LCA Mathematical Applications cheat sheet

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiply coefficients and add exponents, divide coefficients and subtract exponents.

Rules of exponents: a^m · a^n = a^(m+n); a^m / a^n = a^(m−n); (a^m)^n = a^(mn); a^(−n) = 1/a^n; a^0 = 1 for a ≠ 0.

Rational exponents: a^(m/n) = the n-th root of a^m = (n-th root of a)^m, valid for a ≥ 0 when n is even.

Percent change = (new − old)/old × 100; successive changes multiply factors, so +20% then −20% yields 0.96 of the original.

Complex numbers: i² = −1; (a + bi)(c + di) = (ac − bd) + (ad + bc)i; divide by multiplying numerator and denominator by the conjugate.

Absolute value equations |x| = k give x = k or x = −k for k > 0, no solution for k < 0, and x = 0 for k = 0.

Unit conversion by dimensional analysis: multiply by ratios equal to 1 so unwanted units cancel; e.g., 60 mi/h × 5280 ft/mi ÷ 3600 s/h = 88 ft/s.

Vectors in component form: magnitude = √(x² + y²); direction angle = arctan(y/x) adjusted for quadrant; add vectors component-wise.

Slope between two points: m = (y₂ − y₁)/(x₂ − x₁); parallel lines share m, perpendicular lines have m₁ · m₂ = −1.

Linear forms: slope-intercept y = mx + b; point-slope y − y₁ = m(x − x₁); standard Ax + By = C with slope −A/B.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a); discriminant b² − 4ac > 0 gives two real roots, = 0 one, < 0 two complex.

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

Exponential growth/decay: y = a(1 + r)^t for growth and y = a(1 − r)^t for decay, where r is the per-period rate as a decimal.

Logarithms: 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; log_b x = ln x / ln b.

Function composition: (f ∘ g)(x) = f(g(x)); inverse functions satisfy f(f⁻¹(x)) = x, found by swapping x and y and solving.

Systems of two linear equations: solve by substitution or elimination; no solution when slopes are equal and intercepts differ, infinitely many when equations are multiples.

Circle: circumference C = 2πr, area A = πr²; arc length = (θ/360) · 2πr and sector area = (θ/360) · πr² for central angle θ in degrees.

Triangle area = ½bh; Heron's formula with semiperimeter s: A = √[s(s−a)(s−b)(s−c)].

Pythagorean theorem a² + b² = c²; common triples include 3-4-5, 5-12-13, 8-15-17, and 7-24-25.

Volume: prism/cylinder V = Bh; pyramid/cone V = ⅓Bh; sphere V = (4/3)πr³; surface area of a sphere = 4πr².

Similar figures: corresponding lengths scale by k, areas by k², and volumes by k³.

Distance and midpoint in the plane: d = √[(x₂−x₁)² + (y₂−y₁)²]; midpoint = ((x₁+x₂)/2, (y₁+y₂)/2).

Degrees and radians: 180° = π radians; convert degrees to radians by multiplying by π/180.

Right-triangle ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent; Law of Sines a/sin A = b/sin B.

Mean = sum of values / count; median = middle value of ordered data; mode = most frequent value; the mean is pulled toward outliers more than the median.

Standard deviation measures typical distance from the mean; roughly 68% of normal data lie within 1 SD and 95% within 2 SD of the mean.

Probability of an event = favorable outcomes / total equally likely outcomes; P(not A) = 1 − P(A).

Addition rule: P(A or B) = P(A) + P(B) − P(A and B); for mutually exclusive events the intersection term is 0.

Independent events: P(A and B) = P(A) · P(B); conditional probability P(A|B) = P(A and B)/P(B).

Permutations count ordered arrangements: nPr = n!/(n−r)!; combinations count unordered selections: nCr = n!/[r!(n−r)!].

Expected value = Σ(value × probability); a fair game has expected value 0 for net winnings.

Correlation r ranges from −1 to 1 and measures linear association only; correlation does not establish causation.

Rate problems: distance = rate × time; work rates add as reciprocals, so combined time = 1/(1/a + 1/b) for two workers.

Mixture problems balance the solute: (concentration₁ × volume₁) + (concentration₂ × volume₂) = final concentration × total volume.

Linear models apply when a quantity changes by a constant amount per unit; exponential models apply when it changes by a constant percent per unit.

Proportional reasoning: set up a/b = c/d and cross-multiply; check that units on both sides of the equation match.

Break-even occurs where revenue equals cost: solve R(x) = C(x); profit is maximized where marginal revenue equals marginal cost.

Rounding and tolerance: report results to the precision of the given data and state units; percent error = |measured − accepted|/accepted × 100.

Optimization under constraints: identify the objective function and constraints, then test vertices of the feasible region for linear problems.

Sanity-check a model by testing endpoints and units; a model predicting impossible values (negative lengths, populations exceeding limits) signals a domain restriction.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiply coefficients and add exponents, divide coefficients and subtract exponents.

