EMAT0103 – Differential Calculus Quiz 1 Reviewer

Practice flashcards covering key definitions and vocabulary from the EMAT0103 Differential Calculus Quiz 1 Reviewer on Algebraic, Inverse, Trigonometric, Exponential, and Logarithmic Functions.

Calculus

The branch of mathematics dealing with continuously changing properties; derived from the Latin word meaning "pebble" or "little stone."

Differential Calculus

A branch of calculus that deals with very small changing entities, specifically focus on rates of change and slopes.

Integral Calculus

A branch of calculus that deals with overall or accumulated changes.

Function

A relation from a set of inputs to a set of outputs where each input is associated with exactly one output, written as $f(x)$.

Independent variable

The input value in a function, typically represented by $x$.

Dependent variable

The output value in a function, typically represented by $y$, whose value depends on the independent variable.

Domain

The set of all possible input ($x$) values for a function.

Range

The set of all possible output ($y$) values for a function.

Open Interval

A set containing all real $x$ such that $a < x < b$, denoted as $(a, b)$.

Closed Interval

A set containing all real $x$ such that $a \leq x \leq b$, denoted as $[a, b]$.

Function Composition

The process of calculating $(f \circ g)(x) = f(g(x))$, where the output of $g(x)$ becomes the input of $f(x)$.

Inverse Functions

A function $f^{-1}$ that "undoes" the original function $f$, satisfying $(f \circ f^{-1})(x) = (f^{-1} \circ f)(x) = x$.

Algebraic Expression

A mathematical phrase composed of variables, constants, coefficients, and algebraic operations.

Monomial

An algebraic expression consisting of one term where exponents are non-negative integers.

Binomial

An algebraic expression consisting of two monomials.

Trinomial

An algebraic expression consisting of three monomials.

Polynomial

An algebraic expression consisting of one or more monomials.

Multinomial

An algebraic expression for which exponents can be positive or negative, consisting of one or more terms.

Algebraic Equation

An algebraic expression that is set equal to a specific value.

Rational Expression

A fraction in which both the numerator and the denominator are polynomials.

Transcendental Functions

Functions that cannot be expressed as a finite combination of algebraic operations and rely on infinite processes such as power series.

Euler's number ($e$)

An irrational number approximately equal to $2.71828182845905...$ used as the base for the natural exponential function.

Trigonometric Functions

Functions of an angle related to right triangles, including $\sin(x)$, $\cos(x)$, $\tan(x)$, $\csc(x)$, $\sec(x)$, and $\cot(x)$.

Exponential Function

A function of the form $f(x) = a \cdot b^{x}$, where $b > 0$ and $b \neq 1$.

Logarithmic Function

The inverse of an exponential function, defined as $y = \log_b(x) \Leftrightarrow b^y = x$.

Common Logarithm

A logarithm with a base of $10$, denoted as $\log_{10}(x)$ or $\log(x)$.

Natural Logarithm

A logarithm with a base of $e$, denoted as $\log_e(x)$ or $\ln(x)$.

Vertex de a Quadratic Function

The turning point of a parabola defined by the coordinates $x = -\frac{b}{2a}$ and $y = f(-\frac{b}{2a})$.

Degree

A unit of angle measurement where a full revolution is equal to $360^{\circ}$.

Radian

A unit of angle measurement where a full revolution is equal to $2\pi\,rad$.

Gradians (Gons)

A unit of angle measurement where a full revolution is equal to $400\,grads$ or $400\,gons$.

Mils

A unit of angle measurement where a full revolution is equal to $6,400\,mils$.

ASTC Rule

A rule (All Students Take Calculus) used to identify which trigonometric functions are positive in each quadrant: I (All), II (Sin), III (Tan), IV (Cos).

Discrete Exponential Model

A model for growth or decay that occurs once per period, represented by $y = a(1 \pm r)^{t}$.

Continuous Exponential Model

A model for growth or decay that occurs at every instant, represented by $P = P_0 e^{kt}$.

Newton's Law of Cooling

A model for temperature change over time represented by the formula $T = T_s + (T_0 - T_s)e^{-kt}$, where $T_s$ is the surrounding temperature.