Advanced Calculus: Definitions and Examples

Advanced Calculus-Exam 1

• This exam is 50 minutes long.
• Calculators, notes, books, or aids are not allowed.
• Showing work and explaining reasoning is necessary.

1. Definitions (20 points)

• A sequence $(s_n)$ converges to $s$ if for every $\epsilon > 0$, there exists $N \in \mathbb{N}$ such that for all $n > N$, $|s_n - s| < \epsilon$.
• The series $\sum_{k=1}^{\infty} a_k$ converges to $L$ if the sequence of partial sums $S_n = \sum_{k=1}^{n} a_k$ converges to $L$.
• The function $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous at $x=c$ if for every $\epsilon > 0$, there exists $\delta > 0$ such that if $|x - c| < \delta$, then $|f(x) - f(c)| < \epsilon$.
• The function $f: \mathbb{R} \rightarrow \mathbb{R}$ is differentiable at $x=c$ if the limit $\lim_{h \to 0} \frac{f(c+h) - f(c)}{h}$ exists.

2. Examples (15 points)

• A bounded sequence that does not converge is $s_n = (-1)^n$.
• A series $\sum_{k=1}^{\infty} a_k$ that converges, but $\sum_{k=1}^{\infty} |a_k|$ diverges, is the alternating harmonic series $\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}$.
• A function $f: \mathbb{R} \rightarrow \mathbb{R}$ that is continuous everywhere, but differentiable nowhere, is the Weierstrass function.

3. Continuity Proof (20 points)

• To prove $f+g$ is continuous at $x=c$, we must show for every $\epsilon>0$, there exists $\delta>0$ such that if $|x-c|<\delta$, then $|(f+g)(x)-(f+g)(c)|<\epsilon$.
• Since $f$ and $g$ are continuous at $x=c$, for every $\epsilon/2 > 0$, there exist $\delta_1 > 0$ and $\delta_2 > 0$ such that if $|x-c| < \delta_1$, then $|f(x) - f(c)| < \epsilon/2$, and if $|x-c| < \delta_2$, then $|g(x) - g(c)| < \epsilon/2$.
• Let $\delta = \min(\delta_1, \delta_2)$. If $|x-c| < \delta$, then $|f(x) - f(c)| < \epsilon/2$ and $|g(x) - g(c)| < \epsilon/2$.
• Then, $|(f+g)(x) - (f+g)(c)| = |f(x) + g(x) - f(c) - g(c)| \leq |f(x) - f(c)| + |g(x) - g(c)| < \epsilon/2 + \epsilon/2 = \epsilon$. Thus, $f+g$ is continuous at $x=c$.

4. Differentiability Implies Continuity Proof (15 points)

• Assume $f:\mathbb{R} \rightarrow \mathbb{R}$ is differentiable at $x=c$.
• This means $\lim_{x \to c} \frac{f(x)-f(c)}{x-c} = f'(c)$ exists.
• Write $f(x) = f(c) + (x-c) \cdot \frac{f(x)-f(c)}{x-c}$ for $x \neq c$.
• Taking the limit as $x \rightarrow c$, $\lim_{x \to c} f(x) = \lim_{x \to c} f(c) + \lim_{x \to c} (x-c) \cdot \lim_{x \to c} \frac{f(x)-f(c)}{x-c}$.
• Therefore, $\lim_{x \to c} f(x) = f(c) + 0 \cdot f'(c) = f(c)$. This means $f$ is continuous at $x=c$.

5. Series Convergence (30 points)

a) $\sum_{k=1}^{\infty} \frac{1}{k \cdot 2^k}$

• Using the Ratio Test: Let $a_k = \frac{1}{k \cdot 2^k}$. Then $\frac{a_{k+1}}{a_k} = \frac{k \cdot 2^k}{(k+1) \cdot 2^{k+1}} = \frac{k}{2(k+1)}$.
• $\lim_{k \to \infty} |\frac{a_{k+1}}{a_k}| = \lim_{k \to \infty} \frac{k}{2(k+1)} = \lim_{k \to \infty} \frac{1}{2(1 + \frac{1}{k})} = \frac{1}{2}$.
• Since the limit is $\frac{1}{2} < 1$, the series converges by the Ratio Test.

