PreCalculus Formulas Sequences and Series: Binomial Theorem
⎛ n⎞ ( a + b) = ∑ ⎜ ⎟ a n− k b k k =0 ⎝ k ⎠ n
n
Find the rth term
⎛ n ⎞ n−( r −1) r −1 b ⎜ r − 1⎟ a ⎝ ⎠ Functions: To find the inverse function: 1. Set function = y 2. Interchange the variables 3. Solve for y
Arithmetic Last Term
Geometric Last Term
an = a1 + (n − 1)d
an = a1r
Geometric Partial Sum
Arithmetic Partial Sum
⎛1− rn ⎞ S n = a1 ⎜ ⎟ ⎝ 1− r ⎠
⎛a +a ⎞ Sn = n ⎜ 1 n ⎟ ⎝ 2 ⎠ f -1 (x)
n −1
Composition of functions:
( f g )( x) = f ( g ( x)) ( g f )( x) = g ( f ( x)) (f
[r (cosθ + i sin θ )]n = r n (cos niθ + i sin niθ )
r = a 2 + b2 b θ = arctan a
x = r cosθ y = r sin θ
a + bi i = −1 i 2 = −1
( r ,θ ) → ( x , y )
Determinants:
3 5 4 3
= 3i3 − 5i4
Use your calculator for 3x3 determinants.
−1
f )( x) = x
Algebra of functions: ( f + g )( x) = f ( x) + g ( x) ; ( f − g )( x) = f ( x) − g ( x)
( f i g )( x) = f ( x)i g ( x) ; ( f / g )( x) = f ( x) / g ( x), g ( x) ≠ 0 Domains:: D( f ( x)) ∩ D( g ( x)) Domain (usable x’s) Asymptotes: (vertical) Watch for problems with Check to see if the zero denominators and with denominator could ever be negatives under radicals. zero. x f ( x) = 2 Range (y’s used) x + x−6 Difference Quotient Vertical asymptotes at f ( x + h) − f ( x ) x = -3 and x = 2 h terms not containing a mult. of h will be eliminated.
Complex and Polars: DeMoivre’s Theorem:
Asymptotes: (horizontal) x+3 1. f ( x) = 2 x −2 top power < bottom power means y = 0 (z-axis) 4 x2 − 5 2. f ( x) = 2 3x + 4 x + 6 top power = bottom power means y = 4/3 (coefficients) x3 3. f ( x) = None! x+4 top power > bottom power
Also apply Cramer’s rule to 3 equations with 3 unknowns.
Trig:
Reference Triangles:
o a o sin θ = ; cos θ = ; tan θ = h h a h h a csc θ = ; secθ = ; cotθ = o a o
BowTie
Analytic Geometry: Circle
Ellipse
( x − h) 2 + ( y − k ) 2 = r 2 Remember “completing the square” process for all conics.
Parabola
( x − h) = 4a ( y − k ) ( y − k ) 2 = 4a ( x − h) 2
Polynomials: Remainder Theorem: Substitute into the expression to find the remainder. [(x + 3) substitutes -3]
Descartes’ Rule of Signs 1. Maximum possible # of positive roots → number of sign changes in f (x) 2. Maximum possible # of negative roots → number of sign changes in f (-x)
vertex to focus = a, length to directrix = a, latus rectum length from focus = 2a
( x − h) 2 ( y − k ) 2 + =1 a2 b2 larger denominator → major axis and smaller denominator → minor axis
Hyperbola
( x − h) ( y − k) − =1 2 a b2 2
2
Latus length from focus b2/a
c → focus length where major length is hypotenuse of right triangle. Latus rectum lengths from focus are b2/a a→transverse axis b→conjugate axis c→focus where c is the hypotenuse. asymptotes needed
Synthetic Division Mantra: “Bring down, multiply and add, multiply and add…”
Depress equation
[when dividing by (x - 5), use +5 for synthetic division]
(also use calculator to examine roots)
Analysis of Roots P N C Chart
Upper bounds: All values in chart are + Lower bounds: Values alternate signs No remainder: Root
* all rows add to the degree * complex roots come in conjugate pairs * product of roots - sign of constant (same if degree even, opposite if degree odd) * decrease P or N entries by 2
−b ± b 2 − 4ac x= 2a
Sum of roots is the coefficient of second term with sign changed.
Eccentricity: e = 0 circle 0 < e < 1 ellipse e = 1 parabola e > 1 hyperbola
Induction:
Find P(1): Assume P(k) is true: Show P(k+1) is true:
Rate of Growth/Decay: y = y0 ekt y = end result, y0 = start amount, Be sure to find the value of k first.
Far-left/Far-right Behavior of a Polynomial The leading term (anxn ) of the polynomial determines the far-left/far-right behavior of the graph according to the following chart. (“Parity” of n Æ whether n is odd or even.) LEFT-HAND BEHAVIOR n is even n is odd
anxn
(same as right)
an > 0 RIGHTHAND always positive BEHAVIOR or
(opposite right)
negative x < 0 positive x > 0
Leading Coefficient Test
Product of roots is the constant term (sign changed if odd degree, unchanged if even degree).
