PEI Math 521A cheat sheet

Quadratic formula: for ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac)) / (2a); the discriminant b² − 4ac gives two real roots if positive, one if zero, and a conjugate pair if negative.

Vertex of a parabola y = ax² + bx + c occurs at x = −b/(2a); the vertex form y = a(x − h)² + k has vertex (h, k) and axis of symmetry x = h.

Remainder Theorem: dividing polynomial p(x) by (x − r) leaves remainder p(r); Factor Theorem: (x − r) is a factor exactly when p(r) = 0.

Rational Root Theorem: any rational root p/q in lowest terms of a polynomial with integer coefficients has p dividing the constant term and q dividing the leading coefficient.

Logarithm rules: log_b(xy) = log_b x + log_b y, log_b(x/y) = log_b x − log_b y, log_b(x^k) = k log_b x, and log_b x = ln x / ln b for any valid base b > 0, b ≠ 1.

Exponential growth/decay model: A(t) = A₀e^{kt}, where k > 0 gives growth and k < 0 gives decay; doubling time equals ln 2 / k.

Composition and inverses: (f ∘ g)(x) = f(g(x)); f is invertible on its domain only if it is one-to-one, and then f⁻¹ undoes f with f(f⁻¹(x)) = x on the range of f.

Function transformations: y = a·f(b(x − h)) + k stretches vertically by |a|, compresses horizontally by 1/|b|, shifts right by h, and shifts up by k; negative a or b reflects across an axis.

Radian–degree conversion: π radians = 180°; arc length s = rθ and sector area A = ½r²θ when θ is measured in radians.

Unit circle definitions: cos θ and sin θ are the x- and y-coordinates of the point at angle θ; tan θ = sin θ / cos θ, and cos²θ + sin²θ = 1 for every θ.

Sum formulas: sin(A ± B) = sin A cos B ± cos A sin B and cos(A ± B) = cos A cos B ∓ sin A sin B; double angle: sin 2A = 2 sin A cos A, cos 2A = cos²A − sin²A.

Law of Sines: a/sin A = b/sin B = c/sin C = 2R (R the circumradius); Law of Cosines: c² = a² + b² − 2ab cos C, which reduces to Pythagoras when C = 90°.

Triangle area: A = ½ab sin C; Heron's formula A = √(s(s − a)(s − b)(s − c)) with semiperimeter s = (a + b + c)/2.

Circle facts: central angle measure equals intercepted arc measure, an inscribed angle is half its intercepted arc, and a tangent is perpendicular to the radius at the point of contact.

Volume formulas: cylinder V = πr²h, cone V = ⅓πr²h, sphere V = (4/3)πr³, sphere surface area S = 4πr².

Similar figures with scale factor k: lengths scale by k, areas by k², volumes by k³; in right triangles the altitude to the hypotenuse creates two triangles similar to the original.

Limit definition of derivative: f′(x) = lim_{h→0} [f(x + h) − f(x)]/h; the derivative exists only where this two-sided limit exists.

Power, product, quotient, chain: d/dx xⁿ = nx^{n−1}; (fg)′ = f′g + fg′; (f/g)′ = (f′g − fg′)/g²; d/dx f(g(x)) = f′(g(x))·g′(x).

Derivatives to memorize: d/dx sin x = cos x, d/dx cos x = −sin x, d/dx eˣ = eˣ, d/dx ln x = 1/x for x > 0, d/dx tan x = sec²x.

Critical points occur where f′(x) = 0 or f′ is undefined; f″ > 0 there means a local minimum, f″ < 0 a local maximum, and a sign change of f′ without a sign change of f″ indicates an inflection point.

Fundamental Theorem of Calculus: if F′ = f then ∫_a^b f(x) dx = F(b) − F(a); also d/dx ∫_a^x f(t) dt = f(x) when f is continuous.

Integration rules: ∫xⁿ dx = x^{n+1}/(n+1) + C for n ≠ −1; ∫(1/x) dx = ln|x| + C; ∫eˣ dx = eˣ + C; substitution reverses the chain rule.

Mean Value Theorem: if f is continuous on [a, b] and differentiable on (a, b), some c in (a, b) satisfies f′(c) = [f(b) − f(a)]/(b − a).

