Calculus Unit 1: Real Numbers, Functions, Sequences, and Differential Calculus

Real Numbers

• The set of real numbers, $\mathbb{R}$, is a set endowed with two binary operations, ADDITION and MULTIPLICATION, and with a total order relation $\le$ (LESS THAN OR EQUAL TO).

• Addition and multiplication associate to every pair of elements of $\mathbb{R}$ a unique element of $\mathbb{R}$; thus, $\mathbb{R}$ is a closed set with respect to them:   $\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$   $(a, b) \rightarrow (a + b)$   $(a, b) \rightarrow (a \cdot b)$

• The quadruple $(\mathbb{R}, +, \cdot, \le)$ satisfying the axioms of operations and ordering is called an ordered field.

Axioms of Real Numbers

• Axioms Related to Operations:

  • Commutative Property: For all $a, b \in \mathbb{R}$, $a + b = b + a$ and $a \cdot b = b \cdot a$.

  • Associative Property: For all $a, b, c \in \mathbb{R}$, $(a + b) + c = a + (b + c)$ and $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.

  • Distributive Property: For all $a, b, c \in \mathbb{R}$, $a \cdot (b + c) = a \cdot b + a \cdot c$

  • Existence of Zero (Additive Identity): There exists $0 \in \mathbb{R}$ such that for all $a \in \mathbb{R}$, $a + 0 = a$

  • Existence of Multiplicative Identity: There exists $1 \in \mathbb{R}$ such that for all $a \in \mathbb{R}$, $a \cdot 1 = a$

  • Existence of Additive Inverses: For every $a \in \mathbb{R}$, there exists $(-a) \in \mathbb{R}$ such that $a + (-a) = 0$

  • Existence of Multiplicative Inverses: For every $a \neq 0$, there exists $a^{-1} \in \mathbb{R}$ such that $a \cdot a^{-1} = 1$

• Axioms Related to Ordering:

  • Trichotomy: For all $a, b \in \mathbb{R}$, either $a \le b$ or $b \le a$

  • Antisymmetry: For all $a, b \in \mathbb{R}$, if $a \le b$ and $b \le a$, then $a = b$

  • Translation Invariance: For all $a, b, c \in \mathbb{R}$, if $a \le b$, then $a + c \le b + c$

  • Compatibility of Order with Operations: For all $a, b \in \mathbb{R}$, if $0 \le a$ and $0 \le b$, then $0 \le a + b$ and $0 \le a \cdot b$

• Derived Properties and Formal Proofs:

  • Transitive Property: For all $a, b, c \in \mathbb{R}$, if $a \le b$ and $b \le c$, then $a \le c$

  • Proof: From translation invariance, $a \le b \implies a + (-a) \le b + (-a) \implies 0 \le b - a$. Similarly, $b \le c \implies 0 \le c - b$. By compatibility of order, $0 \le (b - a) + (c - b) = c - a$. By translation invariance, $0 + a \le (c - a) + a \implies a \le c$

  • Uniqueness of Zero Element:

  • Proof: Assume by contradiction that two additive identities $0_1 \neq 0_2$ exist. Then $0_2 + 0_1 = 0_2$ and $0_1 + 0_2 = 0_1$. By commutativity, $0_1 + 0_2 = 0_2 + 0_1$, which implies $0_1 = 0_2$, a contradiction.

  • Product with Zero: For every $a \in \mathbb{R}$, $0 \cdot a = 0$

  • Proof: By distributivity, $a + (a \cdot 0) = a \cdot 1 + a \cdot 0 = a \cdot (1 + 0) = a \cdot 1 = a = a + 0$. Adding $(-a)$ to both sides gives $a \cdot 0 = 0$

  • Non-existence of $0^{-1}$:

  • Proof: Assume $0^{-1}$ exists. Then $0 \cdot 0^{-1} = 1$. But $0 \cdot 0^{-1} = 0$ from the previous property, giving $0 = 1$, a contradiction.

