Advanced Calculus Exam 1

Advanced Calculus-Exam 1

• Instructions: 50 minute time limit with no outside resources.
• Required: Show all work and reasoning.

Definitions

• (sₙ) converges to s: Definition of convergence.
• Σₖ=₁^∞ aₖ converges to L: Definition of series convergence.
• f: ℝ → ℝ is continuous at x=c: Definition of continuity.
• f: ℝ → ℝ is differentiable at x=c: Definition of differentiability.

Examples

• A bounded sequence that does not converge: Give an example or explain why it does not exist.
• A series Σₖ=₁^∞ aₖ which converges, but such that Σₖ=₁^∞ |aₖ| diverges: Give an example or explain why it does not exist.
• A function f: ℝ → ℝ that is continuous everywhere but differentiable nowhere: Give an example or explain why it does not exist.

Proofs

• Prove: If f: ℝ → ℝ and g: ℝ → ℝ are both continuous at x=c, then f+g is continuous at x=c.
• Suppose: f: ℝ → ℝ is differentiable at x=c. Prove: f is continuous at x=c.

Convergence/Divergence Determination

• Determine: Whether the following converge or diverge, state test, and show all work.
• Σₖ=₁^∞ 1/(k ⋅ 2ᵏ)
• Σₖ=₁^∞ k³/3ᵏ
• Σₖ=₁^∞ (-1)ᵏ ⋅ (1/√k)

Chemical Principles

Gas Pressure

• Pressure: Force per unit area.
• Formula: Pressure = Force/Area
• SI unit of pressure: Pascal (Pa) defined as 1 N/m².
• 1 bar equals 10⁵ Pa or 100 kPa.
• Atmospheric pressure: Pressure exerted by Earth's atmosphere.
• Standard atmospheric pressure: 1 atm = 760 mmHg = 760 torr = 101.325 kPa = 1.01325 bar.

Gas Laws

Boyle's Law

• Volume of a gas is inversely proportional to its pressure.
• Formula: P ∝ 1/V, PV = k₁.
• For a given amount of gas at constant temperature: P₁V₁ = P₂V₂.

Charles's Law

• Volume of a gas is directly proportional to its absolute temperature.
• Formula: V ∝ T, V = k₂T, V/T = k₂.
• For a given amount of gas at constant pressure: V₁/T₁ = V₂/T₂.
• Use absolute temperature (Kelvin): K = °C + 273.15.

Avogadro's Law

• Volume of a gas is directly proportional to the number of moles.
• Formula: V ∝ n, V = k₃n, V/n = k₃.
• For a gas at constant temperature and pressure: V₁/n₁ = V₂/n₂.

Ideal Gas Law

• Combination of Boyle's, Charles's, and Avogadro's laws.
• Formula: PV = nRT, where R is the gas constant.
• Gas constant values: R = 0.08206 L⋅atm/mol⋅K = 8.314 J/mol⋅K.
• Ideal gas: A hypothetical gas that obeys the ideal gas law exactly.

Standard Temperature and Pressure (STP)

• STP: 0 °C (273.15 K) and 1 atm.
• Standard molar volume of an ideal gas: 22.4 L at STP.

Applications of Ideal Gas Law

Gas Density and Molar Mass

• Density: Mass per unit volume (d = m/V).
• Formula for density: d = PM/RT.
• Formula for molar mass: M = mRT/PV.

Gas Mixtures and Partial Pressures

Dalton's Law of Partial Pressures

• Total pressure of a gas mixture: Sum of the partial pressures of each gas.
• Formula: P_T = P₁ + P₂ + P₃ + ...
• Formula: P_T = n_T(RT/V)
• Mole fraction of a gas: Ratio of moles of that gas to total moles.
• Partial pressure of a gas: Product of its mole fraction and total pressure.

Cardiovascular System

Heart Structure

• Size: Approximately the size of a fist.
• Location: Thoracic cavity

Pericardium

• Membrane that surrounds and protects the heart.

