Advanced Calculus Exam 1

Advanced Calculus - Exam 1

• This exam assesses understanding of fundamental concepts in advanced calculus.
• It requires stating definitions, providing examples, and proving theorems.
• The exam includes questions on convergence of sequences and series, continuity, differentiability, and convergence tests.

Question 1: Definitions

• A sequence $(s_n)$ converges to $s$ when its terms get arbitrarily close to $s$ as $n$ approaches infinity.
• The series $\sum_{k=1}^{\infty} a_k$ converges to $L$ when the sequence of its partial sums approaches $L$.
• A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous at $x=c$ when the limit of $f(x)$ as $x$ approaches $c$ is equal to $f(c)$.
• A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is differentiable at $x=c$ when the limit of the difference quotient exists at $x=c$.

Question 2: Examples

• A bounded sequence that does not converge could oscillate between two values, such as $(-1)^n$.
• A series $\sum_{k=1}^{\infty} a_k$ that converges but $\sum_{k=1}^{\infty} |a_k|$ diverges is an alternating series like $\sum_{k=1}^{\infty} \frac{(-1)^k}{k}$.
• A function $f: \mathbb{R} \rightarrow \mathbb{R}$ continuous everywhere but differentiable nowhere is the Weierstrass function.

Question 3: Proof

• The proof involves demonstrating that for any given $\epsilon > 0$, there exists a $\delta > 0$ such that if $|x - c| < \delta$, then $|(f+g)(x) - (f+g)(c)| < \epsilon$.
• Utilizing the continuity of $f$ and $g$ at $x=c$, one can find $\delta$ values to bound $|f(x) - f(c)|$ and $|g(x) - g(c)|$ by $\epsilon/2$, satisfying the continuity condition for $f+g$.

Question 4: Proof

• To prove that differentiability implies continuity, one starts with the definition of the derivative.
• Then manipulate the expression $f(x) - f(c)$ to show it approaches zero as $x$ approaches $c$, thus proving continuity.

Question 5: Series Convergence

• To determine convergence or divergence of series, apply suitable tests.
• Common tests include the ratio test, root test, comparison test, and alternating series test.
• Key steps involve stating the test used, setting up the limit, and showing all the work to reach a conclusion.

Chemical Principles - The Properties of Gases

• Explores gas pressure, gas laws, the ideal gas law, gas mixtures, and related concepts.

5.1 Gas Pressure

• Pressure is defined as force per unit area ($Pressure = \frac{Force}{Area}$).
• The SI unit for pressure is the pascal (Pa), where $1 Pa = 1 \frac{N}{m^2}$.
• Atmospheric pressure is the pressure exerted by the Earth's atmosphere.
• Standard atmospheric pressure is the typical pressure at sea level.
• $1 \text{ atm} = 760 \text{ mmHg} = 760 \text{ torr} = 101.325 \text{ kPa} = 1.01325 \text{ bar}$

5.2 The Gas Laws

• These laws relate volume to pressure, temperature, and number of moles.

Boyle's Law

• The volume of a gas is inversely proportional to its pressure at constant temperature: $P \propto \frac{1}{V}$, leading to $P_1V_1 = P_2V_2$.

Charles's Law

• The volume of a gas is directly proportional to its absolute temperature at constant pressure: $V \propto T$, leading to $\frac{V_1}{T_1} = \frac{V_2}{T_2}$.
• Use Kelvin for absolute temperature: $K = \degree C + 273.15$.

Avogadro's Law

• The volume of a gas is directly proportional to the number of moles at constant temperature and pressure: $V \propto n$, leading to $\frac{V_1}{n_1} = \frac{V_2}{n_2}$.

5.3 The Ideal Gas Law

Ideal Gas Law

• Combines Boyle's, Charles's, and Avogadro's laws: $PV = nRT$, where $R$ is the gas constant.
• $R = 0.08206 \frac{L \cdot atm}{mol \cdot K} = 8.314 \frac{J}{mol \cdot K}$
• An ideal gas is a hypothetical gas that obeys the ideal gas law exactly.

Standard Temperature and Pressure

• STP is defined as $0 \degree C$ ($273.15 K$) and $1 \text{ atm}$.
• The standard molar volume of an ideal gas is $22.4 \text{ L}$ at STP.

5.4 Applications of the Ideal Gas Law

Gas Density and Molar Mass

• Density is mass per unit volume: $d = \frac{m}{V}$.
• Density can be calculated using the ideal gas law: $d = \frac{PM}{RT}$, where $M$ is the molar mass.
• Molar mass can be determined using $M = \frac{mRT}{PV}$.

5.5 Gas Mixtures and Partial Pressures

Dalton's Law of Partial Pressures

• For a mixture of gases, the total pressure is the sum of the partial pressures of each gas: $P_T = P_1 + P_2 + P_3 +...$.
• The mole fraction of a gas is the ratio of the number of moles of that gas to the total number of moles: $\chi_1 = \frac{n_1}{n_T}$.
• The partial pressure of a gas is the product of its mole fraction and the total pressure: $P_1 = \chi_1 P_T$.

