V6Stetigkeit und Differentialrechnung einer Variablen

General Course Information and Motivation

• Course Details: Lecture 6, titled "Stetigkeit, Differentialrechnung einer Variablen" (Continuity, Differential Calculus of one variable), part of Mathematics I. Delivered on May 20, 2025 by Stephan Block of the AG Bionanogrenzflächen, Institute for Physical and Theoretical Chemistry, Department of BCP, Freie Universität Berlin.

• Motivational Example: Usain Bolt's World Record Run:     * The 2008 Beijing run is used to illustrate the transition from average speed to functional analysis.     * Data Representation: The distance covered (position) is plotted as a function of time: $s = f(t)$.     * Observation: From $4\,\text{s}$ onwards, the distance function is approximately linear.     * Average Velocity (\bar{v}): Calculated as the change in distance over the change in time:         $\bar{v} = \frac{\Delta s}{\Delta t} = \frac{s(t_2) - s(t_1)}{t_2 - t_1} = \frac{f(t_2) - f(t_1)}{t_2 - t_1}$.

Learning Objectives and Outline

• Main Topics:     1. Continuity (Stetigkeit): Definition and examples.     2. Difference Quotient and 1D Differential Calculus.     3. Applications (specifically root-finding).

• Specific Learning Goals:     * Ability to evaluate the continuity of functions.     * Differentiating functions of a single variable.     * Calculating function zeros (roots) using the iterative Newton's method (Newton-Verfahren).

Definitions and Theory of Continuity (Stetigkeit)

• Formal Definition: Let $f: I \rightarrow J$ be a real function and $a \in I$.     1. Continuity at a point: The function $f$ is called continuous at point $a$ if the limit $\lim_{x \rightarrow a} f(x)$ exists and equals the function value at that point: $\lim_{x \rightarrow a} f(x) = f(a)$.     2. Global Continuity: A function $f$ is continuous on its domain $I$ if it is continuous at every point $a \in I$.     3. Discontinuity: A function is discontinuous at point $a$ if it fails to satisfy the continuity condition at that point.

• Formal Remark on Limits: Previously, limits were discussed only in the context of sequences. Formally, to calculate $\lim_{x \rightarrow a} f(x)$, one should choose a sequence $(x_n)$ that converges to $a$. However, for practical purposes, it is sufficient to imagine moving through the domain toward $a$ and observing the behavior of the function close to $a$.

• Visual Intuition: Continuity is often equated with the ability to draw the graph of a function without lifting the pen from the paper.

Criteria and Examples of Discontinuity

• Causes of Discontinuity: A function is discontinuous at point $a$ if any of these three cases occur:     1. The left-sided and right-sided limits exist and are equal, but they do not match the function value $f(a)$.     2. The left-sided and right-sided limits exist but are not equal to each other (jump).     3. The left-sided and/or right-sided limit does not exist.

• Graphical Examples provided in the transcript:     * Case 1 (Limit exists \neq functional value):         $f(x): \mathbb{R} \rightarrow \mathbb{R}$ defined as         $\begin{cases} x^2, & x \neq 2 \ 1, & x = 2 \end{cases}$         Here, the limit as $x \rightarrow 2$ is $2^2 = 4$, but the value defined is $f(2) = 1$.     * Case 2 (Limits differ):         $f(x): \mathbb{R} \rightarrow \mathbb{R}$ defined as         $\begin{cases} 8, & x < 2 \ x^2, & x > 2 \end{cases}$         The left limit is $8$ and the right limit is $4$.     * Case 3 (Limit does not exist):         $f(x): \mathbb{R} \rightarrow \mathbb{R}$ defined as         $\begin{cases} 2\sin(x), & x < 2 \ x^2, & x > 2 \end{cases}$         (Note: The transcript illustrates non-existence often through undefined behavior or infinite oscillation at the boundary point).

Fundamentals of Differential Calculus

• Core Concept: Differentiation is the local linearization of a real function ($f: I \rightarrow \mathbb{R}$). The function is approximated by a linear function $f(x) \approx Ax + B$ in the vicinity of a point $x_0$.