b) $\sum_{k=1}^{\infty} \frac{k^3}{3^k}$

• Using the Ratio Test: Let $a_k = \frac{k^3}{3^k}$. Then $\frac{a_{k+1}}{a_k} = \frac{(k+1)^3}{3^{k+1}} \cdot \frac{3^k}{k^3} = \frac{(k+1)^3}{3k^3}$.
• $\lim_{k \to \infty} |\frac{a_{k+1}}{a_k}| = \lim_{k \to \infty} \frac{(k+1)^3}{3k^3} = \lim_{k \to \infty} \frac{k^3 + 3k^2 + 3k + 1}{3k^3} = \lim_{k \to \infty} \frac{1 + \frac{3}{k} + \frac{3}{k^2} + \frac{1}{k^3}}{3} = \frac{1}{3}$.
• Since the limit is $\frac{1}{3} < 1$, the series converges by the Ratio Test.

c) $\sum_{k=1}^{\infty} (-1)^k \cdot \frac{1}{\sqrt{k}}$

• Using the Alternating Series Test: Let $a_k = \frac{1}{\sqrt{k}}$. We check if $a_k$ is decreasing and $\lim_{k \to \infty} a_k = 0$.
• $a_k = \frac{1}{\sqrt{k}}$ is decreasing because $\sqrt{k}$ is increasing.
• $\lim_{k \to \infty} \frac{1}{\sqrt{k}} = 0$.
• Therefore, the given series converges by the Alternating Series Test.

Chemical Principles

5.1 Gas Pressure

• Pressure is defined as force per unit area: $Pressure = \frac{Force}{Area}$.
• SI unit of pressure is the pascal (Pa): $1 Pa = 1 \frac{N}{m^2}$.
• A related unit is the bar: $1 bar = 10^5 Pa = 100 kPa$.
• Atmospheric pressure is the pressure exerted by the Earth's atmosphere.
• Standard atmospheric pressure (1 atm) is 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

Boyle's Law

• The volume of a gas is inversely proportional to its pressure: $P \propto \frac{1}{V}$.
• $P = k_1 \frac{1}{V}$, and $PV = k_1$.
• For a given amount of gas at constant temperature: $P_1V_1 = P_2V_2$.

Charles's Law

• The volume of a gas is directly proportional to its absolute temperature: $V \propto T$.
• $V = k_2T$, and $\frac{V}{T} = k_2$.
• For a given amount of gas at constant pressure: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$.
• Temperature must be in Kelvin: $K = \degree C + 273.15$

Avogadro's Law

• The volume of a gas is directly proportional to the number of moles: $V \propto n$.
• $V = k_3n$, and $\frac{V}{n} = k_3$.
• For a gas at constant temperature and pressure: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$.

5.3 The Ideal Gas Law

Ideal Gas Law

• Combining Boyle's, Charles's, and Avogadro's laws gives the ideal gas law: $PV = nRT$.
• $R$ 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

• Standard temperature and pressure (STP) is defined as 0 °C and 1 atm.
• $T = 0 \degree C = 273.15 K$ and $P = 1 \text{ atm}$.
• The standard molar volume of an ideal gas is the volume occupied by one mole of the gas at STP, 22.4 L.

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 as: $d = \frac{PM}{RT}$.
• Molar mass of a gas can be determined by: $M = \frac{mRT}{PV}$.

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: $P_T = P_1 + P_2 + P_3 +...$
• $P_T = n_T \frac{RT}{V}$.
• The mole fraction of a gas is the ratio of the number of moles of that gas to the total number of moles: $\chi_1 = \frac{n_1}{n_T}, \quad \chi_2 = \frac{n_2}{n_T}, \quad \chi_3 = \frac{n_3}{n_T}, \quad...$
• The partial pressure of a gas is the product of its mole fraction and the total pressure: $P_1 = \chi_1 P_T, \quad P_2 = \chi_2 P_T, \quad P_3 = \chi_3 P_T, \quad...$

