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47 Terms

1
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(a/b)>0 when
signs of a and b are the same
2
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|a|=b when
a=b or -a=b
3
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|a|>b when
a>b or a
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|a|
\-b
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|a| can be rewritten as
a, x≥0

\-a, x
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√(x²)
|x| or ±x
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f(x)= |x-a|
x-a, x≥a

\-(x-a), x
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a³+b³
(a+b)(a²-ab+b²)
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a³-b³
(a-b)(a²+ab+b²)
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the slopes of 2 parallel lines are
equal
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the slopes of 2 normal lines (perpendicular) are
negative reciprocals
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the equation of a line with slope m passing through (a,b) is
y-b=m(x-a)
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the equation of a line with slope m and y-intercept b is
y=mx+b
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the distance between 2 points (a,b) and (c,d) is
√((a-c)²+(b-d)²)
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the restriction on the domain of (a/b) is
b≠0
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the restriction on the domain of √(x) is
x≥0
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the restriction on the domain of ln(x) is
x>0
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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)
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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)
20
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sin(2x)=
2sinxcosx
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sinxcosx=
(1/2)sin(2x)
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cos(2x)=
1-2sin²x

2cos²x-1

cos²x-sin²x
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the pythagorean identities are
sin²x+cos²x=1

tan²x+1=sec²x

cot²x+1=csc²x
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the quotient identities are
tanx=(sinx/cosx)

cotx=(cosx/sinx)
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the reciprocal identities are
cscx=(1/sinx)

secx=(1/cosx)

cotx=(1/tanx)
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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
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in order to approach a limit from the right of a point c, write
lim

x→c⁺
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in order to approach a limit from the left of a point c, write
lim

x→c⁻
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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
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lim 1/(x-a) =

x→a⁺
\+∞
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lim 1/(x-a) =

x→a⁻
\-∞
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(a constant)/(a very small number)
\+∞ or -∞
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vertical asymptotes occur when
the denominator equals 0, but the numerator is not equal to 0
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holes in a graph occur when
both the numerator and denominator are equal to 0 and there is not a vertical asymptote
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lim (sinx/x) =

x→0
1
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lim (sinx/x) =

x→∞
0
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lim (1-cosx)/x =

x→0
0
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lim {\[f(x)\]²}

x→c
{lim \[f(x)\]}²

x→c
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lim {f\[g(x)\]}

x→c
f {lim \[g(x)\]}

x→c
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0 \* (nonzero/0) is
not equal to zero; simplify to find the limit
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horizontal asymptotes occur when

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

x→±∞

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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

\
43
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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
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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.
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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
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indeterminate forms
0/0

∞/∞

∞-∞ or (-∞)-(-∞) or ∞+(-∞);

1^∞ or 1^-∞

0^0

∞^0 or (-∞)^0

0\*∞ or 0\*(-∞)
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not indetererminate
∞+∞=∞

(-∞)+(-∞)=-∞

0/∞ =0

a\*∞=∞

a/∞ = 0

∞/0 \[check vertical asymptotes\]