  • Sign Rule: For all $a, b \in \mathbb{R}$, $(-a) \cdot (-b) = a \cdot b$

  • Proof: $(-a) \cdot (-b)$ is the additive inverse of $a \cdot (-b)$, so $(-a) \cdot (-b) = -[a \cdot (-b)] = -[-(b \cdot a)] = b \cdot a = a \cdot b$

• Derived Order Relations and Operations:

  • Greater or equal: $a \ge b \iff b \le a$

  • Strict inequalities: $a < b \iff (a \le b \text{ and } a \neq b)$; $a > b \iff (a \ge b \text{ and } a \neq b)$

  • Subtraction: $a - b := a + (-b)$

  • Division: $\frac{a}{b} := a \cdot b^{-1}$ for $b \neq 0$

Subsets of Real Numbers and Completeness

• Nested Subsets:

  • Natural numbers: $\mathbb{N} = \{1, 2, 3, 4, \dots\}$

  • Integers: $\mathbb{Z} = \{0, \pm 1, \pm 2, \dots\}$

  • Rational numbers: $\mathbb{Q} = \left\{\frac{p}{q} : p, q \in \mathbb{Z}, q \neq 0\right\}$

  • Inclusion hierarchy: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$

• Powers with Positive Integer Exponents:

  • For $x \in \mathbb{R}$ and $n \in \mathbb{N}$: $x^n := x \cdot x \cdot \dots \cdot x$ ($n$ times)

  • Algebraic properties:     $x^n \cdot x^m = x^{n+m}$     $x^n \cdot y^n = (x \cdot y)^n$     $(x^n)^m = x^{n \cdot m}$

• Irrationality of $\sqrt{2}$:

  • Proposition: There exists no $a \in \mathbb{Q}$ such that $a^2 = 2$

  • Proof: Assume by contradiction $a = \frac{p}{q} \in \mathbb{Q}$ with $\gcd(p, q) = 1$ and $a^2 = 2$. Then $p^2 = 2q^2$, which implies $p^2$ is even, so $p$ must be even ($p = 2s$). Substituting gives $4s^2 = 2q^2 \implies q^2 = 2s^2$, meaning $q$ must also be even. This contradicts $\gcd(p, q) = 1$.

• Bounded Sets, Extremum Properties, and Dedekind Axiom:

  • Bounded Above: A set $A \subset \mathbb{R}$ is bounded above if there exists $M \in \mathbb{R}$ such that $a \le M$ for all $a \in A$. $M$ is an upper bound. If $M \in A$, then $M = \max A$

  • Bounded Below: A set $A \subset \mathbb{R}$ is bounded below if there exists $m \in \mathbb{R}$ such that $a \ge m$ for all $a \in A$. $m$ is a lower bound. If $m \in A$, then $m = \min A$

  • Bounded Set: $A$ is bounded if it is bounded both above and below.

  • Supremum ($\sup A$): The smallest upper bound of $A$

  • Characterization: $L = \sup A \iff$ (1) $a \le L$ for all $a \in A$; (2) for every $\epsilon > 0$, there exists $a_\epsilon \in A$ such that $a_\epsilon > L - \epsilon$

  • Infimum ($\inf A$): The greatest lower bound of $A$

  • Characterization: $l = \inf A \iff$ (1) $a \ge l$ for all $a \in A$; (2) for every $\epsilon > 0$, there exists $a_\epsilon \in A$ such that $a_\epsilon < l + \epsilon$

  • Unbounded Extensions: If $A$ is not bounded above, $\sup A = +\infty$; if not bounded below, $\inf A = -\infty$

  • Dedekind Axiom (Axiom of Completeness): Every non-empty subset of $\mathbb{R}$ that is bounded above admits a supremum in $\mathbb{R}$. Every non-empty subset of $\mathbb{R}$ that is bounded below admits an infimum in $\mathbb{R}$.

  • Density of Rational Numbers: For any two real numbers $a, b \in \mathbb{R}$ with $a < b$, there exists a rational number $r \in \mathbb{Q}$ such that $a < r < b$.

