MB Applied Math 40S cheat sheet

Compound interest: A = P(1 + i)^n, where i = annual rate ÷ compounding periods per year and n = years × periods per year.

Continuous compounding: A = Pe^(rt), the limit of (1 + i)^n as compounding frequency grows without bound.

Effective annual rate: EAR = (1 + i)^n − 1, converting a nominal rate with n compoundings per year into a comparable yearly rate.

Annuity present value: PV = R[1 − (1 + i)^(−n)]/i, the lump sum equivalent to n payments of R at rate i per period.

Annuity future value: FV = R[(1 + i)^n − 1]/i, the accumulated value of a series of equal payments made at period end.

Loan payment: R = PV · i / [1 − (1 + i)^(−n)]; the payment rises with rate and falls as the amortization period lengthens.

Rule of 72: doubling time ≈ 72 ÷ annual percentage rate, a quick estimate accurate within a few percent for rates near 8%.

Real return uses (1 + nominal) ÷ (1 + inflation) − 1; purchasing power falls whenever inflation exceeds the nominal return.

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

Addition rule: P(A or B) = P(A) + P(B) − P(A and B); subtract the overlap so joint outcomes are not double-counted.

Conditional probability: P(A|B) = P(A and B)/P(B); independent events satisfy P(A and B) = P(A)P(B).

Expected value: E(X) = Σ x·P(x); a game is fair when E(X) = 0 and favourable to the player when E(X) > 0.

Binomial probability: P(r successes) = nCr · p^r · (1 − p)^(n − r), used for fixed n, two outcomes, constant p, independent trials.

Normal data: about 68% within 1 standard deviation, 95% within 2, 99.7% within 3; z = (x − μ)/σ standardizes a score.

Complement rule: P(not A) = 1 − P(A); use it whenever counting the complement is easier than counting the event directly.

Mean is sensitive to outliers while median and mode resist them; standard deviation measures spread about the mean.

A relation is a function only if each domain value pairs with exactly one range value; the vertical line test detects failures.

Quadratic vertex form y = a(x − h)^2 + k places the vertex at (h, k); the axis of symmetry is x = h.

Quadratic formula x = [−b ± √(b^2 − 4ac)]/(2a); the discriminant b^2 − 4ac gives two, one, or no real roots.

Transformations: y = af(b(x − h)) + k stretches vertically by a, compresses horizontally by 1/b, shifts right h and up k.

Exponential growth/decay y = a·b^x: b > 1 grows, 0 < b < 1 decays; the y-intercept is a and the asymptote is y = 0.

Logarithms invert exponentials: log_b(xy) = log_b x + log_b y and log_b(x^p) = p·log_b x, used to solve for unknown exponents.

Inverse functions undo each other; reflect y = f(x) in the line y = x, and f is invertible only if it passes the horizontal line test.

Polynomial end behaviour follows the leading term: even degree matches ends, odd degree opposes them, and the sign of the leading coefficient sets direction.

Degrees to radians: multiply by π/180; radians to degrees: multiply by 180/π. A full rotation is 2π radians or 360°.

Arc length s = rθ and sector area A = ½r^2θ, valid only when θ is measured in radians.

Right-triangle ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent (SOH-CAH-TOA).

Sine law: a/sin A = b/sin B = c/sin C; use it with a known side opposite a known angle (AAS, ASA, SSA).

Cosine law: c^2 = a^2 + b^2 − 2ab·cos C; use it for SAS or SSS, and it reduces to Pythagoras when C = 90°.

In the ambiguous SSA case, the height is h = b·sin A: no triangle if a < h, one if a = h or a ≥ b, two if h < a < b.

Unit circle: (cos θ, sin θ) on the circle of radius 1; sin is positive in quadrants I and II, cos in I and IV, tan in I and III.

Pythagorean identity sin^2 θ + cos^2 θ = 1 and tan θ = sin θ/cos θ support simplification and verification of identities.

Circle: circumference C = 2πr and area A = πr^2; a chord, tangent, and radius relationships follow from the perpendicular from centre bisecting a chord.

Inscribed angle theorem: an angle inscribed in a circle is half the central angle subtending the same arc; an angle in a semicircle is 90°.

Volume formulas: cylinder V = πr^2h, cone V = ⅓πr^2h, sphere V = (4/3)πr^3, prism V = base area × height.

Surface area of a sphere is 4πr^2; a cone's lateral area is πrl where l is the slant height √(r^2 + h^2).

Similar figures: corresponding lengths scale by k, areas by k^2, and volumes by k^3; set up proportions on matching sides.

Scale drawings: map distance ÷ scale factor gives real distance; convert units before computing so labels stay consistent.

Sine-area formula for any triangle: Area = ½ab·sin C, using two sides and the included angle.

Dimensional analysis: multiply by conversion ratios equal to 1 so unwanted units cancel, e.g. km/h × 1000/3600 gives m/s.