Elementary Mathematics III Lecture Notes Flashcards

A comprehensive set of flashcards covering functions, limits, derivatives, integration, and their applications as presented in the Elementary Mathematics III lecture transcript.

Function

A rule $f: X \rightarrow Y$ which assigns or associates to each element $x$ of the set $X$ a unique element $y$ of the set $Y$, commonly written as $y = f(x)$.

Domain

The set $X$ of all $x$-values allowed by the function $f: X \rightarrow Y$, denoted by $Dom(f)$ or $D(f)$.

Range

The set of all admissible $y$-values as $x$ goes through all values in its domain, denoted by $Ran(f)$ or $R(f)$.

Codomain

The set $Y$ in the definition of a function $f: X \rightarrow Y$, which contains all possible image values.

Injective Function

Also known as one-to-one, it is a function where $f(x_1) = f(x_2)$ implies $x_1 = x_2$, meaning different inputs always map to different outputs.

Surjective Function

Also known as onto, it is a function where every element in the codomain $Y$ has at least one preimage $x$ in the domain $X$.

Bijective Function

A function that is both injective (one to one) and surjective (onto).

Composite Function

The result of combining two functions $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ to form $h(x) = (g \text{ o } f)(x) = g(f(x))$.

Inverse Function

A function $f^{-1}$ that exists only for bijective functions such that $f^{-1}(f(x)) = x$ and $f(f^{-1}(y)) = y$.

Monotonic Function

A function which is either increasing, strictly increasing, decreasing, or strictly decreasing on a given interval.

Odd Function

A function that satisfies the condition $f(-x) = -f(x)$ for all $x$ in its domain.

Even Function

A function that satisfies the condition $f(-x) = f(x)$ for all $x$ in its domain.

Parity

The property of a function based on its evenness or oddness.

Floor Function

Represented by $\text{bxc}$, its value for every real number $x$ is the greatest integer which is less than or equal to $x$.

Ceiling Function

Represented by $\text{dxe}$, its value for every real number $x$ is the smallest integer that is greater than or equal to $x$.

Periodic Function

A function where there exists $T > 0$ such that $f(x + T) = f(x)$ for every $x$ in the domain; $T$ is called the period.

Zeros of a Function

The values of $x$ for which the function $f(x)$ takes the value zero.

Singularities

For a rational function $f(x) = \frac{P(x)}{Q(x)}$, these are the values of $x$ for which the denominator $Q(x) = 0$.

Limit from the Left

The limit $l$ reached as $x$ tends to $b$ from the left, written as $\text{lim}_{x \rightarrow b^{-}} f(x) = l$.

Continuity at a Point

A function $f(x)$ is continuous at $x = a$ if $\text{lim}_{x \rightarrow a} f(x)$ exists and equals $f(a)$.

Indeterminate Form

A form such as $0/0$ or $\frac{\text{inf}}{\text{inf}}$ that a ratio of functions takes as the variable approaches a specific value.

L'hospital's Rule

A rule stating that if $f(x)/g(x)$ result in an indeterminate form, its limit can be calculated as $\text{lim}_{x \rightarrow x_0} \frac{f'(x)}{g'(x)}$.

Derivative

The limiting value $M$ of the slope of a tangent line to the graph of $f$ at point $P$, defined as $M = \text{lim}_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$.

First Principles

The process of finding the derived function $f'(x)$ using the definition of the derivative and the limiting process.

Differential Operator

The symbol $\frac{dy}{dx}$ first used by Liebnitz to denote the derivative of a function.

Chain Rule

A method for differentiating composite functions, expressed as $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$.

Leibnitz Rule

A formula for the $n$-th derivative of the product of two functions $U(x)$ and $V(x)$, involving combinations $nCr$.

Implicit Function

A function not expressed solely in terms of the independent variable, defined by conditions it satisfies rather than an explicit formula.

Natural Logarithmic Function

A function of the form $f(x) = \text{log}_e u(x)$, also denoted as $\text{ln } u$.

Critical Point

A value $c$ in the domain where the derivative $f'(c) = 0$, representing a turning point where the tangent is horizontal.

Point of Inflexion

A point on a curve where the second derivative equals zero and the concavity of the graph changes.

Rolle's Theorem

States that if $f$ is continuous on $[x_1, x_2]$, differentiable on $(x_1, x_2)$, and $f(x_1) = f(x_2)$, then there exists at least one $p$ in $(x_1, x_2)$ such that $f'(p) = 0$.

Mean Value Theorem

States that for a continuous and differentiable function on an interval $[a, b]$, there is a point $p$ such that $f(b) - f(a) = (b - a)f'(p)$.

Antiderivative

The perspective of integration as finding a function whose derivative is given.

Indefinite Integral

Represented by $\text{integral } f(x)dx = F(x) + C$, where $C$ is an arbitrary constant.

Integration by Parts

A method based on the product rule for derivatives: $\text{integral } u \frac{dv}{dx} dx = uv - \text{integral } v \frac{du}{dx} dx$.

Partial Fractions

A method of resolving a rational function into a sum of simpler fractions to facilitate integration.

Definite Integral

The limit of the sum of area strips as the number of strips tends to infinity, written as $\text{integral}_a^b f(x)dx$.

Arc Length

The length of a curve $y = f(x)$ between two points, calculated as $\text{integral}_a^b [1 + (f'(x))^2]^{1/2} dx$.

Solid of Revolution

A solid figure generated by rotating a region under a curve around an axis, such as the $x$-axis or $y$-axis.

Moment of Inertia

The sum of the product of each mass and the square of its distance from a given line, often denoted by $I$.

Radius of Gyration

The distance $k$ from a given line at which a single mass equal to the total mass would have the same moment of inertia.

Work

The integral of force over displacement, calculated as $W = \text{integral}_a^b F(s)ds$.

Ordinary Differential Equation (ODE)

An equation involving ordinary differential coefficients like $\frac{dy}{dx}$ or $\frac{d^2y}{dx^2}$.

General Solution

The form of a solution to a differential equation that includes unknown constants representing all possible solutions.

Half-life

The time it takes for a substance to disintegrate to half of its original quantity.

Marginal Cost

The rate of change of the total cost with respect to the number of items produced: $\frac{dC}{dx}$.

Marginal Revenue

The rate of change of total revenue realized from the production and sale of units of a commodity.