• Existence of $\sqrt[n]{a}$:

  • Theorem (Existence of Zeros of a Polynomial): Let $P(x) = a_k x^k + a_{k-1} x^{k-1} + \dots + a_1 x + a_0$. If $s < t$ are real numbers such that $P(s) < 0$ and $P(t) > 0$, then there exists $y \in \mathbb{R}$ with $s < y < t$ such that $P(y) = 0$

  • Definition: For $a \in \mathbb{R}^+$, the $n$-th root $\sqrt[n]{a}$ is the unique positive real number $y \in \mathbb{R}^+$ such that $y^n = a$

  • Theorem (Existence and Uniqueness): For every $a \in \mathbb{R}^+$ and $n \in \mathbb{N}$, there exists a unique $y \in \mathbb{R}^+$ such that $y^n = a$

Powers and Logarithms

• Exponent Generalizations:

  • Integer Exponent: For $x \in \mathbb{R} \setminus \{0\}$ and $n \in \mathbb{Z}$, $x^{-n} := \frac{1}{x^n}$ and $x^0 := 1$

  • Rational Exponent: For $x > 0$ and $q = \frac{m}{n} \in \mathbb{Q}$, $x^{1/n} := \sqrt[n]{x}$ and $x^{m/n} := (\sqrt[n]{x})^m = \sqrt[n]{x^m}$

  • Real Exponent: For base $x > 0$ and $b \in \mathbb{R}$:

  • If $x = 1$, $1^b = 1$

  • If $x > 1$, $x^b := \sup\{x^s : s \in \mathbb{Q}, s \le b\} = \inf\{x^t : t \in \mathbb{Q}, t \ge b\}$

  • If $0 < x < 1$, $x^b := \inf\{x^s : s \in \mathbb{Q}, s \le b\} = \sup\{x^t : t \in \mathbb{Q}, t \ge b\}$

• Logarithms:

  • For base $a > 0, a \neq 1$ and $x > 0$, $\log_a(x)$ is the unique real number satisfying:     $a^{\log_a(x)} = x$

  • Identities and Properties:     $\log_a(a^y) = y$     $\log_a(1) = 0$     $\log_a(x \cdot y) = \log_a(x) + \log_a(y)$     $\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$     $\log_a(x^b) = b \cdot \log_a(x)$     $\log_a\left(\frac{1}{y}\right) = -\log_a(y)$

Topology of the Real Line

• Intervals:

  • Open interval: $(a, b) = \{x \in \mathbb{R} : a < x < b\}$

  • Closed interval: $[a, b] = \{x \in \mathbb{R} : a \le x \le b\}$

  • Half-open intervals: $[a, b) = \{x \in \mathbb{R} : a \le x < b\}$, $(a, b] = \{x \in \mathbb{R} : a < x \le b\}$

  • Unbounded rays: $(a, +\infty)$, $[a, +\infty)$, $\left(-\infty, b\right)$, $\left(-\infty, b\right]$

• Neighborhoods, Open Sets, and Closed Sets:

  • Neighborhood: $I(x_0, r) = \{x \in \mathbb{R} : |x - x_0| < r\} = (x_0 - r, x_0 + r)$

  • Open Set: A subset $A \subset \mathbb{R}$ is open if for every $x \in A$, there exists a neighborhood $I(x, r) \subset A$

  • Closed Set: A subset $D \subset \mathbb{R}$ is closed if its complement $C D = \mathbb{R} \setminus D$ is open.

  • Set Operations:

  • Union of any family of open sets is open; intersection of a finite number of open sets is open.

  • Intersection of any family of closed sets is closed; union of a finite number of closed sets is closed.

  • Infinite intersections of open sets or infinite unions of closed sets are not necessarily open or closed (e.g., $\bigcap_{i \in \mathbb{N}} \left(-\frac{1}{i}, 1\right) = [0, 1)$).

• Accumulation Points and Closure:

  • Accumulation Point: $x \in \mathbb{R}$ is an accumulation point of $A \subset \mathbb{R}$ if every neighborhood of $x$ contains infinitely many points of $A$ distinct from $x$

  • Theorem: A set $D \subset \mathbb{R}$ is closed if and only if it contains all its accumulation points.

  • Closure ($\bar{A}$): The intersection of all closed sets containing $A$ (smallest closed set containing $A$).

