation)
• Properties include:
  • ( f ) is convex if and only if for all ( x_1, x_2, x_3 \in I ) with ( x_1 < x_2 < x_3 ) where the rate of change from $x_1$ and $x_2$ is smaller than the rate of change of $x_2$ and $x_3$. i.e. (\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}} \leq \frac{f(x_{3})-f(x_{2})}{x_{3}-x_{2}}).
  • For function ( f: I \rightarrow \mathbb{R} ) differentiable on ( I ) is convex on ( I ) is and only if its derivative function ( f^\prime ) is increasing on ( I )
  • For function $f$ of $x$ two times differentiable on $I$ is convex on I only if (f^{\prime \prime}(x)>0)
  • If is continuous
• Examples: The square, exponential functions are convex.$ln(x)$ and the square root functions are concave
• Let $I$ be an interval in (\mathbb{R}.)
• A function is convex is it is below it's curve representation and $vice$ $versa$ for concave funnction

Algorithmic Game Theory - Summer 2023

• Key Details: Lecture on May 17, 2023, Lecturer: Nick Gravin, Scribe: Nitish Lakhanpal

Congestion Games

• A congestion game includes a set of players, resources, strategies, and cost functions for each resource.
• Definition: A congestion game is a tuple ((N, R, {S_i}{i \in N}, {c_e}{e \in R})) where:
  • ( N = {1, 2, ..., n} ) is the set of players
  • ( R ) is the set of resources
  • ( S_i \subseteq 2^R ) is the strategy set of player ( i )
  • ( c_e : \mathbb{N} \rightarrow \mathbb{R} ) is the cost function of resource ( e )
• Cost of a strategy is the cost of its used resources for each player. $cost_i(s) = \sum_{e \in s_i} c_e(n_e(s))$

Congestion Game Properties

• Every congestion game has at least one pure Nash equilibrium.
• Potential Function:
  • Rosenthal's potential function is (\Phi(s) = \sum_{e \in R} \sum_{i=1}^{n_e(s)} c_e(i)).

Proof of Existence of Pure Nash Equilibrium

• Uses improving moves to strictly decrease the potential function.

Example (Braess' Paradox)

• Illustrates scenarios where adding capacity to a network can increase overall congestion.

Definition (Price of Anarchy)

• The Price of Anarchy (PoA) is the ratio of the worst Nash equilibrium to the social optimum.
• (PoA = \frac{\text{cost of worst Nash equilibrium}}{\text{cost of social optimum}})

Definition (Price of Stability)

• The Price of Stability (PoS) is the ratio of the best Nash equilibrium to the social optimum.
• (PoS = \frac{\text{cost of best Nash equilibrium}}{\text{cost of social optimum}})

5. Eigenvalues and Eigenvectors

• Explores the concept, definition, and method for finding eigenvalues and eigenvectors of a matrix.

5.1. Motivation

• Determines if there exists a nonzero vector, where (A\vec{x}) is parallel to (\vec{x}).
• Parallel means: (A\vec{x} = \lambda \vec{x}) for some scalar (\lambda).
• If it can be solved that lambda is an eigenvalue of A, and (\vec{x}) is called an eigenvector of (A corresponding to λ).

5.2. Definition

• Given (A) as an {n*n} matrix.
• Scalar (λ) is called an eigenvalue of (A) if there exists a nonzero vector (\vec{x} such that (A\vec{x}) = λ(\vec{x})
• (\vec{x}) is called an eigenvector of (A) corresponding to (\lambda).

Notes

• (\vec{x} \neq \overrightarrow{0}) by definition.
• If (\vec{x}) is an eigenvector of (A) corresponding to (λ), then (c\vec{x}) is also an eigenvector. The inverse is false.
• Because (A(c\vec{x}) = c(A\vec{x}) = c(\lambda \vec{x}) = \lambda(c\vec{x}))

5.3. How to find eigenvalues?

• We call (det(A - \lambda I)) the characteristic polynomial of (A).
• Therefore, to find the eigenvalues of (A), we need to solve the equation (det(A - \lambda I) = 0)

Example 1

• Solve the eigenvalues of the following matrix by letting its characteristic polynomial be equal to 0.
• A = (\begin{bmatrix} 2 & 2 \ 1 & 3 \end{bmatrix})
• (\Rightarrow \lambda_1 = 1, \lambda_2 = 4)