Cardiovascular System

Heart

Structure

• Size of your fist.
• Located in the thoracic cavity.
• Pericardium: The membrane that surrounds and protects the heart.
  • Fibrous pericardium: Tough, inelastic outer layer. Prevents overstretching, provides protection and anchorage.
  • Serous pericardium: Thinner, double layer.
    • Parietal layer: Fused to the fibrous pericardium.
    • Visceral layer: (Epicardium) Layer of the heart wall adhering to the surface.
    • Pericardial cavity: Contains pericardial fluid, lubricating serous fluid to reduce friction.
• Heart Wall
  • Epicardium: (Visceral Layer) Thin, transparent layer with blood vessels, lymphatics, and adipose tissue.
  • Myocardium: Cardiac muscle layer responsible for pumping action; 95% of the heart wall.
  • Endocardium: Thin, smooth lining for chambers and valves; continuous with blood vessel lining.
• Chambers
  • Atria: Two superior chambers receiving blood from veins.
    • Auricles: Pouch-like structures on the anterior surface increasing atrial capacity.
  • Ventricles: Two inferior chambers ejecting blood into arteries.
• Sulci: Grooves on the surface that contain coronary blood vessels and fat.
  • Coronary sulcus: Encircle the heart and mark the boundary between atria and ventricles.
  • Anterior/Posterior interventricular sulcus: Mark the boundary between the ventricles on the anterior/posterior surface
• Right Atrium: Receives blood from three veins.
  • Superior vena cava: From head, neck, upper limbs and chest.
  • Inferior vena cava: From trunk, viscera, and lower limbs.
  • Coronary sinus: From the heart itself.
  • Fossa ovalis: Remnant of the foramen ovale in the fetal heart.
• Right Ventricle: Receives blood from the right atrium; pumps blood into the pulmonary trunk to the lungs.
  • Tricuspid valve: (Right atrioventricular valve) Allows blood passage from right atrium to right ventricle.
  • Chordae tendineae: Connect cusps of tricuspid valve to papillary muscles.
  • Papillary muscles: Cone-shaped trabeculae carneae.
  • Interventricular septum: Separates the right and left ventricles.
  • Pulmonary valve: Allows blood passage from the right ventricle to the pulmonary trunk.
• Left Atrium: Receives blood from the lungs through four pulmonary veins.
  • Bicuspid valve: (Mitral valve, Left atrioventricular valve) Allows blood passage from left atrium to left ventricle.
• Left Ventricle: Receives blood from the left atrium; pumps blood into the aorta to the body.
  • Aortic valve: Allows blood passage from the left ventricle to the aorta.
  • Ligamentum arteriosum: Remnant of the ductus arteriosus, a fetal blood vessel, between the pulmonary trunk and the aorta.
• Thickness of Ventricles: The left ventricle wall is thicker because it pumps blood to the entire body.

Valves and Circulation

• Valves: Ensure one-way blood flow.
  • Atrioventricular valves: Between atria and ventricles.
    • Tricuspid vale: Right side.
    • Bicuspid valve: (Mitral valve) Left side.
  • Semilunar valves: Between ventricles and arteries.
    • Pulmonary valve: Right side.
    • Aortic valve: Left side.
• Pulmonary Circulation: Right ventricle → pulmonary trunk → pulmonary arteries → lungs → pulmonary veins → left atrium.
• Systemic Circulation: Left ventricle → aorta → arteries → arterioles → capillaries → venules → veins → superior vena cava, inferior vena cava, coronary sinus → right atrium

Cardiac Muscle

• Striated: Contains sarcomeres
• Short, branching cells: Allows for rapid spread of signal
• One nucleus per cell
• Intercalated discs: Connect cardiac muscle cells to each other.
  • Desmosomes: Provide strength
  • Gap junctions: Allow action potentials to spread from one cell to another.
• Autorhythmic cells: Specialized cardiac muscle cells that repeatedly and spontaneously generate action potentials.
  • Sinoatrial (SA) node: the main pacemaker of the heart, located in the right atrium
  • Atrioventricular (AV) node: located in the interatrial septum
  • Atrioventricular (AV) bundle: (a.k.a. bundle of His) located in the interventricular septum
  • Right and left bundle branches: located in the interventricular septum
  • Purkinje fibers: located in the apex of the heart and the outer walls of the ventricles

Conduction System

• Action potential initiated by the SA node spreads out through the atria.
• Atria contract.
• Action potential reaches the AV node.
• Action potential passes through the AV bundle.
• Action potential splits into the right and left bundle branches.
• Action potential travels up the Purkinje fibers.
• Ventricles contract.

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

Cardiac Cycle

• Systole: Contraction
• Diastole: Relaxation
• Atrial systole: Atria contract, 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: Beats per minute.
• Stroke volume: Blood ejected per ventricular contraction.
• Cardiac output: Blood ejected per minute.
  • $CO = HR \times SV$.

