Mathematics: Analysis Course Book Notes

UNIT 1: SEQUENCES AND SERIES

Introduction and Basic Definitions

• Sequence: An enumerated collection of real numbers $a_1, a_2, a_3, \dots$ denoted as $(a_n)$. Every single number in the sequence is a sequence element or member.

• Index: A natural number used to number sequence elements, $(n \in \mathbb{N})$ where $\mathbb{N} = \{1, 2, 3, \dots\}$. Sometimes $\mathbb{N}_0 = \{0, 1, 2, 3, \dots\}$ is used, making $a_0$ the first element.

• Characteristics: A sequence is distinguished from a set by its fixed order and the fact that the same element can occur multiple times.

• Specification Methods:

  • Listing: Writing elements separated by commas, e.g., $b_n = 2, 4, 6, 8, \dots$

  • Formula: An equation for a general member, e.g., $b_n = 2 \times n$ for all $n \in \mathbb{N}$.

  • Recursive Formula: Determining subsequent elements based on preceding ones, provided at least the first element is given.

• Graphical Representation: Sequences are plotted in a coordinate system with index $n$ on the x-axis and values $a_n$ on the y-axis. Elements are individual points; connecting them to form a continuous graph is generally not allowed.

Recursive Sequences and Modeling Examples

• Fibonacci Sequence: Formed recursively with $f_n = f_{n-2} + f_{n-1}$ for $n > 2$, with $f_1 = 0$ and $f_2 = 1$. Elements: $0, 1, 1, 2, 3, 5, 8, 13, 21, 34, \dots$ This sequence often describes natural growth processes.

• Finite Sequence: A sequence with a finite number of elements, defined by a maximum value for the index $n$. Example: $e_n = \frac{1}{n}$ for $1 \le n \le 10$.

• John’s Forest Growth Model: Initial size $w_1 = 1,000$ hectares. Annual growth of $5\%$ and annual clearing of $70$ hectares. Recursive formula: $w_n = 1.05 \times w_{n-1} - 70$ for $n \ge 2$. Elements: $w_1 = 1,000$, $w_2 = 980$, $w_3 = 959$, $w_4 = 936.95$.

• John’s Well Drilling Model: Cost of first meter $z_1 = 80$ euros, each subsequent meter costs $8\%$ more. Formula: $z_n = 1.08 \times z_{n-1}$ for $12 \ge n \ge 2$. Costs for meter 12: $z_{12} = 186.53 \, euros$.

Monotonicity and Boundedness

• Monotonicity Definitions:

  1. Monotone increasing: $a_{n+1} \ge a_n$ for all $n$.

  2. Strictly monotone increasing: $a_{n+1} > a_n$ for all $n$.

  3. Monotone decreasing: $a_{n+1} \le a_n$ for all $n$.

  4. Strictly monotone decreasing: $a_{n+1} < a_n$ for all $n$.

  5. Constant: $a_{n+1} = a_n$ for all $n$.

• Boundedness Definitions:

  • Bounded above: $a_n \le S$ for all $n \in \mathbb{N}$, where $S$ is the upper bound.

  • Bounded below: $a_n \ge s$ for all $n \in \mathbb{N}$, where $s$ is the lower bound.

  • Bounded: A sequence that is bounded from both above and below.

  • Unbounded: A sequence that lacks either an upper bound, a lower bound, or both.

• Alternating Sequence: A sequence where members are alternately positive and negative, e.g., $d_n = (-1)^n$.

Convergence and Limit Values

• Distance: The unsigned difference between two real numbers: $|x - a|$.

• $\epsilon$-neighborhood: The set of all points $x \in \mathbb{R}$ whose distance from $a$ is less than a given small number $\epsilon \, (x \in \mathbb{R} : |x - a| < \epsilon)$. This represents an open interval $(a - \epsilon, a + \epsilon)$.

• Convergence: A sequence $(a_n)$ is convergent with limit $a \in \mathbb{R}$ if for all $\epsilon > 0$, almost all sequence elements lie in the $\epsilon$-neighborhood of $a$. Notation: $\lim_{n \rightarrow \infty} a_n = a$ or $a_n \rightarrow a$.