  • Bolzano–Weierstrass Theorem: Every bounded subset $E \subset \mathbb{R}$ containing infinitely many points has at least one accumulation point.

Real Functions of One Variable

• Basic Definitions:

  • Function $f: A \rightarrow B$ maps each $x \in A$ (domain) to a unique $y = f(x) \in B$ (codomain).

  • Image $f(A) = \{y \in B : \exists x \in A, f(x) = y\}$

  • Surjective: $f(A) = B$

  • Injective: $x_1 \neq x_2 \implies f(x_1) \neq f(x_2)$

  • Bijective: Both injective and surjective.

  • Graph: $G_f = \{(x, y) \in \mathbb{R} \times \mathbb{R} : y = f(x), x \in A\}$

• Limits of Functions:

  • Finite Limit at Finite Point: $\lim_{x \to x_0} f(x) = L \iff \forall \epsilon > 0, \exists \delta > 0$ such that for all $x \in A$ with $0 < |x - x_0| < \delta$, $|f(x) - L| < \epsilon$

  • Infinite Limits: $\lim_{x \to x_0} f(x) = +\infty \iff \forall M > 0, \exists \delta > 0$ such that $0 < |x - x_0| < \delta \implies f(x) > M$

  • Limits at Infinity: $\lim_{x \to +\infty} f(x) = L \iff \forall \epsilon > 0, \exists M > 0$ such that $x > M \implies |f(x) - L| < \epsilon$

  • Fundamental Fundamental Trigonometric Limit:     $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$          

    

        

  • Proof: For $x \in \left(0, \frac{\pi}{2}\right)$, comparing areas gives $A_{\triangle OPA} < A_{\text{sector } OPA} < A_{\triangle OTA} \implies \frac{\sin(x)}{2} < \frac{x}{2} < \frac{\tan(x)}{2} \implies 1 < \frac{x}{\sin(x)} < \frac{1}{\cos(x)} \implies \cos(x) < \frac{\sin(x)}{x} < 1$. By symmetry, it holds for $x \in \left(-\frac{\pi}{2}, 0\right)$. Taking $x \to 0$ yields the limit.

• Limit Theorems and Properties:

  • Uniqueness of Limit: If a limit exists, it is unique.

  • Sign Preservation Theorem: If $\lim_{x \to x_0} f(x) = l > 0$, then there exists a neighborhood $I(x_0, \delta)$ such that $f(x) > 0$ for all $x \in A \cap I(x_0, \delta)$.

  • Squeeze (Sandwich / Comparison) Theorem: If $h(x) \le f(x) \le g(x)$ near $x_0$ and $\lim_{x \to x_0} h(x) = \lim_{x \to x_0} g(x) = l$, then $\lim_{x \to x_0} f(x) = l$          

    

             

    

  • One-Sided Limits: Right-hand limit $\lim_{x \to x_0^+} f(x) = L$ ($x_0 < x < x_0 + \delta$); left-hand limit $\lim_{x \to x_0^-} f(x) = L$ ($x_0 - \delta < x < x_0$). The two-sided limit exists if and only if both one-sided limits exist and are equal.

Monotone and Continuous Functions

• Monotone Functions:

  • Monotone increasing: $x_1 < x_2 \implies f(x_1) \le f(x_2)$

  • Monotone decreasing: $x_1 < x_2 \implies f(x_1) \ge f(x_2)$

  • Theorem on One-Sided Limits: If $f: (a, b) \rightarrow \mathbb{R}$ is monotone increasing, then for any $x_0 \in (a, b)$,     $\lim_{x \to x_0^-} f(x) = \sup_{a < x < x_0} f(x), \quad \lim_{x \to x_0^+} f(x) = \inf_{x_0 < x < b} f(x)$

• Continuous Functions:

  • $f$ is continuous at $x_0 \in A$ if $\lim_{x \to x_0} f(x) = f(x_0)$.

  • Trigonometric functions ($\sin(x), \cos(x), \tan(x), \cot(x)$) are continuous on their domains.

  • Exponential function $a^x$ ($a > 0$) is continuous on $\mathbb{R}$, strictly increasing for $a > 1$, strictly decreasing for $0 < a < 1$.

