Calculus and Real Analysis: Derivatives, Limits, and Function Graphing

ANALYSIS OF FUNCTIONS I: INCREASE, DECREASE, AND CONCAVITY

• General Purpose: While graphing utilities offer general shapes, calculus provides precision to determine exact shapes and key features like curvature and precise locations.
• Intuitive Descriptions: The behavior of a function is described by traveling from left to right along its graph.     * Increasing: Moving upward as $x$ increases.     * Decreasing: Moving downward as $x$ increases.     * Constant: Horizontal movement where the value remains unchanged.
• Formal 3.1.1 DEFINITION: Let $f$ be defined on an interval, with $x_1$ and $x_2$ being points in that interval.     * (a) $f$ is increasing on the interval if $f(x_1) < f(x_2)$ whenever $x_1 < x_2$.     * (b) $f$ is decreasing on the interval if $f(x_1) > f(x_2)$ whenever x_1 < x_2.     * (c) $f$ is constant on the interval if $f(x_1) = f(x_2)$ for all points $x_1$ and $x_2$.
• The First Derivative Test for Behavior (Theorem 3.1.2): Let $f$ be continuous on a closed interval $[a, b]$ and differentiable on the open interval $(a, b)$.     * (a) If f'(x) > 0 for every $x$ in $(a, b)$, then $f$ is increasing on $[a, b]$.     * (b) If f'(x) < 0 for every $x$ in $(a, b)$, then $f$ is decreasing on $[a, b]$.     * (c) If $f'(x) = 0$ for every $x$ in $(a, b)$, then $f$ is constant on $[a, b]$.     * Applicability Note: This theorem applies to any interval where $f$ is continuous, including infinite intervals (e.g., $(-\infty, +\infty)$).

CONCAVITY AND SECOND DERIVATIVES

• Direction of Curvature: The first derivative sign reveals direction, but not the curvature.
• Terminology:     * Concave Up: "Holds water"; the graph lies above its tangent lines. The slopes of the tangent lines are increasing.     * Concave Down: "Spills water"; the graph lies below its tangent lines. The slopes of the tangent lines are decreasing.
• 3.1.3 DEFINITION: If $f$ is differentiable on an open interval, it is concave up if $f'$ is increasing on that interval, and concave down if $f'$ is decreasing.
• 3.1.4 THEOREM (Test for Concavity): Let $f$ be twice differentiable on an open interval.     * (a) If f''(x) > 0 for every value of $x$ in the interval, $f$ is concave up.     * (b) If f''(x) < 0 for every value of $x$ in the interval, $f$ is concave down.

INFLECTION POINTS

• 3.1.5 DEFINITION: If $f$ is continuous on an open interval containing $x_0$, and changes concavity at the point $(x_0, f(x_0))$, the point is an inflection point.
• Identifying Candidates: Inflection points often occur where $f''(x) = 0$ or where $f''(x)$ is undefined. However, $f''(x) = 0$ does not guarantee an inflection point (e.g., $f(x) = x^4$ at $x = 0$ has $f''(0) = 0$ but no change in concavity).
• Interpreting Inflection Points in Applications: They mark locations where the rate of change reaches a local maximum or minimum (e.g., the narrowest point of a flask neck when filling with water, where the water level rise rate changes from increasing to decreasing).