Rules of exponents: a^m · a^n = a^(m+n); a^m / a^n = a^(m−n); (a^m)^n = a^(mn); a^(−n) = 1/a^n; a^0 = 1 for a ≠ 0.

Rational exponents: a^(m/n) = the n-th root of a^m = (n-th root of a)^m, valid for a ≥ 0 when n is even.

Percent change = (new − old)/old × 100; successive changes multiply factors, so +20% then −20% yields 0.96 of the original.

Complex numbers: i² = −1; (a + bi)(c + di) = (ac − bd) + (ad + bc)i; divide by multiplying numerator and denominator by the conjugate.

Absolute value equations |x| = k give x = k or x = −k for k > 0, no solution for k < 0, and x = 0 for k = 0.

Unit conversion by dimensional analysis: multiply by ratios equal to 1 so unwanted units cancel; e.g., 60 mi/h × 5280 ft/mi ÷ 3600 s/h = 88 ft/s.

Vectors in component form: magnitude = √(x² + y²); direction angle = arctan(y/x) adjusted for quadrant; add vectors component-wise.

Slope between two points: m = (y₂ − y₁)/(x₂ − x₁); parallel lines share m, perpendicular lines have m₁ · m₂ = −1.

Linear forms: slope-intercept y = mx + b; point-slope y − y₁ = m(x − x₁); standard Ax + By = C with slope −A/B.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a); discriminant b² − 4ac > 0 gives two real roots, = 0 one, < 0 two complex.

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

Exponential growth/decay: y = a(1 + r)^t for growth and y = a(1 − r)^t for decay, where r is the per-period rate as a decimal.

Logarithms: 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; log_b x = ln x / ln b.

Function composition: (f ∘ g)(x) = f(g(x)); inverse functions satisfy f(f⁻¹(x)) = x, found by swapping x and y and solving.

Systems of two linear equations: solve by substitution or elimination; no solution when slopes are equal and intercepts differ, infinitely many when equations are multiples.

Circle: circumference C = 2πr, area A = πr²; arc length = (θ/360) · 2πr and sector area = (θ/360) · πr² for central angle θ in degrees.

Triangle area = ½bh; Heron's formula with semiperimeter s: A = √[s(s−a)(s−b)(s−c)].

Pythagorean theorem a² + b² = c²; common triples include 3-4-5, 5-12-13, 8-15-17, and 7-24-25.

Volume: prism/cylinder V = Bh; pyramid/cone V = ⅓Bh; sphere V = (4/3)πr³; surface area of a sphere = 4πr².

Similar figures: corresponding lengths scale by k, areas by k², and volumes by k³.

Distance and midpoint in the plane: d = √[(x₂−x₁)² + (y₂−y₁)²]; midpoint = ((x₁+x₂)/2, (y₁+y₂)/2).

Degrees and radians: 180° = π radians; convert degrees to radians by multiplying by π/180.

Right-triangle ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent; Law of Sines a/sin A = b/sin B.

Mean = sum of values / count; median = middle value of ordered data; mode = most frequent value; the mean is pulled toward outliers more than the median.

Standard deviation measures typical distance from the mean; roughly 68% of normal data lie within 1 SD and 95% within 2 SD of the mean.

Probability of an event = favorable outcomes / total equally likely outcomes; P(not A) = 1 − P(A).

Addition rule: P(A or B) = P(A) + P(B) − P(A and B); for mutually exclusive events the intersection term is 0.

Independent events: P(A and B) = P(A) · P(B); conditional probability P(A|B) = P(A and B)/P(B).

Permutations count ordered arrangements: nPr = n!/(n−r)!; combinations count unordered selections: nCr = n!/[r!(n−r)!].

Expected value = Σ(value × probability); a fair game has expected value 0 for net winnings.

Correlation r ranges from −1 to 1 and measures linear association only; correlation does not establish causation.

Rate problems: distance = rate × time; work rates add as reciprocals, so combined time = 1/(1/a + 1/b) for two workers.

Mixture problems balance the solute: (concentration₁ × volume₁) + (concentration₂ × volume₂) = final concentration × total volume.

Linear models apply when a quantity changes by a constant amount per unit; exponential models apply when it changes by a constant percent per unit.

Proportional reasoning: set up a/b = c/d and cross-multiply; check that units on both sides of the equation match.

Break-even occurs where revenue equals cost: solve R(x) = C(x); profit is maximized where marginal revenue equals marginal cost.

Rounding and tolerance: report results to the precision of the given data and state units; percent error = |measured − accepted|/accepted × 100.

Optimization under constraints: identify the objective function and constraints, then test vertices of the feasible region for linear problems.

Sanity-check a model by testing endpoints and units; a model predicting impossible values (negative lengths, populations exceeding limits) signals a domain restriction.