Hon Precalculus Sem 2 Finals

Function symmetry

• f(-x) = f(x) → even

• f(-x) = -f(x) → odd

function composition domain

• start w/ domain of inside function + add restrictions from composition

intermediate value theorem

• for function P, if P(a) and P(b) have opposite signs, there exists at least one value c between a and b for which P (c ) = 0

finding possibly polynomial zeros

• h/k where h is a factor of the constant term (term w/out variable) and k is a factor of the coefficient of the leading term

asymptotes of polynomials

• vertical: factor in numerator but not in denominator

• horizontal

  • quotient of num/denom (if both are the same degree)

  • 0 is degree of denom > num

  • if degree of num = degree of denom +1 → slant asymptote

transformations of 1/x

• use P(x)/D(x) = Q(x) + R(x)/D(x)

circle equations

• arc length: s = rθ

• angular speed: w = θ/t

• area of sector: A = ½ r²θ

• linear speed: v = s/t or v = rw

arc trig domain and range

• sin^-1 = d {-1, 1}, r {-pi/2, pi/2}

• cos^-1 = d{-1, 1}, r {0, pi}

• tan^-1 = d {-infinity, infinity}, r {-pi/2, pi/2}

SSA

• ambiguous case: if sin θ * b < a < b

Pythagorean trig identities

• cos² θ + sin² θ = 1

• cot² θ + 1 = csc² θ

• 1 + tan² θ = sec² θ

cofunction trig identities

• sin (pi/2 - θ) = cos θ

• cos (pi/2 - θ) = sin θ

• tan (pi/2 - θ) = cot θ

• csc (pi/2 - θ) = sec θ

• sec (pi/2 - θ) = csc θ

• cot (pi/2 - θ) = tan θ

odd/even trig identities

• sin( - θ) = - sin(θ) → csc ( - θ) = - csc (θ)

• cos ( - θ) = cos (θ) → sec ( - θ) = sec (θ)

• tan ( - θ) = - tan (θ) → cot ( - θ) = - cot (θ)

sum/difference trig identities

• sin(u ± v) = sin(u)cos(v) ± cos(u)sin(v)

• cos(u ± v) = cos(u)cos(v) ∓ sin(u)sin(v)

• tan(u ± v) = sin(u ± v)/cos(u ± v)

doubles trig identities

• sin 2x = 2sin x cos x

• cos 2x = cos² x - sin² x

• tan 2x = 2tan x/(1 - tan² x)

polar to rectangular

• x = r cos θ

• y = r sin θ

• r² = x² + y²

• tan θ = y/x

polar lines

• vertical: r = a sec θ

• horizontal: r = a csc θ

• origin: θ = …

polar circles

• r = a

• r = a sin θ

• r = a cos θ

roses

• r = a sin (nθ) or r = a cos (nθ)

• a = petal length

• n petals if n is odd, 2n petals if n is even

• petals start @ θ = 0 for cos

limacon

• r = a ± b sin θ or a ± b cos θ

• large petal: up or right if +, down or left if -

• min: a - b

• max: a + b

• shapes

  • inner loop: a/b < 1

  • cardioid: a/b = 1

  • dimple: 1 < a/b < 2

  • convex/ shield: a/b >= 2

Spirals

• spiral out: r = a θ

• spiral in: r = a/θ

lemniscates

• r² = a² cos 2θ or r² = a² sin 2θ

polar symmetry

• replace θ with - θ → symmetrical around the polar axis

• replace r with -r or θ with θ + pi → symmetrical around the pole

• replace θ with pi - θ → symmetrical around θ = pi /2

complex numbers

• modulus (|z|) = sqrt ( a² + b²)

• polar form - z = r(cos θ + i sin θ)

  • r = |z|

  • tan θ = b/a

• multiplication: z₁ z₂ = r₁ r₂ [cos (θ₁ + θ₂) + i sin (θ₁ + θ₂)]

• division: z₁ / z₂ = r₁ / r₂ [cos (θ₁ - θ₂) + i sin (θ₁ - θ₂)]

• De Moivre’s Theorem: zⁿ = rⁿ (cos nθ + i sin nθ)

• nth roots

  • z has n distinctive nth roots

  • w_k = r^1/n [cos ((θ + 2kpi)/n) + i sin ((θ + 2kpi)/n)]

  • k = 0, 1, 2, …, n-1

parametrics

• line

  • x = x₀ + at, y = y₀ + bt where m = b/a

• parabola

  • normal = (x = at², y = 2at) or (y = at², x = 2at)

  • motion - x = vcos θ t, y = -16t² + vsinθt + h

    • v = speed, θ = angle at launch, h = height at launch

• circle

  • x = h + rcost

  • y = k + rsint

• ellips

  • x = h + acost

  • y = k + bcost

vectors

• magnitude = |v| = sqrt (a² + b²)

