MB Applied Math 40S flashcards

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

Compound interest formula

A = P(1 + i)^n, where P is principal, i is the interest rate per compounding period, and n is the total number of compounding periods.

Effective annual rate

The yearly rate that produces the same growth as a nominal rate compounded more often: EAR = (1 + i)^n − 1, with n periods per year.

Present value of annuity

PV = R[1 − (1 + i)^(−n)]/i gives the single lump sum today equivalent to n equal payments of R at periodic rate i.

Future value of annuity

FV = R[(1 + i)^n − 1]/i accumulates n equal end-of-period payments of R at periodic rate i into one total value.

Rule of 72

An estimate of doubling time for invested money: divide 72 by the annual percentage growth rate. It is reasonably accurate for rates near 8%.

Amortized loan payment

R = PV · i / [1 − (1 + i)^(−n)] gives the equal periodic payment that repays a loan of PV over n periods at rate i.

Combination notation nCr

nCr = n!/[r!(n − r)!] counts the number of unordered selections of r items from n distinct items, where arrangement order does not matter.

Permutation notation nPr

nPr = n!/(n − r)! counts ordered arrangements of r items chosen from n distinct items, so different orders count as different outcomes.

Addition rule for probability

P(A or B) = P(A) + P(B) − P(A and B). The subtraction removes outcomes counted twice when events can occur together.

Conditional probability

P(A|B) = P(A and B)/P(B) gives the probability of A given that B has occurred, restricted to the reduced sample space of B.

Expected value

E(X) = Σ x·P(x), the long-run average outcome. A game with expected value zero is fair; positive favours the player.

Binomial probability

P(r) = nCr · p^r · (1 − p)^(n − r) gives the probability of exactly r successes in n independent trials with constant success probability p.

z-score standardization

z = (x − μ)/σ states how many standard deviations a value lies from the mean, allowing comparison across different normal distributions.

Empirical rule percentages

For normal data, roughly 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three.

Vertical line test

A graph represents a function only if no vertical line crosses it more than once, meaning each input has at most one output.

Vertex form of quadratic

y = a(x − h)^2 + k has vertex (h, k) and axis of symmetry x = h; the sign of a determines whether it opens up or down.

Discriminant meaning

For ax^2 + bx + c = 0, the value b^2 − 4ac gives two real roots if positive, one if zero, and none if negative.

Exponential function form

y = a·b^x has y-intercept a; growth occurs when b > 1 and decay when 0 < b < 1, with y = 0 as a horizontal asymptote.

Inverse function test

A function has an inverse function only if it passes the horizontal line test; the inverse graph is the reflection across the line y = x.

Logarithm product rule

log_b(xy) = log_b x + log_b y converts multiplication into addition, which allows solving exponential equations for an unknown exponent.

Function definition

A relation in which every value in the domain is paired with exactly one value in the range, so repeated outputs are allowed but repeated inputs are not.

Degree-radian conversion

Multiply degrees by π/180 to obtain radians, or multiply radians by 180/π to obtain degrees. A full circle is 2π radians.

Arc length formula

s = rθ gives the arc length on a circle of radius r subtended by central angle θ, which must be measured in radians.

Sector area formula

A = ½r^2θ gives the area of a circular sector with radius r and central angle θ measured in radians, half the arc length times radius.

Sine law

a/sin A = b/sin B = c/sin C relates sides to opposite angles in any triangle; it applies to AAS, ASA, and the ambiguous SSA case.

Cosine law

c^2 = a^2 + b^2 − 2ab·cos C finds a side or angle in any triangle from SAS or SSS data, generalizing the Pythagorean theorem.

Ambiguous SSA case

With two sides and a non-included angle, compare the given side to h = b·sin A: zero, one, or two triangles may satisfy the data.

Unit circle coordinates

A point on the unit circle at angle θ has coordinates (cos θ, sin θ), so sine and cosine range from −1 to 1.

Pythagorean identity

sin^2 θ + cos^2 θ = 1 holds for every angle and follows from the unit circle; with tan θ = sin θ/cos θ it simplifies expressions.

Inscribed angle theorem

An angle inscribed in a circle measures half the central angle that subtends the same arc, so an angle in a semicircle is 90°.

Sphere volume and area

A sphere of radius r has volume (4/3)πr^3 and surface area 4πr^2; both scale by the cube and square of the radius factor.

Similar figure scaling

When two figures are similar with length ratio k, their areas compare as k^2 and their volumes as k^3, so proportions must match like quantities.