• Null Sequence: A convergent sequence whose limit value is $0$.

• Divergence: A sequence that does not converge. If elements become arbitrarily large, notation is $\lim_{n \rightarrow \infty} a_n = \infty$. Note that $\infty$ is not a real number limit.

• Theorems on Convergence:

  • Every convergent sequence is bounded.

  • Every unbounded sequence is divergent (contraposition).

  • Every bounded and monotone sequence converges.

• Rules of Convergence (assuming $a_n \rightarrow a$ and $b_n \rightarrow b$):

  • $\lim_{n \rightarrow \infty} (a_n + b_n) = a + b$

  • $\lim_{n \rightarrow \infty} (a_n - b_n) = a - b$

  • $\lim_{n \rightarrow \infty} (a_n \times b_n) = a \times b$

  • $\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = \frac{a}{b}$ (where $b \ne 0, b_n \ne 0$)

  • $\lim_{n \rightarrow \infty} (a_n)^r = a^r$ for $r \in \mathbb{R}$

Arithmetic and Geometric Sequences

• Arithmetic Sequence: The difference $d$ of two consecutive members is constant: $d = a_{n+1} - a_n$. General formula: $a_n = a_1 + (n - 1) \times d$.

  • $d > 0$: Strictly monotone increasing.

  • $d < 0$: Strictly monotone decreasing.

  • $d = 0$: Constant.

• Geometric Sequence: The quotient $q$ of two consecutive members is constant: $q = \frac{a_{n+1}}{a_n}$. General formula: $a_n = a_1 \times q^{n-1}$.

  • $q > 1$: Strictly monotone increasing.

  • $0 < q < 1$: Strictly monotone decreasing and converges to $0$.

  • $q = 1$: Constant.

  • $q < 0$: Alternating.

Series: Definition and Convergence

• Series: A special sequence $(s_n)$ resulting from the stepwise addition of elements of a sequence $(a_n)$.

• $n$-th Partial Sum: The element $s_n = \sum_{i=1}^{n} a_i = a_1 + a_2 + \dots + a_n$.

• Recursive form: $s_{n+1} = s_n + a_{n+1}$.

• Infinite Series: Summation continues to infinity, written as $\sum_{i=1}^{\infty} a_i$.

• Convergence Condition: An infinite series $\sum a_i$ can only converge if $(a_n)$ is a null sequence. This condition is necessary but not sufficient (e.g., the harmonic series $\sum \frac{1}{i}$ diverges despite having a null sequence).

• Arithmetic Series: The sum of terms of an arithmetic sequence. Formula: $s_n = \frac{n}{2}(a_1 + a_n)$.

• Geometric Series: The sum of terms of a geometric sequence. Formula: $s_n = a_1 \times \frac{q^n - 1}{q - 1}$.

  • Convergence: A geometric series converges if and only if $|q| < 1$. The limit is $\lim_{n \rightarrow \infty} s_n = \frac{a_1}{1 - q}$.

Euler’s Number and Specific Limits

• Euler’s Number $e$: Defined as $\lim_{n \rightarrow \infty} (1 + \frac{1}{n})^n = 2.71828\dots$ It is also the limit of the series $\sum_{k=0}^{\infty} \frac{1}{k!}$, where $k!$ (k factorial) is the product $1 \times 2 \times 3 \times \dots \times k$. By convention, $0! = 1$.

• Limit Theorems (for $n \rightarrow \infty$):

  • $\lim \frac{x}{n} = 0$

  • $\lim x^n = 0$ for $|x| < 1$

  • $\lim \sqrt[n]{n} = 1$

  • $\lim \sqrt[n]{x} = 1$ for $x > 0$

  • $\lim (1 + \frac{x}{n})^n = e^x$

  • $\lim (1 - \frac{1}{n})^n = \frac{1}{e}$

• Leibniz Series: Converges to $\frac{\pi}{4}$. Defined as $\sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} = 1 - \frac{1}{3} + \frac{1}{5} - \dots$

• Power Series: $P(x) = \sum_{k=0}^{\infty} a_k x^k = a_0 + a_1 x + a_2 x^2 + \dots$ The partial sums of a power series are polynomials.