L'Hôpital's rule applies to 0/0 or ∞/∞ indeterminate forms: lim f/g = lim f′/g′, provided the latter limit exists; other indeterminate forms must first be rewritten.

Conditional probability: P(A|B) = P(A ∩ B)/P(B) for P(B) > 0; independence means P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A).

Bayes' theorem: P(A|B) = P(B|A)P(A) / P(B), where P(B) = Σ P(B|Aᵢ)P(Aᵢ) over a partition of the sample space.

Binomial distribution: P(X = k) = C(n, k)p^k(1 − p)^{n−k}, with mean np and standard deviation √(np(1 − p)); trials must be independent with constant p.

Normal calculations standardize with z = (x − μ)/σ; the empirical rule gives roughly 68% of values within 1σ, 95% within 2σ, and 99.7% within 3σ of the mean.

Central limit theorem: for independent samples of size n from a population with mean μ and standard deviation σ, the sample mean is approximately normal with mean μ and standard error σ/√n as n grows.

Expected value of a discrete variable is E(X) = Σ x·P(x); variance is Var(X) = E(X²) − [E(X)]², and Var(aX + b) = a²Var(X).

A confidence interval for a mean with known σ is x̄ ± z*·σ/√n; increasing n or lowering the confidence level narrows the interval.

Correlation r lies in [−1, 1] and measures only linear association; a small p-value is evidence against the null hypothesis, but it is not the probability that the null is true.

Counting: multiplication principle for sequential choices, permutations P(n, r) = n!/(n − r)!, and combinations C(n, r) = n!/(r!(n − r)!), which ignore order.

Pigeonhole principle: placing n items into k boxes forces at least one box to hold ⌈n/k⌉ items; used to prove existence without construction.

Inclusion–exclusion: |A ∪ B| = |A| + |B| − |A ∩ B|, extended to three sets by subtracting pairwise intersections and adding back the triple intersection.

Induction template: prove the base case, assume the statement for n = k, then derive it for n = k + 1 using that assumption; strong induction assumes all cases up to k.

A statement is logically equivalent to its contrapositive (¬Q → ¬P) but not to its converse (Q → P) or inverse (¬P → ¬Q); a single counterexample refutes a universal claim.

De Morgan's laws: ¬(P ∧ Q) ≡ ¬P ∨ ¬Q and ¬(P ∨ Q) ≡ ¬P ∧ ¬Q; implication P → Q is equivalent to ¬P ∨ Q.

A graph has an Euler circuit exactly when it is connected and every vertex has even degree; it has an Euler path when exactly zero or two vertices have odd degree.

Recurrence relations of the form aₙ = raₙ₋₁ + c have closed forms; the Fibonacci-style relation aₙ = aₙ₋₁ + aₙ₋₂ needs two initial conditions to be determined.

Modeling cycle: translate the real situation into variables and assumptions, build the model, solve it mathematically, then validate the result against the original context and revise.

Linear model y = mx + b fits constant-rate change; exponential y = abˣ fits constant percent change; logistic y = L/(1 + ae^{−bx}) fits growth that saturates at carrying capacity L.

Least-squares regression line has slope b = r(s_y/s_x) and passes through (x̄, ȳ); residual = observed − predicted, and residuals should show no pattern if the fit is adequate.

Rate problems: set up a relation between quantities changing in time and differentiate with respect to t, e.g. for a cone, V = ⅓πr²h gives dV/dt = ⅓π(2rh·dr/dt + r²·dh/dt).

Optimization: write the target quantity as a function of one variable using the constraint, differentiate, solve for critical points, and check endpoints of the feasible interval.

Unit and dimensional analysis: multiplying by conversion ratios that equal 1 changes units without changing value; a dimensionally inconsistent equation signals an error in the model.

Sensitivity and reasonableness: round only at the end, state units, and test whether the answer falls in a plausible range; extrapolating far beyond the data range weakens any conclusion.

Mixture and work models: for two rates r₁ and r₂ working together, combined rate is r₁ + r₂ and time is 1/(r₁ + r₂) when each alone takes 1/rᵢ time units.