Blood Vessels

• Arteries: Carry blood away from the heart.
  • Elastic arteries: Largest; walls contain elastin
    • Aorta:
    • Pulmonary trunk.
  • Muscular Arteries: Medium Sized
    • Brachial artery:
    • Radial artery:
  • Arterioles: Deliver blood to capillaries.
• Capillaries: Exchange between blood and tissues.
  • Capillary beds: Networks of capillaries
  • Precapillary sphincters: Control blood flow.
• Venules: Collect blood from capillaries.
• Veins: Carry blood back to the heart.
  • Valves: Prevent backflow
  • Skeletal muscle pump: Promotes blood return to the heart from the limbs.
• Blood pressure: Pressure in blood vessels.
  • Systolic blood pressure: Pressure during ventricular contraction.
  • Diastolic blood pressure: Pressure during ventricular relaxation.
  • Pulse pressure: The difference between systolic and diastolic blood pressure - $Pulse Pressure = Systolic BP - Diastolic BP$
  • Mean arterial pressure (MAP): Average arterial blood pressure.
    • $MAP = Diastolic BP + \frac{1}{3} (Systolic BP - Diastolic BP)$

Blood

• Plasma: Liquid portion, contains water, proteins, electrolytes, hormones, gases, waste products
• Formed elements: Cells and cell fragments
  • Red blood cells: Erythrocytes transport oxygen/carbon dioxide.
  • White blood cells: Leukocytes protect from disease
    • Neutrophils: Phagocytize bacteria
    • Lymphocytes: Involved in immune responses
      • T cells: Attack viruses, fungi, transplanted cells, cancer cells, and some bacteria
      • B cells: Develop into plasma cells that produce antibodies
      • Natural killer (NK) cells: Attack a wide variety of infectious microbes and certain spontaneously arising tumor cells
    • Monocytes: Develop into macrophages.
    • Eosinophils: Combat histamine effects, phagocytize antigen-antibody complexes.
    • Basophils: Liberate heparin, histamine, and serotonin; intensify inflammatory response
  • Platelets: Thrombocytes release chemicals that promote blood clotting.

Blood Groups

• Based on antigens on RBC surface
• ABO blood groups:
  • Type A: A antigen, anti-B antibodies
  • Type B: B antigen, 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
• Transfusion reactions: Antibodies bind to antigens.
• Universal recipient: Type AB positive.
• Universal donor: Type O negative.

Disorders

• Hypertension: High blood pressure.
• Atherosclerosis: Arterial hardening due to plaque.
• Myocardial infarction: Heart attack. Death of cardiac muscle tissue due to lack of blood supply.
• Strok: Death of brain tissue due to lack of blood supply
• Anemia: RBC or hemoglobin deficiency.
• Leukemia: Cancer of blood forming tissues: abnormal WBCs.
• Arrhythmia: Irregular heartbeat.

Fonction Exponentielle

I. Definition

Definition and fundamental property

The exponential function, denoted exp, is the only differentiable function on $\mathbb{R}$ such that:

• $\exp'(x) = \exp(x)$
• $\exp(0) = 1$
• Fundamental Property:* For all real numbers $a$ and $b$: $\qquad \exp(a + b) = \exp(a) \times \exp(b)$

Notation $e^x$

For any real $x$, $\exp(x) = e^x$ $e$ is the number such that $\exp(1) = e \approx 2.718$.

• 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. Study of the exponential function

Sign and variations

The exponential function is strictly positive on $\mathbb{R}$.

$\qquad$ For all real numbers $x, e^x > 0$

The exponential function is strictly increasing on $\mathbb{R}$.

Limits

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

Derivative

The exponential function is differentiable on $\mathbb{R}$ and its derivative is itself.

$\qquad (e^x)' = e^x$

More generally, if $u$ is a differentiable function on an interval $I$:

$\qquad (e^{u(x)})' = u'(x)e^{u(x)}$

Variation Table

Graphic representation

Exponential function graph $y = e^x$.

III. Equations with exponentials

Fundamental Theorem

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

For all real numbers $a$ and $b$:

$\qquad e^a = e^b \Leftrightarrow a = b$

Inequalities

Since the exponential function is strictly increasing on $\mathbb{R}$, for all real numbers $a$ and $b$:

• $e^a < e^b \Leftrightarrow a < b$
• $e^a > e^b \Leftrightarrow a > b$

Examples

Solving the equations and inequalities:

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

• Solutions :*

1. $x = 3$
2. $x = \sqrt{2}$ ou $x = -\sqrt{2}$
3. $x > 1$

Automic Radius

• Atomic radius is a measure of the size of an atom.
• The covalent radius is the distance between the nuclei of two identical atoms bonded together.
• The metallic radius is defined as half the distance between the nuclei of two adjacent atoms in a solid metal.
• The Van der Waals radius is defined as half the distance between the nuclei of two adjacent atoms in a solid nonmetallic element.

Trends in Atomic Radius

Within a Group

• Atomic radius increases from top to bottom.
• Increasing due to the addition of new electron shells.

Within a Period

• Atomic radius decreases from left to right.
• Decreasing due to an increase in the effective nuclear charge.

Factors Affecting Atomic Radius

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

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

• It is an important factor in determining 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.