UNIT 2: FUNCTIONS AND INVERSE FUNCTIONS

Terms and Definitions

• Function (Map/Mapping): A rule $f: A \rightarrow B$ that uniquely assigns an element from set $B$ to each element from set $A$.

• Domain ($A$): The set from which input values are taken.

• Co-domain ($B$): The set in which the function takes its values.

• Argument (Input value/Pre-image): An element $x \in A$.

• Function Value (Image/Output): The result of applying the function, written as $f(x)$.

• Image of $f$ ($Im(f)$ ): The set of all actual function values achieved.

• Uniqueness: Each argument has exactly one function value, but multiple arguments can map to the same function value.

• Identity Function ($id$): Maps each element to itself, $id(x) = x$.

• Constant Function ($const$): Maps all arguments to a single value $c$, $const(x) = c$.

• Absolute Value Function ($|x|$): Maps negative inputs to positive outputs: $|x| = x$ if $x \ge 0$ and $|x| = -x$ if $x < 0$.

Graphical Representation and Elementary Functions

• Graph: Illustration of point pairs $(x, f(x))$ in a two-dimensional coordinate system with an x-axis (domain) and y-axis (co-domain).

• Linear Functions: $f(x) = a \times x + b$. $a$ is the slope; $b$ is the y-axis intercept.

• Quadratic Functions: $f(x) = a \times x^2 + b \times x + c$, with $a \ne 0$.

  • $a = 1, b = 0, c = 0$ is the Normal Parabola.

  • $a > 0$: Open upward; $a < 0$: Open downward.

  • $c$: Vertical shift.

  • $b$: Vertex shift along the x-axis by $-\frac{b}{2a}$ units.

• Polynomial Functions of Degree $n$: $f(x) = \sum_{k=0}^{n} a_k x^k$.

Composition and Properties of Functions

• Composition: If $f: A \rightarrow B$ and $g: C \rightarrow D$, and $f(x) \in C$, then $(g \circ f)(x) = g(f(x))$. Read as "g after f." It is not commutative ($g \circ f \ne f \circ g$).

• Surjective: Every element in the co-domain $B$ has at least one pre-image in $A$.

• Injective: Different inputs result in different outputs ($x_1 \ne x_2 \implies f(x_1) \ne f(x_2)$).

• Bijective: A function that is both surjective and injective.

Inverse Functions and Invertibility

• Invertibility: A function $f: A \rightarrow B$ is invertible if there exists a function $g: B \rightarrow A$ such that $g \circ f = id$ and $f \circ g = id$. The unique inverse is denoted $f^{-1}$.

• Condition: A function must be bijective to be invertible.

• Graphical property: The graph of an inverse function is the reflection of the original graph across the identity line $y = x$.

• Root Function: The inverse of the restricted quadratic function ($f: \mathbb{R}^+ \rightarrow \mathbb{R}^+, f(x) = x^2$) is $g(x) = \sqrt{x}$.

Exponential and Logarithmic Functions

• General Exponential Function: $f(x) = a^x$ with positive base $a \in \mathbb{R}^+ \setminus \{0\}$.

  • $a > 1$: Strictly monotone increasing.

  • $0 < a < 1$: Strictly monotone decreasing.

  • Asymptote: The x-axis is an asymptote (values approach but never reach $0$).

• Natural Exponential Function ($exp$): Base is Euler’s constant $e \, (f(x) = e^x)$.

• Compound Interest: capital growth after $x$ years with compounding $n$ times per year results in $\lim_{n \rightarrow \infty} (1 + \frac{r}{n})^{nx} = e^{rx}$.

• Logarithm Functions: Inverse of exponential functions, denoted $g(x) = \log_a x$. Only defined for $x > 0$.

