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OpenStem
@openstem · Joined Jul 2026
7420 public items8 groups
Flashcards8 cards
What is the formula for the volume of a rectangular prism (box) with length l, width w, and height h?1 / 8
Multiply the three dimensions together. Doubling any one of them doubles the volume.
Mathematics · L3 · Surface Area & Volume of Solids
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Mathematics · L3 · Surface Area & Volume of SolidsFlashcards7 cards
You know two angles and a side of a triangle (AAS or ASA). Which law finds the missing sides, and why does it work?1 / 7
The Law of Sines. In any triangle, the ratio of a side to the sine of its opposite angle is the same for all three sides:
Mathematics · L3 · Law of Sines & Cosines
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Mathematics · L3 · Law of Sines & CosinesFlashcards8 cards
State the sum formula for an arithmetic series of n terms with first term a₁ and last term aₙ.1 / 8
Sum of the first n terms:
Mathematics · L3 · Arithmetic & Geometric Series
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Mathematics · L3 · Arithmetic & Geometric SeriesFlashcards8 cards
State the Fundamental Counting Principle.1 / 8
If one task can be done in m ways and a second, independent task can be done in n ways, then the two tasks together can be done in m × n ways. Example: 3 shirts and 2 pairs of pants make 3 × 2 = 6 possible outfits.
Mathematics · L3 · Permutations, Combinations & Probability
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Mathematics · L3 · Permutations, Combinations & ProbabilityFlashcards8 cards
What is a rational expression?1 / 8
A rational expression is a fraction whose numerator and denominator are both polynomials, such as (x + 2) / (x − 5).
Mathematics · L3 · Rational Expressions & Equations
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Mathematics · L3 · Rational Expressions & EquationsFlashcards8 cards
What does |x| mean, geometrically, on a number line?1 / 8
|x| is the distance from x to 0 on the number line. Distance is never negative, so |x| ≥ 0 for every real number x. For example, |4| = 4 and |−4| = 4 — both points sit 4 units from zero.
Mathematics · L3 · Absolute Value Equations & Inequalities
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Mathematics · L3 · Absolute Value Equations & InequalitiesFlashcards8 cards
What is the distance formula, and how is it related to the Pythagorean theorem?1 / 8
The distance between two points is the hypotenuse of a right triangle whose legs are the horizontal gap |x₂ − x₁| and the vertical gap |y₂ − y₁| between the points.
Mathematics · L3 · Coordinate Geometry
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Mathematics · L3 · Coordinate GeometryFlashcards10 cards
In y = A·sin(B(x − C)) + D, what does A control, and how do you find its value from the equation?1 / 10
A controls the amplitude — how far the graph rises and falls above and below its midline. Amplitude = |A|. For example, y = 3·sin(x) oscillates between −3 and 3.
Mathematics · L3 · Graphing Trig Functions
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Mathematics · L3 · Graphing Trig FunctionsFlashcards8 cards
What is a matrix?1 / 8
A rectangular arrangement of numbers in rows and columns. Each number in the matrix is called an entry (or element).
Mathematics · L3 · Introduction to Matrices
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Mathematics · L3 · Introduction to MatricesFlashcards8 cards
What is a rational function?1 / 8
A function that can be written as one polynomial divided by another, f(x) = p(x) / q(x), where q(x) is not the zero polynomial. Since division by zero is undefined, the domain excludes every x that makes q(x) = 0.
Mathematics · L3 · Rational Functions & Asymptotes
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Mathematics · L3 · Rational Functions & AsymptotesFlashcards8 cards
What does it mean for two events A and B to be independent?1 / 8
The outcome of A has no effect on the probability of B, and vice versa. Knowing that A happened doesn't change how likely B is.
Mathematics · L3 · Independent, Dependent & Conditional Probability
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Mathematics · L3 · Independent, Dependent & Conditional ProbabilityFlashcards9 cards
What does it mean to write A = {2, 4, 6, 8}?1 / 9
A is the set containing exactly the elements 2, 4, 6, and 8. Order and repetition do not matter — {2, 4, 6, 8} and {8, 6, 4, 2} are the same set.
Mathematics · L3 · Sets & Set Operations
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Mathematics · L3 · Sets & Set OperationsFlashcards9 cards
What is a proposition in logic?1 / 9
A proposition is a statement that is definitely true or definitely false, but not both. "7 is a prime number" is a proposition (it's true). "What time is it?" is not a proposition — questions don't have a truth value.
Mathematics · L3 · Logic & Truth Tables
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Mathematics · L3 · Logic & Truth TablesFlashcards9 cards
In graph theory, what are the two basic building blocks of a graph?1 / 9
Vertices (also called nodes) — the points — and edges — the connections between pairs of vertices. A graph is just a set of vertices plus a set of edges joining some of them.
Mathematics · L3 · Graph Theory Basics
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Mathematics · L3 · Graph Theory BasicsFlashcards9 cards
What is the difference between a recursive definition and an explicit (closed-form) definition of a sequence?1 / 9
A recursive definition builds each term from the term(s) before it, plus a starting base case. An explicit definition gives a formula that plugs n straight in to get any term directly, with no need to know earlier terms.
