Calculus - Integration Techniques

Calculus - Integration

• Indefinite Integral: ∫f(x)dx=F(x)+C, where F(x) is the antiderivative of f(x).
• Antiderivative: d/dx F(x) = f(x)

Fundamental Theorem of Calculus

• ∫ab f(x) dx = F(b) - F(a), where F(x) is the antiderivative of f(x).
• The antiderivative is defined as: d/dx F(x) = f(x)

Basic Integration Rules

• Constant Rule: ∫a f(x) dx = a ∫ f(x) dx
• Sum Rule: ∫ f(x) + g(x) dx = ∫ f(x) dx + ∫ g(x) dx
• Power Rule: ∫ x^n dx = (x^(n+1))/(n+1) + C, n ≠ -1
• Trigonometric Integrals: ∫ sin(x) dx = -cos(x) + C and ∫ cos(x) dx = sin(x) + C

U-Substitution Steps

• Choose a u, usually the "inside function".
• Compute du/dx.
• Solve for dx.
• Substitute u and dx into the integral.
• Evaluate the integral.
• Substitute x back in for u.

Integration by Parts

• ∫ u dv = uv - ∫ v du

Trigonometric Substitution

• For √a^2 - x^2, use x = a sin(θ).
• For √a^2 + x^2, use x = a tan(θ).
• For √x^2 - a^2, use x = a sec(θ).
• Then compute dx, substitute x and dx into the integral, evaluate the integral, and substitute x back in for θ.

Partial Fractions

• Factor the denominator.
• Write the partial fraction decomposition.
• Multiply both sides by the denominator.
• Solve for the unknown coefficients and integrate each term.

Improper Integrals

• ∫a^∞ f(x) dx = lim (t→∞) ∫ab f(x) dx
• ∫-∞b f(x) dx = lim (t→-∞) ∫tb f(x) dx
• ∫-∞^∞ f(x) dx = ∫-∞c f(x) dx + ∫c^∞ f(x) dx

Volumes of Revolution

• Disk Method: V = ∫ab π [f(x)]^2 dx
• Washer Method: V = ∫ab π ([f(x)]^2 - [g(x)]^2) dx
• Shell Method: V = ∫ab 2π x f(x) dx

Algorithmic Trading

• Algorithmic trading uses programs to execute trades based on predefined instructions.
• Aims to achieve profits at speeds humans cannot match, relying on analysis.
• Involves Backtesting and forward testing to validate algorithms.

Common Algorithmic Trading Strategies

• Trend Following: Follows market trends, using moving averages and channel breakouts for entry and trailing stop losses for exits; performs well in trending markets but is susceptible to whipsaws.
• Mean Reversion: Exploits price tendencies to revert to average values, using RSI or Bollinger Bands for entry and profit targets near the mean with stop losses beyond extremes; effective in range-bound markets but risky in trending markets.
• Arbitrage: Exploits price differences in different markets, entering when a price discrepancy exists and exiting when it disappears; characterized by low risk and high frequency, requiring fast execution.
• Statistical Arbitrage: Employs statistical models to identify mispricings between assets, entering on deviation from correlations and exiting upon restoration, requiring large datasets and involving higher risks.
• Market Making: Provides market liquidity by placing buy and sell orders based on order book dynamics, managing inventory with dependency on speed.
• High-Frequency Trading (HFT): Executes numerous orders at extremely high speeds based on complex, event-driven triggers with rapid profit targets, requiring co-location, advanced tech, and is heavily regulated.
• Event-Driven Trading: Reacts to news or economic releases, entering on news sentiment analysis and exiting with short-term price targets, relying on information and rapid execution but with vulnerability to misinformation.
• Execution Algorithms: Optimizes large orders like VWAP and TWAP, used by institutional investors, for example, to minimize market impact, completing orders as the exit signal.

Key Algorithmic Trading Considerations

• Backtesting is crucial.
• Avoid overfitting strategies to past data.
• Implement risk management.
• Factor in transaction costs.
• Ensure adaptability of algorithms to changing markets.