• Step 1: The Secant Slope (Difference Quotient):     * The slope of the secant line passing through $(x_0, f(x_0))$ and $(x_0 + \Delta x, f(x_0 + \Delta x))$ is:         $\frac{\Delta f}{\Delta x} = \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x}$     * Geometric Interpretation: This represents the average increase/slope over the interval $\Delta x$.

• Step 2: The Tangent Slope (The Derivative):     * The slope at the specific point $x_0$ is the limit of the difference quotient as $\Delta x \rightarrow 0$:         $f'(x_0) = \frac{df(x)}{dx}\Big|<em>{x_0} = \lim</em>{\Delta x \rightarrow 0} \frac{\Delta f}{\Delta x} = \lim_{\Delta x \rightarrow 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x}$     * Geometric Interpretation: This represents the slope of the tangent to the function $f$ at point $x_0$.

• Linear Approximation: If the derivative exists, the function can be approximated near $x_0$ as:     $f(x_0 + \Delta x) \approx f(x_0) + f'(x_0) \cdot \Delta x$

Standard Derivatives of Basic Functions

• Power Functions: For $f(x) = x^p$ (with $x \in (0, \infty)$, $p \in \mathbb{R}$):     $f'(x) = p \cdot x^{p-1}$     * Example derivation for $x^2$: $\lim_{\Delta x \rightarrow 0} \frac{(x + \Delta x)^2 - x^2}{\Delta x} = \lim_{\Delta x \rightarrow 0} \frac{x^2 + 2x\Delta x + \Delta x^2 - x^2}{\Delta x} = 2x$.

• Exponential Functions:     * For $f(x) = a^x$ (with a > 0): $f'(x) = a^x \cdot \ln(a)$.     * For the natural exponential function $f(x) = e^x$: $f'(x) = e^x$.

• Logarithmic Functions: For $f(x) = \ln(x)$ (with $x \in (0, \infty)$):     $f'(x) = \frac{1}{x}$

• Trigonometric Functions:     * $f(x) = \sin(x) \rightarrow f'(x) = \cos(x)$     * $f(x) = \cos(x) \rightarrow f'(x) = -\sin(x)$

Essential Rules and Conditions

• Differentiability and Continuity: A function can only be differentiated at points where it is continuous. Continuity is a necessary condition for differentiability, but not a sufficient one (e.g., functions with "kinks" or sharp corners are continuous but not differentiable at the kink).

• The Symbol $d$: The notation $dx$ refers to an infinitesimally small change in $x$. It implies that while $x_0 + dx$ is practically indistinguishable from $x_0$, the difference $(x_0 + dx) - x_0$ is sufficiently non-zero to prevent the limit quotient from diverging.

• Rechenregeln (Rules of Calculation):     * Sum Rule: $(f + g)'(x) = f'(x) + g'(x)$     * Scaling Rule: $(c \cdot f)'(x) = c \cdot f'(x)$ where $c$ is a constant.     * Product Rule: $(f \cdot g)'(x) = f'(x)g(x) + f(x)g'(x)$     * Quotient Rule: $(\frac{f}{g})'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$ (provided $g(x) \neq 0$).     * Chain Rule: For $f(x) = g(h(x))$, the derivative is $f'(x) = g'(h(x)) \cdot h'(x)$ (Exterior derivative $\times$ interior derivative).

Applications of Differential Calculus

• Curve Discussion: The first derivative corresponds to the slope of the tangent. This is used to analyze function behavior and estimate how much the function value changes when moving from $x_0$ to $x_0 + dx$.

• Numerical Equation Solving: Many equations lack analytical solutions (e.g., $e^x = -x^2$). Finding $x$ such that the equation holds is equivalent to finding the zeros of the function $f(x) = e^x + x^2$.

• Newton's Method (Newton-Verfahren):     * Based on the identity for continuous functions: $f(x_0 + \Delta x) \approx f(x_0) + \Delta x \cdot f'(x_0)$.     * To find a zero, we set $f(x_0 + \Delta x) = 0$. This leads to the approximation:         $0 \approx f(x_0) + \Delta x \cdot f'(x_0)$         $\Delta x \approx -\frac{f(x_0)}{f'(x_0)}$     * Iteration: By choosing a starting point $x_0$ and iterating the equation $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$, one can converge on the root of the function. The approximation improves as $\Delta x$ becomes smaller (i.e., as $x_n$ gets closer to the actual zero point).