Mathematics · L3 · Recursive Sequences & Fibonacci
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Mathematics · L3 · Recursive Sequences & FibonacciFlashcards10 cards
What does the expression a mod n mean?1 / 10
It is the remainder left over when a is divided by n. For example, 14 mod 5 = 4, because 14 = 2 × 5 + 4.
Mathematics · L3 · Modular Arithmetic
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Mathematics · L3 · Modular ArithmeticFlashcards10 cards
What two things must you prove to show a statement P(n) is true for every natural number n?1 / 10
Mathematics · L3 · Mathematical Induction
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Mathematics · L3 · Mathematical InductionFlashcards10 cards
What does Big-O notation describe about an algorithm?1 / 10
It describes how the running time (or number of steps) grows as the input size n grows — not the exact number of seconds, just the shape of the growth as n gets larger. Constant factors and small details are ignored.
Mathematics · L3 · Algorithms & Big-O Basics
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Mathematics · L3 · Algorithms & Big-O BasicsFlashcards8 cards
What two pieces of information define a vector?1 / 8
A vector is defined by its magnitude (length) and its direction. It's often drawn as an arrow — the length of the arrow shows the magnitude, and the way it points shows the direction.
Mathematics · L3 · Introduction to Vectors
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Mathematics · L3 · Introduction to VectorsFlashcards8 cards
What is a piecewise function?1 / 8
A function defined by two or more different rules, each applying only to a specific interval of the domain. To evaluate it at some x, you first find which interval x falls into, then apply only that piece's rule.
Mathematics · L3 · Piecewise Functions & Step Functions
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Mathematics · L3 · Piecewise Functions & Step FunctionsFlashcards7 cards
What does a direct proof of 'If P then Q' actually do?1 / 7
It assumes P is true, then uses definitions, algebra, or known facts to derive Q in a straight chain of steps — no detour through the negation of anything.
Mathematics · L3 · Proof Techniques: Direct, Contrapositive & Contradiction
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Mathematics · L3 · Proof Techniques: Direct, Contrapositive & ContradictionFlashcards9 cards
What does GCD(a, b) mean?1 / 9
The greatest common divisor: the largest whole number that divides evenly into both a and b. For example, GCD(12, 18) = 6, since 6 is the biggest number that divides both 12 and 18.
Mathematics · L3 · Number Theory: GCD, LCM & the Euclidean Algorithm
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Mathematics · L3 · Number Theory: GCD, LCM & the Euclidean AlgorithmFlashcards7 cards
What is the polar coordinate system, and what do r and θ represent?1 / 7
Instead of locating a point by horizontal and vertical distances (x, y), polar coordinates locate it by a distance and an angle: r is the distance from the pole (the origin), and θ is the angle measured counterclockwise from the polar axis (the positive x-axis).
Mathematics · L3 · Polar Coordinates & Graphing
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Mathematics · L3 · Polar Coordinates & GraphingFlashcards8 cards
State the formula for sin(A + B).1 / 8
Note this is NOT sin A + sin B — a common mistake. The sine of a sum mixes both angles' sines and cosines.
Mathematics · L3 · Sum, Difference & Double-Angle Identities
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Mathematics · L3 · Sum, Difference & Double-Angle IdentitiesFlashcards9 cards
What is the difference between a population and a sample?1 / 9
The population is the entire group you want to learn about — every student in a school, every voter in a country. A sample is a smaller subset of the population that is actually measured or surveyed.
Mathematics · L3 · Sampling & Statistical Inference Basics
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Mathematics · L3 · Sampling & Statistical Inference BasicsFlashcards8 cards
What does i² equal, and why does that rule matter for complex numbers?1 / 8
i² = −1. Every time a product of two complex numbers produces an i² term, that rule is what lets you turn it into a real number — it is the single fact that makes all complex arithmetic work.
Mathematics · L3 · Complex Number Operations
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Mathematics · L3 · Complex Number OperationsFlashcards8 cards
What does it mean to describe a curve "parametrically"?1 / 8
Instead of writing y directly in terms of x, both coordinates are written in terms of a third variable, the parameter t (often thought of as time):
Mathematics · L3 · Parametric Equations & Motion
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Mathematics · L3 · Parametric Equations & MotionFlashcards11 cards
State the ε–δ definition of lim(x→a) f(x) = L.1 / 11
For every ε > 0 there exists δ > 0 such that 0 < |x − a| < δ implies |f(x) − L| < ε. In words: f(x) can be made arbitrarily close to L by taking x sufficiently close (but not equal) to a.
Mathematics · L4 · Calculus I — Limits & Derivatives
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Mathematics · L4 · Calculus I — Limits & DerivativesFlashcards8 cards
State the substitution rule (u-substitution).1 / 8
If u = g(x), then ∫f(g(x))g′(x)dx = ∫f(u)du. Rewrite the integrand in terms of u; convert dx via du = g′(x)dx.
Mathematics · L4 · Calculus — Integration Techniques
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Mathematics · L4 · Calculus — Integration TechniquesFlashcards10 cards
What is a dot product u·v, and what does it measure?1 / 10
u·v = Σ uᵢvᵢ = |u||v|cos θ. It measures the component of one vector in the direction of the other. u·v = 0 iff u ⊥ v.
Mathematics · L4 · Linear Algebra — Vectors & Matrices
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Mathematics · L4 · Linear Algebra — Vectors & Matrices