• Classification of Points of Discontinuity:

  • Jump Discontinuity (First Kind): Finite one-sided limits exist, but $\lim_{x \to x_0^+} f(x) \neq \lim_{x \to x_0^-} f(x)$. Jump $s(x_0) = f(x_0^+) - f(x_0^-)$.

  • Examples: Floor function $\lfloor x \rfloor$ and Ceiling function $\lceil x \rceil$              

    

                 

    

  • Discontinuity of the Second Kind: At least one of the one-sided limits is infinite.

  • Example: $f(x) = \frac{1}{x}$ or $f(x) = a^{1/x}$ at $x_0 = 0$              

    

  • Discontinuity of the Third Kind: At least one of the one-sided limits does not exist.

  • Example: $f(x) = \sin\left(\frac{1}{x}\right)$ at $x_0 = 0$              

    

• Invertibility and Notable Limits:

  • Invertibility Theorem: A continuous function $f$ on an interval $I$ is invertible if and only if it is strictly monotone. Its inverse $f^{-1}$ is continuous and strictly monotone.

  • Euler's Number and Notable Limits:     $e := \lim_{x \to +\infty} \left(1 + \frac{1}{x}\right)^x$     $\lim_{x \to 0} (1 + x)^{1/x} = e$     $\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1$     $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$     $\lim_{x \to 0} \frac{\log_a(1 + x)}{x} = \frac{1}{\ln(a)}$     $\lim_{x \to 0} \frac{a^x - 1}{x} = \ln(a)$     $\lim_{x \to 0} \frac{1 - \cos(x)}{x^2} = \frac{1}{2}$

Numerical Sequences

• Definition: A numerical sequence is a function $f: \mathbb{N} \rightarrow \mathbb{R}$, denoted $(a_n)_{n \in \mathbb{N}}$ where $a_n = f(n)$.

• Convergence: $\lim_{n \to +\infty} a_n = L \in \mathbb{R} \iff \forall \epsilon > 0, \exists n_\epsilon \in \mathbb{N}$ such that $|a_n - L| < \epsilon$ for all $n \ge n_\epsilon$

• Divergence: $\lim_{n \to +\infty} a_n = +\infty \iff \forall M > 0, \exists k \in \mathbb{N}$ such that $a_n > M$ for all $n \ge k$

• Subsequences and Bolzano–Weierstrass:

  • A subsequence $a_{n_k}$ is formed by a strictly increasing sequence of indices $n_1 < n_2 < \dots < n_k$

  • Theorem (Bolzano–Weierstrass): Every bounded sequence in $\mathbb{R}$ has a convergent subsequence.

• Cauchy Criterion: A sequence $(a_n)$ converges if and only if it is a Cauchy sequence: $\forall \epsilon > 0, \exists n_\epsilon$ such that $|a_m - a_n| < \epsilon$ for all $m, n \ge n_\epsilon$

• Sequence Convergence Tests:

  • Monotonicity Theorem: A monotone increasing sequence bounded above converges to its supremum. A monotone decreasing sequence bounded below converges to its infimum.

  • Ratio Test for Sequences: Let $a_n > 0$ and $\lim_{n \to +\infty} \frac{a_{n+1}}{a_n} = l$

  • If $l < 1$, then $\lim_{n \to +\infty} a_n = 0$

  • If $l > 1$, then $\lim_{n \to +\infty} a_n = +\infty$

  • If $l = 1$, the test is inconclusive.

  • Hierarchy of Infinities for Sequences (for $\alpha > 0, a > 1$):     $\lim_{n \to +\infty} \frac{\log_a(n)}{n^\alpha} = 0$     $\lim_{n \to +\infty} \frac{n^\alpha}{a^n} = 0$     $\lim_{n \to +\infty} \frac{a^n}{n!} = 0$     $\lim_{n \to +\infty} \frac{n!}{n^n} = 0$

  • Special Sequence Limits:     $\lim_{n \to +\infty} \sqrt[n]{n} = 1$     $\lim_{n \to +\infty} \left(\frac{1}{n}\right)^{1/n} = 1$

• Bridge Theorem (Sequential Characterization of Limits):