• Natural Logarithm ($\ln$): Logarithm with base $e$. Properties: $\ln(e^x) = x$ and $e^{\ln(x)} = x$.

• Growth Rate Relation: $a^x = e^{rx}$ with $r = \ln(a)$. $r$ is the continuous growth rate.

Trigonometric Functions

• Unit Circle: Circle with radius $1$ centered at $(0,0)$. Sine and cosine are definitions of coordinates for a point $P$ with angle $\alpha$.

• Radian Measure ($x$): Dimensionless measure of arc length. $x = \frac{\alpha}{360} \times 2\pi$. $360^\circ = 2\pi$, $180^\circ = \pi$, $90^\circ = \frac{\pi}{2}$.

• Properties:

  • Range: $[-1, 1]$ for sine and cosine.

  • Periodicity: Sine and cosine are $2\pi$-periodic ($\sin(x + 2\pi) = \sin(x)$).

  • Pythagorean identity: $(\sin(x))^2 + (\cos(x))^2 = 1$.

• Tangent and Cotangent:

  • $\tan(x) = \frac{\sin(x)}{\cos(x)}$; Domain excluded where $\cos(x) = 0 \, (x = \frac{\pi}{2} + k\pi)$.

  • $\cot(x) = \frac{\cos(x)}{\sin(x)}$; Domain excluded where $\sin(x) = 0 \, (x = k\pi)$.

  • Periodicity: Both are $\pi$-periodic.

• Inverse Trigonometric Functions:

  • Arcsine ($\arcsin$): Inverse of sine on $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

  • Arccosine ($\arccos$): Inverse of cosine on $[0, \pi]$.

  • Arctangent ($\arctan$): Inverse of tangent on $(-\frac{\pi}{2}, \frac{\pi}{2})$.

  • Arccotangent (\arccot): Inverse of cotangent on $(0, \pi)$.

• Oscillation Modeling: $f(x) = a \times \sin(b \times x) + d$.

  • $a$: Amplitude ($\frac{y_{max} - y_{min}}{2}$).

  • $d$: Average level/Vertical shift ($\frac{y_{max} + y_{min}}{2}$).

  • $b$: Period parameter ($\frac{2\pi}{p}$, where $p$ is cycle length).

UNIT 3: DIFFERENTIAL CALCULUS

First Derivative and Slope

• Difference Quotient: The slope of the secant connecting two points $(x, y)$ and $(x_1, y_1)$: $m = \frac{\Delta y}{\Delta x} = \frac{f(x + \Delta x) - f(x)}{\Delta x}$.

• Tangent: The limiting case of the secant as $\Delta x \rightarrow 0$ .

• First Derivative: The slope of the tangent at point $(x, y)$, denoted $f'(x)$ or $\frac{dy}{dx}$.

• Definition: $f'(x) = \lim_{\Delta x \rightarrow 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$.

• Differentiable: A function is differentiable if the differential quotient limit exists, is unique, and is finite.

Differentiation Rules for Elementary Functions

• Power Rule: If $f(x) = a \times x^n + c$, then $f'(x) = a \times n \times x^{n-1}$. Applies to all $n \in \mathbb{R} \setminus \{0\}$.

• Table of Derivatives:

  • $const = 0$

  • $x^n = nx^{n-1}$

  • $\sqrt{x} = \frac{1}{2\sqrt{x}}$

  • $e^x = e^x$

  • $a^x = \ln(a) \times a^x$

  • $\ln(x) = \frac{1}{x}$

  • $\log_a x = \frac{1}{x \ln(a)}$

  • $\sin(x) = \cos(x)$

  • $\cos(x) = -\sin(x)$

  • $\tan(x) = \frac{1}{(\cos(x))^2} = 1 + (\tan(x))^2$

  • $\arcsin(x) = \frac{1}{\sqrt{1 - x^2}}$

  • $\arctan(x) = \frac{1}{1 + x^2}$

General Differentiation Rules

• Summation Rule: $(f + g)' = f' + g'$

• Product Rule: $(f \times g)' = f' \times g + f \times g'$

• Quotient Rule: $(\frac{f}{g})' = \frac{g \times f' - f \times g'}{g^2}$

• Chain Rule: For composite functions $f(x) = g(h(x))$, the derivative is $f'(x) = g'(h) \times h'(x)$ ("outer derivative times inner derivative").