L'HÔPITAL'S RULE AND INDETERMINATE FORMS

• Indeterminate Type 0/0: Limiting forms where both numerator and denominator approach zero. Examples include $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$.
• 6.5.1 THEOREM (L'Hôpital's Rule for 0/0): Suppose $f$ and $g$ are differentiable on an open interval containing $a$ (except possibly at $a$), and $\lim_{x \to a} f(x) = 0$ and $\lim_{x \to a} g(x) = 0$. If $\lim_{x \to a} [f'(x)/g'(x)]$ exists (or is $\pm \infty$), then:     $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$
• Biographical Note: Guillaume François Antoine de L'Hôpital (1661–1704) was a French mathematician who published the first textbook on differential calculus. The rule was actually discovered by his teacher, John Bernoulli.
• Indeterminate Type $\infty/\infty$ (Theorem 6.5.2): Applicable when both numerator and denominator approach $\infty$. The rule remains the same: the limit of the ratio equals the limit of the ratio of the derivatives.
• Analyzing Growth: L'Hôpital's rule helps prove that exponential functions grow faster than polynomial functions: $\lim_{x \to +\infty} \frac{x^n}{e^x} = 0$ for any positive integer $n$.
• Other Indeterminate Types:     * $0 \cdot \infty$: Rewrite as a ratio to use 0/0 or $\infty/\infty$.     * $\infty - \infty$: Combine terms (algebraic manipulation) to form a ratio.     * $0^0, \infty^0, 1^\infty$: Use the natural logarithm to simplify: If $y = f(x)^{g(x)}$, then $\ln(y) = g(x) \ln[f(x)]$. Evaluate $\lim \ln(y) = L$, then $\lim y = e^L$.

LIMITS OF FUNCTIONS (REAL ANALYSIS)

• Historical Evolution:     * 1680s (Newton and Leibniz): Formulated early notions of function and "quantities being close". Newton used the term "fluent"; Leibniz used "calculus" and "infinitesimally small" numbers.     * 1748 (Euler): Published "Introducito in Analysin Infinitorum", discussing power series and exponential/trig functions.     * 1821 (Cauchy): Set standards for rigor but the limit definition remained slightly verbal.     * Karl Weierstrass: Established the precise $\delta-\epsilon$ language used today.
• 4.1.1 DEFINITION (Cluster Point): A point $c \in \mathbb{R}$ is a cluster point of set $A$ if every $\delta-$neighborhood of $c$ contains at least one point of $A$ distinct from $c$.
• 4.1.4 DEFINITION (Limit of a Function): For $f: A \to \mathbb{R}$ and cluster point $c$. A real number $L$ is the limit of $f$ at $c$ if for every \epsilon > 0, there exists \delta > 0 such that if $x \in A$ and 0 < |x - c| < \delta, then |f(x) - L| < \epsilon.
• Uniqueness (Theorem 4.1.5): If a limit exists, it is unique.
• Sequential Criterion (Theorem 4.1.8): $\lim_{x \to c} f = L$ iff for every sequence $(x_n)$ in $A \setminus {c}$ that converges to $c$, the sequence $(f(x_n))$ converges to $L$.

SUCCESSIVE DIFFERENTIATION AND SINGULAR POINTS

• Notation:     * First derivative: $\frac{dy}{dx}$, $y_1$, $y'$, or $f'(x)$.     * $n$-th derivative: $\frac{d^n y}{dx^n}$, $y_n$, or $f^{(n)}(x)$.
• Leibnitz's Theorem: Used for the $n$-th derivative of a product of two functions ($u$ and $v$):     $D^n(uv) = (D^n u)v + \binom{n}{1} D^{n-1}u Dv + \binom{n}{2} D^{n-2}u D^2v + \dots + u D^nv$
• Singular/Multiple Points:     * Multiple Point: A point through which more than one branch of a curve passes.     * Node: A double point where two real branches pass with distinct tangents.     * Cusp: A double point where two branches have coincident tangents.     * Conjugate Point (Isolated Point): A point whose coordinates satisfy the equation, but no other real points of the curve exist in its immediate neighborhood.
• Tangents at the Origin: For a rational integral algebraic curve passing through the origin, tangents are found by equating the terms of the lowest degree to zero.
• Curve Tracing Procedure:     1. Symmetry: Check axes and quadrants.     2. Origin: Check if it passes through $(0,0)$ and find tangents.     3. Solve for $y$: Observe behavior as $x$ increases.     4. Consider all values of $x$: Including behavior toward $\infty$.     5. Imaginary values: Identify regions where the curve does not exist.     6. Asymptotes: Find linear relations for large $x, y$.     7. Special points: Maxima, minima, intercepts.     8. Inflection Points: Locating via $d^2y/dx^2 = 0$.