Cardiovascular System

• The cardiovascular system includes the heart, blood vessels, and blood.

Heart Structure

• The heart is about the size of your fist and located in the thoracic cavity.
• Pericardium: protects and surrounds the heart.
  • Fibrous pericardium: outer layer, prevents overstretching.
  • Serous pericardium: thinner, double layer.
    • Parietal layer: fused to the fibrous pericardium.
    • Visceral layer (epicardium): adheres to the heart surface.
    • Pericardial cavity: contains pericardial fluid to reduce friction.
• Heart Wall
  • Epicardium (visceral layer): a thin, transparent outer layer with blood vessels.
  • Myocardium: the cardiac muscle layer responsible for pumping.
  • Endocardium: a smooth lining for chambers and valves, continuous with blood vessel lining.
• Chambers
  • Atria: receive blood from veins; auricles increase atrial capacity.
  • Ventricles: eject blood into arteries.
• Sulci: grooves containing coronary blood vessels and fat.
  • Coronary sulcus: marks the boundary between atria and ventricles.
  • Anterior/Posterior interventricular sulci: mark the boundary between the ventricles.
• Right Atrium: receives blood from superior vena cava, inferior vena cava, and coronary sinus; includes fossa ovalis.
• Right Ventricle: receives blood from the right atrium; pumps blood into the pulmonary trunk.
  • Includes the Tricuspid valve, Chordae tendineae, Papillary muscles, Interventricular septum, Pulmonary valve.
• Left Atrium: receives blood from pulmonary veins through four pulmonary veins.
  • Includes the Bicuspid valve.
• Left Ventricle: receives blood from the left atrium, pumps blood into the aorta.
  • Includes the Aortic valve, and the Ligamentum arteriosum.
• Ventricular Thickness: the left is thicker due to systemic circulation pumping.

Valves and Circulation

• Valves ensure one-way blood flow.
  • Atrioventricular valves: Tricuspid (right), Bicuspid (left).
  • Semilunar valves: Pulmonary (right), Aortic (left).
• Pulmonary circulation: Right ventricle → lungs → left atrium.
• Systemic Circulation: Left ventricle → body → right atrium.

Cardiac Muscle

• Characteristics: striated, short branching cells, one nucleus, intercalated discs.
• Includes Desmosomes (provide strength), and Gap junctions (allow action potentials).
• Autorhythmic cells: generate action potentials spontaneously:
• SA node (main pacemaker)
• AV node
• AV bundle
• Bundle branches
• Purkinje fibers

Conduction System

• Action potential from SA node spreads through atria, atria contracts, reaches the AV node, spreads to the AV bundle, Bundle branches, Purkinje fibers, and ventricles contract.

Electrocardiogram (ECG/EKG)

• P wave: atrial depolarization.
• QRS complex: ventricular depolarization.
• T wave: ventricular repolarization.
• P-Q interval: time for the action potential to travel through the atria, AV node, and remaining conducting system.
• S-T segment: time during which the ventricles are contracting and emptying.
• Q-T interval: time from start of ventricular depolarization to end of ventricular repolarization.

Cardiac Cycle

• Systole: contraction,
• Diastole: relaxation.
• Atrial systole: blood is forced into the ventricles.
• Ventricular systole: ventricles contract, blood is forced into the pulmonary trunk and aorta.
• Atrial diastole: atria relax, blood flows in from the vena cava and pulmonary veins
• Ventricular diastole: ventricles relax, blood flows in from the atria
• Heart Rate: the number of heartbeats per minute
• Stroke Volume: the amount of blood ejected from the ventricle with each contraction
• Cardiac Output: the amount of blood ejected from the ventricle per minute
• $CO = HR \times SV$

Blood Vessels

• Arteries: carry blood away from the heart:
  • Elastic arteries: largest, help propel blood.
    • Aorta, Pulmonary trunk.
  • Muscular arteries: medium, vasoconstriction and vasodilation.
    • Brachial, Radial artery
  • Arterioles: small, deliver blood to capillaries.
• Capillaries: exchange between blood and tissues happens here.
  • Capillary beds: networks of capillaries.
  • Precapillary sphincters: control blood flow.
• Venules: collect blood from capillaries.
• Veins: carry blood back to the heart: - Includes Valves and a Skeletal muscle pump that helps to pump blood back to the heart
• Blood pressure: systolic blood pressure or ventricular contraction, diastolic blood pressure or ventricular relaxation
  • Pulse Pressure : $Pulse Pressure = Systolic BP - Diastolic BP$
  • Mean arterial pressure: : $MAP = Diastolic BP + \frac{1}{3} (Systolic BP - Diastolic BP)$