Higher Derivatives

• Second Derivative ($f''(x)$ or $\frac{d^2f}{dx^2}$ ): The derivative of the first derivative.

• $n$-th Derivative: Result of differentiating a function $n$ times.

• Polynomials: Each derivative reduces the degree by $1$ until it reaches $0$. All subsequent derivatives are zero.

• Exponential ($e^x$): All higher derivatives are identically $e^x$.

Taylor Series and Taylor Polynomial

• Concept: Approximating real-valued, and infinitely differentiable functions using a power series around an expansion point (usually $x = 0$).

• Definition: The Taylor series of $f$ at $x=0$ is $T_f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!} x^k$.

• Taylor Polynomial of degree $n$: The partial sum $T_{f,n}(x) = \sum_{k=0}^{n} \frac{f^{(k)}(0)}{k!} x^k$.

• Examples:

  • $e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}$

  • $\sin(x) = \sum_{k=0}^{\infty} \frac{(-1)^k x^{2k+1}}{(2k+1)!} = x - \frac{x^3}{6} + \frac{x^5}{120} - \dots$

  • $\cos(x) = \sum_{k=0}^{\infty} \frac{(-1)^k x^{2k}}{(2k)!} = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \dots$

Curve Sketching

• Criteria for analysis:

  • Domain & Co-domain: Valid input and output sets.

  • Symmetry: Even if $f(-x) = f(x)$ (y-axis symmetry); Odd if $f(-x) = -f(x)$ (origin symmetry).

  • Zeros: Solutions to $f(x) = 0$.

  • Singularities: Points where the function is undefined (e.g., denominator is zero).

  • Asymptotic Behavior: Limit behavior for $x \rightarrow \pm\infty$.

  • Monotonicity: Increasing if $f'(x) \ge 0$; Decreasing if $f'(x) \le 0$.

  • Extremes: Local maximum if $f'(x_0) = 0$ and $f''(x_0) < 0$; Local minimum if $f'(x_0) = 0$ and $f''(x_0) > 0$.

  • Turning Points: Where $f''(x_0) = 0$ and $f'''(x_0) \ne 0$.

  • Saddle Point: Where $f'(x_0) = 0, f''(x_0) = 0,$, and $f'''(x_0) \ne 0$.

Partial Derivatives

• Multidimensional Functions: Functions of several variables, $f(x_1, x_2, \dots, x_n)$. For example, kinetic energy $W_{kin}(m, v) = \frac{mv^2}{2}$.

• First-order Partial Derivative: Differentiating with respect to one variable while treating all others as constants. Notation: $\frac{\partial f}{\partial x_1}$ or $f_{x_1}'$.

• Interpretations: The slope of a surface at a point along the direction of a specific coordinate axis.

• Symmetry of Second Derivatives: Usually $f_{xy}'' = f_{yx}''$.

UNIT 4: INTEGRAL CALCULUS

Indefinite Integral and Anti-derivatives

• Integral Calculus: The process of reversing differentiation to find the original function, called the anti-derivative or primitive function $F(x)$. Notation: $F'(x) = f(x)$.

• Integration Constant ($C$): Because constants disappear during differentiation, an anti-derivative is only unique up to a real constant $C$.

• Indefinite Integral: The set of all primitive functions, denoted $\int f(x) \, dx = F(x) + C$.

• Integration Rules:

  • Factor Rule: $\int a \times f(x) \, dx = a \int f(x) \, dx$.

  • Summation Rule: $\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx$.