• direction = tan θ = a₁ / a₂

• component form: v = < a₁, a₂ >

• i and j form: v = ai + bj

• unit vector: vector of magnitude 1, u = v/|v|

• horizontal + vertical components → < |v| cos θ, |v| sin θ >

• component of u along v (scalar): comp_vu = (u * v)/ |v|

• projection of u onto v (vector): proj_vu = [(u * v)/ |v|] * (v/|v|)

• Work = force * distance

• equation of a sphere w/ center (h, k, l) + radius r: (x - h)² + (y - k)² + (z - l)² = r²

parabola

• up/down: 4p(y - k) = (x - h)²

  • 4p > 0 opens up, 4p < 0 opens down

  • vertex = (h, k)

  • directrix = y = k - p

  • focus = (h, k + p)

• sideways: 4p(x - h) = (y - k)²

  • 4p > 0 opens right, 4p < 0 opens left

  • vertex = (h, k)

  • directrix = x = h - p

  • focus = (h + p, k)

• geo definition: all points are equidistant from the focus and directrix

• eccentricity = c/a = 1

ellipse

• major axis x: (x-h)²/a² + (y - k)²/b² = 1

• major axis y: (y - k)²/a² + (x - h)²/b² = 1

• major axis - longest, 2a

• a² = b² + c²

• eccentricity = e = c/a = 0 < e < 1

• geo definition: sum of distance from point to foci is constant

hyperbolas

• major axis x ( )( ): (x-h)²/a² - (y-k)²/b² = 1

  • asymptotes = +- b/a (x-h)

• major axis y: (y - k)²/a² - (x - h)²/b² = 1

  • asymptotes = +- a/b (x - h)

• c² = a² + b²

• geo definition: |difference| of distance to foci is constant

polar conics

• r = ed/(1 + e sinθ) → hill, d above

• r = ed/(1 - e sinθ) → valley, d below

• r = ed/(1 - e cosθ) → c, d to the left

• r = ed/(1 + e cosθ) → reverse c, d to the right

rotated conics

• cot 2θ = (A - C)/B

• discriminant: B² - 4AC

  • = 0 → parabola

  • < 0 → ellipse

  • > 0 → hyperbola

degenerate conics

• appears as a point, a line, or 2 intersecting lines

modelling

• compound interest: A(t) = P(1 + r/n)^nt

  • P = principal

  • r = interest rate/year

  • n = # of times interest is compounded per year

  • t = # of years

• continuously compounded interest: A(t) = Pe^rt

• logistic growth: P(t) = d/(1 + ke^-ct)

• doubling time: n(t) = n₀2^t/a

  • a = doubling time

• relative growth rate: n(t) = n₀e^rt

  • r = relative rate of growth (proportion)

• radioactive decay model: m(t) = m₀e^-rt

  • r = ln 2/ h where h = half-life

arithmetic sequence

• a_n = a + (n - 1)d

• partial sum: S_n = n((a + a_n)/2)

geometric sequence

• a_n = a r^{n - 1}

• partial sum: S_n = a (1-rⁿ / 1-r)

• sum of infinite series (only if |r| < 1): S = a/ (1-r)

row echelon form

• ax + by + cz = d

• ey + fz = g

• hz = i

reduced row echelon form

• x = a

• y = b

• z = c

minor

• M_ij = det. of matrix w/out row i and column j

cofactor

• A_ij = (-1)^{i + j} * M_ij

determinant of A

= a₁₁ A₁₁ + a₁₂A₁₂ + … + a_1nA_1n

ways to display data

• pie - good for percentages + fewer categories, bad at comparing data sets

• bar charts - shows frequency, good for data set comparison

mean/median/mode

• mean = average

  • always affected by outliers

• median: middle # or average of 2 middle #s

  • usually not affected by outliers

• mode: # that appears most often

Empirical Rule

• ~ 68% of the data is between μ - σ and μ + σ

• ~ 95% of the data is between μ - 2σ and μ + 2σ

• ~ 99.7% of the data is between μ - 3σ and μ + 3σ

• μ = mean

• σ = standard deviation

Five number summary

• min, Q1, median, Q3, max

  • Q1 = # in the middle of the min and median

  • Q3 = # in the middle of the median and max

• outliers (should be excluded): #s < Q1 - 1.5 * IQR or > Q3 + 1.5 * IQR

  • IQR = Q3 - Q1

subsets of an n-set

• set w/ n elements has 2ⁿ different subsets

distinguishable permutations

• when some combinations are the same

• n!/(a! b! c!…)

permutation

• P_{(n, r)} = n! / (n - r)!

• order matters

combination

C_(n,r) = n! / r! (n-r)!

union of events

• if looking for P(E or F)

• P (E or F) = P(E) + P(F) - P(E and F)

intersection of events

• P(F | E) = probability of F given E happened = P(F and E) / P(E)

Binomial probability

• P(r successes in n trials) = C_{(n, r)} p^r (1 - p)^{n - r}