Capítulo 1 Conjuntos

• The notion of a set is primitive and undefined in mathematics.
• A set is a collection of objects (called elements)

Representation of a Set

• By enumeration. List all elements between braces, separated by commas.
• By comprehension. Enunciate the characteristic property of its elements
• By Venn diagram. Represent the set by a region limited by a closed curve, not necessarily circular; its elements are represented inside that region.

Relation of Pertinence

• $x ∈ A$ which means x belongs to A
• $x ∉ A$ which means x does not belong to A

Special Sets

• Unitary Set: A set that has only one element.
• Empty Set: A set that has no elements. $∅$ or $\lbrace \rbrace$.
• Universe Set: Contains all elements that can be used in a particular context.

Equality between sets

• Sets A and B are equal if every element of A is an element of B and every element of B is an element of A.

Set of Parts

• The set of parts of A, indicated by P(A), is the set formed by all subsets of A.
• If a set A has n elements, then the number of elements in the set of parts of A is. $#P(A) = 2^n$

Relations Between Sets

• Inclusion: Set A is contained in Set B if every element A is also an element of B. $A \subset B$

Operations with Sets

• Union: The union of two sets A and B, indicated by $\A \cup B$, is the set formed by elements that belong to A or to B or to both.
• Intersection: The intersection of two sets A and B, indicated by $A \cap B$, is the set formed by all elements which belong to A and B at the same time.
• Difference The difference between two sets A and B, indicated by $A - B (or A \setminus B )$, is the set formed by all elements that belong to A and do not belong to B.
• Complement Let A and B be two sets with $A \subset B$. The complement of A with respect to B, indicated by $C_B^A$, is the difference set $B - A$.

Physics

Vectors - Adding Vectors Mathematically

• Vectors are added by summing their components.
• For vectors $\vec{A} = A_x \hat{i} + A_y \hat{j}$ and $\vec{B} = B_x \hat{i} + B_y \hat{j}$, the sum is $\vec{A} + \vec{B} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j}$.

Adding Vectors Geometrically

• Draw vectors starting from the origin.
• Move the tail of the second vector to the head of the first.
• The resultant vector goes from the origin to the head of the second vector.

Vector Decomposition Utilizes trigonometry to resolve forces into components.

• $\sin(\theta) = opposite / hypotenuse$
• $\cos(\theta) = adjacent / hypotenuse$
• $\tan(\theta) = opposite / adjacent$

Example of Vector Decomposition

• A force vector $\vec{F}$ with magnitude 10 N at an angle of 30° above the x-axis decomposes into $F_x = 10\cos(30°) = 8.66 N$ and $F_y = 10\sin(30°) = 5 N$.

Dot Product

• $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta)$

Cross Product

• $\vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin(\theta) \hat{n}$, where $\hat{n}$ is a unit vector perpendicular to both vectors, found with the right-hand rule.

Bernoulli's Principle

• Faster-moving air has lower pressure.
• Airplane wings are designed to have faster airflow on top for lower pressure.
• This creates lift because the pressure is reduced.
• Carburetors use a narrow section to speed up airflow creating low pressure.
• Low pressure is used to draw fuel into the airstream, for combustion.
• Blowing air across a tube reduces pressure, causing liquid to rise and create a spray.

Summary Table

• Bernoulli's Principle - Airplane Wings - Lift (Upward Force)
• Bernoulli's Principle - Carburetors - Fuel-Air Mixture Creation
• Bernoulli's Principle - Atomizer/Sprayer - Fine Spray

Cardiovascular System

• Arteries: Carry blood away from the heart.
• Tunica intima (innermost): Endothelium.
• Tunica media (middle): Smooth muscle (vasoconstriction and vasodilation)
• Tunica externa (outermost): Connective tissue.
• Capillaries: Smallest vessels with single-cell walls (endothelium) for gas and nutrient exchange.
• Veins: Carry blood to the heart, thinner walls, and have valves to prevent backflow.

Blood Pressure

• Force of blood against vessel walls, measured in mm Hg.
• Systolic: Heart beats during contraction.
• Diastolic: Heart at rest. Normal is roughly 120/80 mm Hg.