  • Let $f: A \rightarrow \mathbb{R}$ and $x_0$ be an accumulation point of $A$. Then $\lim_{x \to x_0} f(x) = L$ if and only if for every sequence $(x_n) \subset A \setminus \{x_0\}$ converging to $x_0$, $\lim_{n \to +\infty} f(x_n) = L$

Global Properties of Continuous Functions

• Zero Theorem (Intermediate Value Theorem for Zeros):

  • Let $f: [a, b] \rightarrow \mathbb{R}$ be continuous such that $f(a) \cdot f(b) < 0$. Then there exists $x_0 \in (a, b)$ such that $f(x_0) = 0$

  • Proof (Bisection Method): Bisect $[a, b]$ at $c_1 = \frac{a+b}{2}$. If $f(c_1) = 0$, finished. Else choose subinterval $[a_1, b_1]$ where endpoints have opposite signs. Iterating yields nested intervals $[a_n, b_n]$ with $(b_n - a_n) = \frac{b-a}{2^n}$. By monotonicity and boundedness, $a_n \to \bar{x}$ and $b_n \to \bar{x}$. By continuity, $0 \ge f(a_n)f(b_n) \to f(\bar{x})^2 \ge 0 \implies f(\bar{x}) = 0$

• Intermediate Value Theorem:

  • If $f: I \rightarrow \mathbb{R}$ is continuous on an interval $I$, then $f$ attains every value between $\inf_I f$ and $\sup_I f$

• Weierstrass Theorem:

  • If $f: [a, b] \rightarrow \mathbb{R}$ is continuous on a closed, bounded interval, then $f$ attains a maximum and a minimum on $[a, b]$.

Differential Calculus for Functions of One Variable

• The Derivative:

  • Definition: $f'(x_0) := \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}$

  • Geometric Interpretation: Slope of the tangent line to the graph of $f$ at $(x_0, f(x_0))$. Tangent line equation: $y = f(x_0) + f'(x_0)(x - x_0)$          

    

• Algebra of Derivatives:

  • Sum rule: $(f \pm g)' = f' \pm g'$

  • Product rule: $(f \cdot g)' = f' \cdot g + f \cdot g'$

  • Quotient rule: $\left(\frac{f}{g}\right)' = \frac{f' \cdot g - f \cdot g'}{g^2}$

  • Chain rule: $(g \circ f)'(x_0) = g'(f(x_0)) \cdot f'(x_0)$

• Derivatives of Elementary Functions:

  • Powers: $\frac{d}{dx}(x^\alpha) = \alpha x^{\alpha - 1}$

  • Trigonometric: $\frac{d}{dx}(\sin(x)) = \cos(x)$; $\frac{d}{dx}(\cos(x)) = -\sin(x)$

  • Exponential and Logarithmic: $\frac{d}{dx}(e^x) = e^x$; $\frac{d}{dx}(a^x) = a^x \ln(a)$; $\frac{d}{dx}(\ln(x)) = \frac{1}{x}$; $\frac{d}{dx}(\log_a(x)) = \frac{1}{x \ln(a)}$

  • General Exponential Power: $\frac{d}{dx}\left(f(x)^{g(x)}\right) = f(x)^{g(x)} \left(g'(x) \ln(f(x)) + g(x) \frac{f'(x)}{f(x)}\right)$

• Differentiability and Non-Differentiability Points:

  • Theorem: If $f$ is differentiable at $x_0$, then $f$ is continuous at $x_0$.

  • Corner Point: $f$ is continuous at $x_0$, and finite one-sided derivatives exist but are unequal: $f'_+(x_0) \neq f'_-(x_0)$.