• Fibrous Pericardium: Outer layer, tough, inelastic, prevents overstretching. Protection and anchorage

• Serous Pericardium: Thinner, more delicate, double layer

  • Parietal Layer: Fused to fibrous pericardium
  • Visceral Layer: (Epicardium). Adheres to the surface of heart. Lubrication that reduces friction.
  • Pericardial cavity- In between parietal and visceral layers

Heart Wall

• Epicardium: (Visceral layer). Thin, transparent layer, contains blood vessels, lymphatics, and adipose tissue
• Myocardium: Cardiac muscle layer, responsible for pumping action, 95% of the heart wall
• Endocardium: Thin layer, smooth lining for the chambers and covers valves

Chambers

• Atria: Two superior chambers, receive blood from veins. Auricles increase capacity of the atria.
• Ventricles: Two inferior chambers, eject blood into arteries.

Sulci

• Grooves on the heart's surface containing coronary blood vessels and fat.
• Coronary Sulcus: separates atria from ventricles.
• Anterior Interventricular Sulcus: separates ventricles on the anterior side.
• Posterior Interventricular Sulcus: separates ventricles on the posterior side.

Right Atrium

• Receives blood from three veins.
• Superior Vena Cava: Brings blood from the head, neck, upper limbs, and chest.
• Inferior Vena Cava: Bring blood from the trunk, viscera, and lower limbs.
• Coronary Sinus: Brings blood from the heart itself.
• Fossa Ovalis: a remnant of the foramen ovale.

Right Ventricle

• Receives blood from the right atrium pumps blood to the lungs via the pulmonary trunk.
• Tricuspid Valve: Blood flows between the right atrium and right ventricle
• Chordae Tendineae: Cords connecting the cusps of the tricuspid valve to the papillary muscles.
• Papillary Muscles: Cone-shaped trabeculae carneae
• Interventricular Septum: Separates right and left ventricles.
• Pulmonary Valve: Blood flows from the right ventricle to the pulmonary trunk.

Left Atrium

• Receives blood from the lungs through pulmonary veins.
• Bicuspid Valve: (AKA mitral). Blood moves from the left atrium to the left ventricle.

Left Ventricle

• Receives blood from the left atrium and pumps blood into the aorta to the body.
• Aortic Valve: Blood passes from the left ventricle to the aorta through this.
• Ligamentum Arteriosum: Remnant of the ductus arteriosus.
• Wall Thickness: The left ventricle is thicker than the right because it pumps blood to the entire body

Valves and Circulation

Valves:

• Ensure one-way flow of blood through the heart.

• Atrioventricular Valves: Between atria and ventricles

  • Tricuspid valve: right side
  • Bicuspid valve: (AKA mitral valve) left side

• Semilunar Valves: Between ventricles and arteries

  • Pulmonary valve: right side
  • Aortic valve: left side

Pulmonary Circulation

• Flow of blood from the right ventricle to the lungs and back to the left atrium.
• Right ventricle → pulmonary trunk → pulmonary arteries → lungs → pulmonary veins → left atrium

Systemic Circulation

• Flow of blood from the left ventricle to the body and back to the right atrium.
• Left ventricle → aorta → arteries → arterioles → capillaries → venules → veins → superior vena cava, inferior vena cava, coronary sinus → right atrium

Cardiac Muscle

• Striated with sarcomeres.

• Short, branching cells for rapid signal spread.

• Autorhythmic cells generate action potentials.

• Intercalated discs: connect cardiac muscle cells to each other

  • Desmosomes: provide strength
  • Gap junctions: allow action potentials to spread from one cell to another

• Sinoatrial (SA) Node: The main pacemaker

• Atrioventricular (AV) Node

Conduction System

• Action potential initiated by the SA node spreads out through the atria.
• Action potential travels through the AV node, bundle, splits into the right and left bundle branches, then up the Purkinje fibers, which leads to contraction.