  • Basic integrals:

    • $\int a \, dx = ax + C$

    • $\int x^a \, dx = \frac{1}{a+1} x^{a+1} + C$ (for $a \ne -1$)

    • $\int e^x \, dx = e^x + C$

    • $\int \frac{1}{x} \, dx = \ln|x| + C$

Integration Techniques

• Logarithmic Integration: For integrands in the form $\frac{f'(x)}{f(x)}$: $\int \frac{f'(x)}{f(x)} \, dx = \ln|f(x)| + C$.

• Partial Integration: Reversal of the product rule: $\int f(x) \times g'(x) \, dx = f(x) \times g(x) - \int f'(x) \times g(x) \, dx$.

• Continuity: A function continuous on the interval $[a, b]$ can be integrated on that interval.

The Definite Integral

• Definite Integral: Represents the surface area between the function graph and the x-axis between limits $a$ and $b$. It results in a real number.

• Fundamental Theorem of Calculus: $\int_a^b f(x) \, dx = [F(x)]_a^b = F(b) - F(a)$.

• Rules for Limits:

  • Exchanging: $\int_a^b f \, dx = - \int_b^a f \, dx$

  • Decomposition: $\int_a^b f \, dx = \int_a^c f \, dx + \int_c^b f \, dx$ for $a \le c \le b$

• Area Calculation:

  • Areas below the x-axis result in negative integral values; the absolute value must be taken for area calculation.

  • If a function crosses the x-axis, the integral must be split at the zero points.

Solids of Revolution and Arc Length

• Volume ($V$): Created when a surface rotates around the x-axis: $V = \pi \int_a^b (f(x))^2 \, dx$.

• Arc Length ($L$): The length of a curve on $[a, b]$: $L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx$.

• Lateral Surface ($M$): The surface area of the body of revolution: $M = 2\pi \int_a^b f(x) \sqrt{1 + (f'(x))^2} \, dx$.

UNIT 5: DIFFERENTIAL EQUATIONS

Introduction and Basic Terms

• Differential Equation (D.E.): An equation establishing a relationship between a function and its derivatives. The solution is a function, not a number.

• Order: Determined by the highest derivative present (e.g., only $y'$ means first-order).

• Ordinary Differential Equation: A D.E. depending on only one variable (usually $x$ or $t$).

• Linear D.E.: Function and its derivatives appear only linearly (no squares, roots, etc.).

• Homogeneous vs. Non-homogeneous: A homogeneous D.E. has a zero on the "right side" ($s(x) = 0$). Non-homogeneous has a non-zero function or constant.

Solutions of First-Order Linear D.E.s

• General Solution: The entire set of functions solving the D.E., usually containing an arbitrary constant factor $a$.

• Initial Condition/Initial Value Problem: Providing a specific point $(x_0, y_0)$ to find a Special Solution from the general group.

• Slope/Direction Field: A graphical sketch showing the directions of solution curves.

• Basic Homogeneous Case: For $y' = c \times y$, the general solution is $y(t) = a e^{ct}$.

• Non-homogeneous Case: Solve by adding the general solution of the homogeneous version ($y_h$) and one special solution of the non-homogeneous version ($y_s$): $y(x) = y_h(x) + y_s(x)$.

Modeling Applications

• Limited Growth: $y'(t) = c(K - y(t))$, where $K$ is an upper capacity limit. General solution: $y(t) = K + a e^{-ct}$. With initial condition $y(0) = y_0$, solution is $y(t) = K + (y_0 - K)e^{-ct}$.

• Newton’s Law of Cooling: Tea temperature $T(t)$ with ambient temperature $T_U$ follows $T'(t) = -k(T(t) - T_U)$. Solution: $T(t) = T_U + (T_0 - T_U)e^{-kt}$.

Outlook: Partial Differential Equations (PDEs)

• Definition: Equations involving partial derivatives of a function with multiple variables (e.g., space and time).

• Examples:

  • Laplace Equation: $u_{xx}'' + u_{yy}'' = 0$. Describes equilibrium states, such as a constant temperature gradient in a metal rod.

  • Heat Conduction Equation: General form for thermal transfer.

  • Others: Maxwell equations (electrodynamics), Schrödinger equation (quantum mechanics), Navier-Stokes (hydrodynamics).