ch 1 terms

(a/b)>0 when

(a/b)>0 when

signs of a and b are the same

|a|=b when

a=b or -a=b

|a|>b when

a>b or a<-b

|a|<b when

-b<a<b

|a| can be rewritten as

a, x≥0

-a, x<0

√(x²)

|x| or ±x

f(x)= |x-a|

x-a, x≥a

-(x-a), x<a

a³+b³

(a+b)(a²-ab+b²)

a³-b³

(a-b)(a²+ab+b²)

the slopes of 2 parallel lines are

equal

the slopes of 2 normal lines (perpendicular) are

negative reciprocals

the equation of a line with slope m passing through (a,b) is

y-b=m(x-a)

the equation of a line with slope m and y-intercept b is

y=mx+b

the distance between 2 points (a,b) and (c,d) is

√((a-c)²+(b-d)²)

the restriction on the domain of (a/b) is

b≠0

the restriction on the domain of √(x) is

x≥0

the restriction on the domain of ln(x) is

x>0

a function is even if

symmetric to y-axis, or when the point (a,b) is on the graph, so is the point (-a,b), or f(-x)=f(x)

a function is odd if

symmetric to origin, or when the point (a,b) is on the graph, so is the point (-a,-b), or f(-x)=-f(x)

sin(2x)=

2sinxcosx

sinxcosx=

(1/2)sin(2x)

cos(2x)=

1-2sin²x

2cos²x-1

cos²x-sin²x

the pythagorean identities are

sin²x+cos²x=1

tan²x+1=sec²x

cot²x+1=csc²x

the quotient identities are

tanx=(sinx/cosx)

cotx=(cosx/sinx)

the reciprocal identities are

cscx=(1/sinx)

secx=(1/cosx)

cotx=(1/tanx)

when substitution yields (0/0) in a limit, you should try

factoring, expanding, common denominator, multiply by the conjugate, one of the 2 known trig identities, L’Hopital’s rule

in order to approach a limit from the right of a point c, write

lim

x→c⁺

in order to approach a limit from the left of a point c, write

lim

x→c⁻

limits fail to exist when

the function approaches a different value from the left and right, the function is unbounded (approaches +∞ or -∞), or the function oscillates

lim 1/(x-a) =

x→a⁺

+∞

lim 1/(x-a) =

x→a⁻

-∞

(a constant)/(a very small number)

+∞ or -∞

vertical asymptotes occur when

the denominator equals 0, but the numerator is not equal to 0

holes in a graph occur when

both the numerator and denominator are equal to 0 and there is not a vertical asymptote

lim (sinx/x) =

x→0

1

lim (sinx/x) =

x→∞

0

lim (1-cosx)/x =

x→0

0

lim {[f(x)]²}

x→c

{lim [f(x)]}²

x→c

lim {f[g(x)]}

x→c

f {lim [g(x)]}

x→c

0 * (nonzero/0) is

not equal to zero; simplify to find the limit

horizontal asymptotes occur when

lim f(x) = L ; the asymptote is y=L

x→±∞

If f(x)= P(x)/Q(x), with P(x) and Q(x) as polynomial functions, the possible asymptotes are

1. HA: y=0, if the degree of numerator is less than the degree of denominator

2. HA: y=ratio of leading coefficients, if the numerator and denominator have the same degree

3. OA: y=quotient after long division

the 3 conditions for continuity are

1. lim f(x) = lim f(x)

   x→c⁻ x→c⁺

2. f(c) is defined

3. lim f(x) = f(c)

   x→c

the Squeeze Theorem for a limit says that

if f(x)≤g(x)≤h(x), and limx→c f(x)= L= limx→c h(x), then limx→c g(x)=L.

the Intermediate Value Theorem states

If f(x) is continuous on the closed interval [a,b] and k is between f(a) and f(b), then there exists at least one number c in [a,b] such that f(c)=k

indeterminate forms

0/0

∞/∞

∞-∞ or (-∞)-(-∞) or ∞+(-∞);

1^∞ or 1^-∞

0^0

∞^0 or (-∞)^0

0*∞ or 0*(-∞)

not indetererminate

∞+∞=∞

(-∞)+(-∞)=-∞

0/∞ =0

a*∞=∞

a/∞ = 0

∞/0 [check vertical asymptotes]