Factors:

• Blood volume: Increase (BP Increase)
• Heart rate: Increase (BP Increase)
• Blood viscosity: Increase (BP Increase)
• Peripheral resistance: Increase (BP Increase)

Blood:

• Fluid matrix: plasma
• Plasma: 90% water with dissolved substances
• Formed elements: (erythrocytes) carry oxygen, (leukocytes) fight infection, and (thrombocytes) for clotting.

Blood Types

• A: A antigens, anti-B antibodies
• B: B antigens, anti-A antibodies
• AB: A and B antigens, no antibodies
• 0: No antigens, A and B antibodies

Rh Factor

• Rh Positive: Is present
• Rh Negative: Absent

Functions Vectoriales de Variable Real

• Functions assign a vector to each real number in its domain.
• $\vec{r}(t) = (x(t), y(t), z(t))$, where $x(t)$, $y(t)$ and $z(t)$ are real functions.

Gráfica

• A graph is a set of points $(x(t), y(t), z(t))$, the function represents a hélice in 3 dimensions.

Derivada

• Se calcula componente a componente

Integral

• La integral de la función vectorial $\vec{r}(t)$ se define as. $\qquad \int \vec{r}(t) dt = \left( \int x(t) dt, \int y(t) dt, \int z(t) dt \right)$

Longitud de Arco

• $\qquad L = \int_a^b ||\vec{r}'(t)|| dt = \int_a^b \sqrt{(x'(t))^2 + (y'(t))^2 + (z'(t))^2} dt$
• Funciones vectoriales are used in physics and engeneerings

Lab 6: VSEPR and Molecular Geometry

• Part 1: Exploring VSEPR Theory with Models
• Molecule Formula | Lewis Structure | Electron Groups Around Central Atom | Bonding Groups | Lone Pairs | Electron Geometry | Molecular Geometry | Bond Angle | Is the Molecule Polar?
• Use VSEPR theory to predict the shapes of the following molecules and ions:

1. $PO_4^{3-}$
2. $H_3O^+$
3. $XeO_3$
4. $SCl_2$
5. $ICl_4^-$
6. $SF_4$

Post-Lab Questions

• Determine the polarity of a molecule and physical properties of a substance

Physics Equations

Mechanics

• Kinematics equations such as $\bar{v}=\frac{\Delta x}{\Delta t}$ and dynamics equations such as $\Sigma \vec{F}=m \vec{a}$

Momentum and Energy

• $p=m v$
• $K=\frac{1}{2} m v^{2}$

Thermodynamics

• $Q=m c \Delta T$
• $Q=m L$

Wave and Optics

• $v=f \lambda$
• $T=\frac{1}{f}$

Electricity and Magnetism

• Coulomb's law: $F=k \frac{q_{1} q_{2}}{r^{2}}$
• $V=I R$

Chapter 14: Vibration

Introduction Vibration is a periodic motion

Simple Harmonic Motion

• Consider a particle oscillating between two extreme positions, A and B, with an equilibrium position at point O.

• The displacement $x$ is the distance of the particle from O at any instant.

• Equation:$\qquad \ddot{x} = - \omega^2 x$

Terms

• Amplitude (A): The maximum displacement from equilibrium position

• Period (T): The time for one complete cycle $\qquad T = \frac{2 \pi}{\omega}$

• Frequency (f): The number of cycles per unit time (units: Hz or $s^{-1}$)

$\qquad f = \frac{1}{T} = \frac{\omega}{2 \pi}$

Vibration of a Spring-Mass System

Undamped Free Vibration

• Consider a mass $m$ attached to a spring with stiffness $k$.
• When the mass is displaced and released, it oscillates with a natural frequency.

Damped Free Vibration

• Damping Ratio ($\zeta$): $\qquad \zeta = \frac{c}{c_c}$

Forced Vibration

• Occurs when the forcing frequency $\omega$ is close to the natural frequency $\omega_n$, resulting in a large amplitude vibration.

Vibration Isolation

• Used to reduce the transmission of vibration from a source to a receiver
• Achieved by using isolators (e.g., springs, rubber mounts) between the source and receiver.