PEI Math 521A flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the solutions are x = (−b ± √(b² − 4ac))/(2a). The discriminant b² − 4ac decides whether roots are real and distinct, repeated, or complex.

Discriminant

The quantity b² − 4ac from a quadratic ax² + bx + c. If positive, two distinct real roots; if zero, one repeated real root; if negative, two nonreal conjugate roots.

Vertex form

A quadratic written as y = a(x − h)² + k, whose graph is a parabola with vertex (h, k) and axis of symmetry x = h. It reveals the maximum or minimum value k directly.

Remainder theorem

Dividing a polynomial p(x) by x − r gives remainder p(r). Consequently x − r is a factor of p exactly when p(r) = 0, which lets you test candidate factors quickly.

Logarithm product rule

For valid bases, log_b(xy) = log_b x + log_b y. This converts multiplication into addition, and the companion rules handle quotients and powers: log_b(x/y) and k·log_b x.

Inverse function condition

A function has an inverse only if it is one-to-one on its stated domain. Then f⁻¹ satisfies f(f⁻¹(x)) = x for x in the range of f, and graphs reflect across y = x.

Radian measure

One radian is the angle subtending an arc equal in length to the radius; π radians equal 180°. Arc length is s = rθ and sector area is ½r²θ for θ in radians.

Law of cosines

For a triangle with sides a, b, c and angle C opposite side c: c² = a² + b² − 2ab·cos C. It handles any triangle and reduces to the Pythagorean theorem when C = 90°.

Law of sines

In any triangle, a/sin A = b/sin B = c/sin C, where each side is opposite its named angle. The common ratio also equals twice the circumradius of the triangle.

Double angle identities

sin 2A = 2 sin A cos A and cos 2A = cos²A − sin²A, which also equals 2cos²A − 1 or 1 − 2sin²A. They follow from the sine and cosine sum formulas.

Inscribed angle theorem

An angle whose vertex lies on a circle measures half the intercepted arc. A diameter subtends a right angle at any point on the circle, a frequent shortcut in geometry problems.

Similar figure scaling

For similar figures with linear scale factor k, corresponding lengths scale by k, areas by k², and volumes by k³. Doubling every dimension multiplies volume by eight.

Sphere volume

A sphere of radius r has volume (4/3)πr³ and surface area 4πr². Both formulas are standard reference results needed for solids and related-rate problems.

Derivative limit definition

The derivative at x is lim_{h→0} [f(x + h) − f(x)]/h, the limit of secant slopes. It equals the slope of the tangent line where the two-sided limit exists.

Chain rule

For a composition, d/dx f(g(x)) = f′(g(x))·g′(x). Differentiate the outer function at the inner value, then multiply by the inner function's derivative.

Product and quotient rules

Product: (fg)′ = f′g + fg′. Quotient: (f/g)′ = (f′g − fg′)/g², valid where g ≠ 0. Both follow from the derivative limit definition.

Second derivative test

At a critical point where f′ = 0, f″ > 0 indicates a local minimum and f″ < 0 a local maximum. If f″ = 0, the test is inconclusive.

Fundamental Theorem of Calculus

If F is an antiderivative of continuous f, then ∫ from a to b of f(x)dx = F(b) − F(a). It links accumulated area to antiderivatives and underlies most integration.

Mean Value Theorem

If f is continuous on [a, b] and differentiable on (a, b), some c in (a, b) has f′(c) = [f(b) − f(a)]/(b − a). The tangent slope matches the secant slope somewhere.

L'Hôpital's rule

For a 0/0 or ∞/∞ indeterminate limit, lim f/g = lim f′/g′, provided the second limit exists. Other indeterminate forms must be rewritten first into one of these two.

Conditional probability

P(A|B) = P(A ∩ B)/P(B) for P(B) > 0. Events are independent exactly when P(A|B) = P(A), equivalently when P(A ∩ B) = P(A)P(B).

Bayes' theorem

P(A|B) = P(B|A)P(A)/P(B). The denominator expands as a sum of P(B|Aᵢ)P(Aᵢ) over a partition, letting prior probabilities update to posterior ones.

Binomial distribution

For n independent trials each with success probability p, P(X = k) = C(n,k)p^k(1 − p)^{n−k}. The mean is np and the standard deviation is √(np(1 − p)).

Central limit theorem

For large independent samples of size n, the sample mean is approximately normally distributed about the population mean μ with standard error σ/√n, regardless of population shape.

Expected value

For a discrete random variable, E(X) = Σ x·P(x), a probability-weighted average. Variance is E(X²) − [E(X)]², and Var(aX + b) = a²Var(X).

Combinations formula

C(n, r) = n!/(r!(n − r)!) counts subsets of size r chosen from n distinct items when order does not matter; permutations P(n, r) = n!/(n − r)! count ordered arrangements.

Pigeonhole principle

Placing n items into k containers guarantees some container holds at least ⌈n/k⌉ items. It proves existence of a repeated category without identifying which one.

Contrapositive equivalence

P → Q is logically equivalent to ¬Q → ¬P, its contrapositive. The converse Q → P and inverse ¬P → ¬Q are not equivalent to the original statement.

Mathematical induction

Prove the base case, assume the statement holds for n = k, then prove it for n = k + 1. Strong induction instead assumes all cases from the base up through k.

Euler circuit condition

A connected graph has an Euler circuit exactly when every vertex has even degree. An Euler path exists when exactly zero or two vertices have odd degree.

Exponential vs logistic growth

Exponential y = abˣ grows at constant percent rate without bound. Logistic y = L/(1 + ae^{−bx}) starts similarly but levels off at carrying capacity L, modeling limited resources.

Least-squares regression

The best-fit line minimizes squared vertical residuals and passes through (x̄, ȳ), with slope b = r(s_y/s_x). A residual is observed minus predicted value.