  • Example: $f(x) = |x|$ at $x_0 = 0$              

    

  • Inflection Point with Vertical Tangent: $f$ is continuous at $x_0$, and $\lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h} = +\infty$ or $-\infty$

  • Example: $f(x) = \sqrt[3]{x}$ at $x_0 = 0$              

    

  • Cusp: $f$ is continuous at $x_0$, and one-sided derivatives are infinite with opposite signs: $f'_+(x_0) = \pm \infty$ and $f'_-(x_0) = \mp \infty$

  • Example: $f(x) = \sqrt[3]{|x|}$ at $x_0 = 0$              

    

Extrema, Mean Value Theorems, and L'Hôpital

• Extrema and Mean Value Theorems:

  • Fermat's Theorem: If $f$ is differentiable on $(a, b)$ and $x_0 \in (a, b)$ is a local maximum or minimum, then $f'(x_0) = 0$

  • Rolle's Theorem: If $f: [a, b] \rightarrow \mathbb{R}$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists $\xi \in (a, b)$ such that $f'(\xi) = 0$

  • Lagrange's Mean Value Theorem: If $f: [a, b] \rightarrow \mathbb{R}$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists $\xi \in (a, b)$ such that     $\frac{f(b) - f(a)}{b - a} = f'(\xi)$

  • Cauchy's Mean Value Theorem: If $f, g$ are continuous on $[a, b]$, differentiable on $(a, b)$, and $g'(x) \neq 0$, then there exists $\xi \in (a, b)$ such that     $\frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(\xi)}{g'(\xi)}$

  • Monotonicity Criterion: $f'(x) \ge 0 \iff f$ is increasing; $f'(x) \le 0 \iff f$ is decreasing.

• Second Derivative Test and Convexity:

  • Second Derivative Test for Extrema: If $f'(x_0) = 0$

  • $f''(x_0) < 0 \implies x_0$ is a local maximum.

  • $f''(x_0) > 0 \implies x_0$ is a local minimum.

  • Convexity Criterion: $f$ is convex on $I \iff f'(x)$ is increasing $\iff f''(x) \ge 0$ for all $x \in I$

  • Concavity Criterion: $f$ is concave on $I \iff f'(x)$ is decreasing $\iff f''(x) \le 0$ for all $x \in I$

  • Flex (Inflection) Point: Point where concavity changes.

• L'Hôpital's Theorem:

  • Let $f, g$ be differentiable on $(a, b) \setminus \{x_0\}$ with $g'(x) \neq 0$. If $\lim_{x \to x_0} f(x) = \lim_{x \to x_0} g(x) = 0$ (or $\pm \infty$) and $\lim_{x \to x_0} \frac{f'(x)}{g'(x)} = L$, then:     $\lim_{x \to x_0} \frac{f(x)}{g(x)} = \lim_{x \to x_0} \frac{f'(x)}{g'(x)} = L$

  • Indeterminate forms handled: $\frac{0}{0}, \frac{\infty}{\infty}, \infty - \infty, 0 \cdot \infty, 0^0, \infty^0, 1^\infty$

Taylor Formula and Expansions

• Little-o Notation: $f(x) = o(g(x))$ as $x \to x_0 \iff \lim_{x \to x_0} \frac{f(x)}{g(x)} = 0$

• Taylor's Theorem with Peano Remainder: If $f \in C^n(I_\delta(x_0))$, then   $f(x) = P_n(x, x_0) + o((x - x_0)^n)$   where the Taylor polynomial $P_n(x, x_0)$ is defined as:   $P_n(x, x_0) := \sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!} (x - x_0)^k$

• Standard Maclaurin Series Expansions ($x_0 = 0$):

  • Exponential: $e^x = \sum_{k=0}^n \frac{x^k}{k!} + o(x^n) = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \dots + \frac{x^n}{n!} + o(x^n)$

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

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

  • Logarithm: $\ln(1 + x) = \sum_{k=1}^n (-1)^{k-1} \frac{x^k}{k} + o(x^n) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots$

  • Geometric Series: \frac{1}{1 - x} = \sum_{k=0}^n x^k + o(x^n) = 1 + x + x^2 + \dots

  • Binomial Series: $(1 + x)^\alpha = \sum_{k=0}^n \binom{\alpha}{k} x^k + o(x^n) = 1 + \alpha x + \frac{\alpha(\alpha - 1)}{2} x^2 + \dots$

  • Arctangent: $\arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \dots + (-1)^n \frac{x^{2n+1}}{2n+1} + o(x^{2n+2})$

  • Tangent: $\tan(x) = x + \frac{x^3}{3} + o(x^4)$