Electrocardiogram (ECG or EKG)

• A recording of the electrical activity of the heart
• P wave: Atrial depolarization
• QRS complex: Ventricular depolarization
• T wave: Ventricular repolarization

Cardiac Cycle

• Systole: Contraction

• Diastole: Relaxation

• Cardiac output (CO) is the amount of blood ejected from the ventricle per minute

  • CO = HR x SV Heart rate (HR) = the number of heartbeats per minute

Blood Vessels

• Arteries: Carry blood away from the heart.
• Arterioles: Deliver blood to capillaries
• Capillaries: Small vessels that allow for exchange of substances between blood and tissues.
• Venules: Small veins that collect blood from capillaries.
• Veins: Carry blood back to the heart.
• Blood Pressure: Pressure of blood against the walls of blood vessels.

Blood

• Plasma: Liquid portion of blood, contains water, proteins, nutrients, etc.

• Formed elements: Cells & fragments

  • Red blood cells: (Erythrocytes). Transport oxygen and carbon dioxide.
  • White blood cells: (Leukocytes).
  • Platelets: (Thrombocytes). Release chemicals that promote blood clotting.

Blood groups

• ABO blood groups

  • Type A: A antigen, anti-B antibodies
  • Type B: B antigens, anti-A antibodies
  • Type AB: A and B antigens, neither anti-A nor anti-B antibodies
  • Type O: neither A nor B antigens, both anti-A and anti-B antibodies

• Rh blood group

  • Rh positive: Rh antigen
  • Rh negative: no Rh antigen

• Universal recipient: type AB positive

• Universal donor: type O negative

Disorders

• Hypertension: High blood pressure.
• Atherosclerosis: Hardening & narrowing of the arteries.
• Myocardial infarction: (Heart Attack). Death of cardiac muscle tissue.
• Stroke: Death of brain tissue.
• Anemia: A deficiency of red blood cells.
• Leukemia: A cancer of the blood-forming tissues.
• Arrhythmia: Irregular heartbeat.

Fonction Exponentielle

Définition et propriété fondamentale

• The exponential function (exp) is the only differentiable function such that exp'(x) = exp(x) and exp(0) = 1.
• Fundamental property: exp(a + b) = exp(a) * exp(b) for all real numbers a and b.

Notation e^x

• For any real number x, exp(x) = e^x.

• e ≈ 2.718.

• Properties include:

  • e^(x+y) = e^x * e^y
  • e^(x-y) = e^x / e^y
  • e^(-x) = 1 / e^x
  • (e^x)^y = e^(xy)

Signe et variations

• The exponential function is strictly positive: e^x > 0 for all real numbers x.
• The exponential function is strictly increasing on ℝ.

Limites

• lim (x→+∞) e^x = +∞
• lim (x→-∞) e^x = 0

Dérivée

• (e^x)' = e^x.
• More generally, (e^(u(x)))' = u'(x)e^(u(x)) if u is differentiable.

Tableau de variations

• Shows function increasing from 0 to +∞ from -∞ to +∞

Représentation graphique

• The graph shows the exponential function y = e^x, always above zero and increasing.

Théorème fondamental

• The exponential function maps ℝ onto ]0; +∞[ bijectively.
• e^a = e^b if and only if a = b for all real numbers a and b.

Inéquations

• If exponential function is strictly increasing on R:
• e^a < e^b if and only if a < b.
• e^a > e^b if and only if a > b.

Exemples

• Examples of equations with exponentials.

Automic Radius

Atomic Radius

• Atomic radius: Measure of the size of an atom. Defined using covalent, metallic, and Van Der Waals radii. Trends explained by principal quantum number, effective nuclear charge, and shielding effect.

Trends in Atomic Radius

• Within a Group: Increases from top to bottom due to the addition of new electron shells.
• Within a Period: Decreases from left to right due to an increase in the effective nuclear charge.

Factors Affecting Atomic Radius

• Principal Quantum Number (n): As n increases, atomic radius increases.
• Effective Nuclear Charge: increases, the atomic radius decreases.
• Shielding Effect: increases, the atomic radius also increases.

Importance

• An important factor in determining the physical and chemical properties of elements.
• Used to predict the reactivity of elements.
• Used in many industrial applications.