Blood

• Plasma: liquid portion containing a multitude of elements.
• Formed elements:
  • Red blood cells: transport oxygen and carbon dioxide
  • White blood cells: protect the body from disease such as the following:
    • Neutrophils: phagocytize bacteria
    • Lymphocytes: involved in immune responses like T cells, B cells, and NK cells
    • Monocytes: develop into macrophages that phagocytize microbes and cellular debris
    • Eosinophils: combat the effects of histamine destroy certain parasitic worms
    • Basophils: liberate heparin, histamine, and serotonin in allergic reactions
  • Platelets: release chemicals that promote blood clotting

Blood Groups

• Based on antigens on red blood cell surfaces.
• ABO blood groups: A, B, AB, O.
• Rh blood group: Rh positive or Rh negative.
• Transfusion reactions: occur when antibodies in the recipient's blood bind to antigens on the donor's red blood cells.
• Universal recipient: type AB positive.
• Universal donor: type O negative.

Disorders

• Hypertension: high blood pressure.
• Atherosclerosis: narrowing of arteries due to plaque buildup.
• Myocardial infarction: heart attack.
• Stroke: death of brain tissue due to lack of blood supply.
• Anemia: deficiency of red blood cells or hemoglobin.
• Leukemia: cancer of blood-forming tissues.
• Arrhythmia: irregular heartbeat.

Fonction Exponentielle

I. Définition

Définition et propriété fondamentale

• The exponential function, noted exp, is the unique differentiable function on $\mathbb{R}$ such that $\exp'(x) = \exp(x)$ and $\exp(0) = 1$.
• Fundamental Property: For all real numbers $a$ and $b$: $\exp(a + b) = \exp(a) \times \exp(b)$

Notation $e^x$

• For all real $x$, $\exp(x) = e^x$, where $e \approx 2,718$.
• Properties:* For all real numbers $x$ and $y$:
• $e^{x+y} = e^x \times e^y$
• $e^{x-y} = \frac{e^x}{e^y}$
• $e^{-x} = \frac{1}{e^x}$
• $(e^x)^y = e^{xy}$

II. Étude de la fonction exponentielle

Signe et variations

• $\exp(x)$ is strictly positive on $\mathbb{R}$: For all real $x, e^x > 0$
• $\exp(x)$ is strictly increasing on $\mathbb{R}$.

Limites

• $\lim\limits_{x \to +\infty} e^x = +\infty$
• $\lim\limits_{x \to -\infty} e^x = 0$

Dérivée

• The derivative of $e^x$ is $e^x$: $(e^x)' = e^x$
• More generally, if $u$ is a differentiable function on an interval $I$: $(e^{u(x)})' = u'(x)e^{u(x)}$

Tableau de variations

• The table shows variations with x from negative infinity to positive infinity, and exp(x) going from 0 to positive infinity, with the function increasing.

Représentation graphique

• Illustrates the exponential function which is always positive and increasing.

III. Équations avec exponentielles

Théorème fondamental

• The exponential function is a bijection from $\mathbb{R}$ to $]0; +\infty[$.

• For all real numbers $a$ and $b$: $e^a = e^b \Leftrightarrow a = b$

Inéquations

• Since $\exp(x)$ is strictly increasing on $\mathbb{R}$:
• $e^a < e^b \Leftrightarrow a < b$
• $e^a > e^b \Leftrightarrow a > b$

Exemples

• Solutions :*

1. $e^{3x-5} = e^{x+1} \Leftrightarrow x = 3$
2. $e^{x^2-3} = \frac{1}{e} \Leftrightarrow x = \sqrt{2}$ or $x = -\sqrt{2}$
3. $e^{2x+1} > e^{-x+4} \Leftrightarrow x > 1$

Atomic Radius

• Atomic radius is a measure of the size of an atom.
• Several different definitions exist depending on the method used to measure it.

Definitions

• Covalent radius: Half the distance between the nuclei of two identical atoms bonded together.
• Metallic radius: Half the distance between the nuclei of two adjacent atoms in a solid metal.
• Van der Waals radius: Half the distance between the nuclei of two adjacent atoms in a solid nonmetallic element.

Trends in Atomic Radius

• The atomic radius increases from top to bottom within a group due to the addition of new electron shells.
• The atomic radius decreases from left to right within a period due to an increase in the effective nuclear charge.

Factors Affecting Atomic Radius

• Principal Quantum Number (n): As the value of $n$ increases, the atomic radius also increases.
• Effective Nuclear Charge ( $Z_{eff}$): As the $Z_{eff}$ increases, the atomic radius decreases.
• Shielding Effect: As the shielding effect increases, the atomic radius also increases.

Examples

• The atomic radius of sodium (Na) is greater than that of chlorine (Cl).
• The atomic radius of potassium (K) is greater than that of sodium (Na).

Importance of Atomic Radius

• Atomic radius determines the physical and chemical properties of elements.
• It can be used to predict the reactivity of elements.
• It is also used in many industrial applications.