the equation of a line with slope m passing through (a,b) is
y-b=m(x-a)
13
New cards
the equation of a line with slope m and y-intercept b is
y=mx+b
14
New cards
the distance between 2 points (a,b) and (c,d) is
√((a-c)²+(b-d)²)
15
New cards
the restriction on the domain of (a/b) is
b≠0
16
New cards
the restriction on the domain of √(x) is
x≥0
17
New cards
the restriction on the domain of ln(x) is
x>0
18
New cards
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)
19
New cards
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
New cards
sin(2x)=
2sinxcosx
21
New cards
sinxcosx=
(1/2)sin(2x)
22
New cards
cos(2x)=
1-2sin²x
2cos²x-1
cos²x-sin²x
23
New cards
the pythagorean identities are
sin²x+cos²x=1
tan²x+1=sec²x
cot²x+1=csc²x
24
New cards
the quotient identities are
tanx=(sinx/cosx)
cotx=(cosx/sinx)
25
New cards
the reciprocal identities are
cscx=(1/sinx)
secx=(1/cosx)
cotx=(1/tanx)
26
New cards
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
27
New cards
in order to approach a limit from the right of a point c, write
lim
x→c⁺
28
New cards
in order to approach a limit from the left of a point c, write
lim
x→c⁻
29
New cards
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
30
New cards
lim 1/(x-a) =
x→a⁺
\+∞
31
New cards
lim 1/(x-a) =
x→a⁻
\-∞
32
New cards
(a constant)/(a very small number)
\+∞ or -∞
33
New cards
vertical asymptotes occur when
the denominator equals 0, but the numerator is not equal to 0
34
New cards
holes in a graph occur when
both the numerator and denominator are equal to 0 and there is not a vertical asymptote
35
New cards
lim (sinx/x) =
x→0
1
36
New cards
lim (sinx/x) =
x→∞
0
37
New cards
lim (1-cosx)/x =
x→0
0
38
New cards
lim {\[f(x)\]²}
x→c
{lim \[f(x)\]}²
x→c
39
New cards
lim {f\[g(x)\]}
x→c
f {lim \[g(x)\]}
x→c
40
New cards
0 \* (nonzero/0) is
not equal to zero; simplify to find the limit
41
New cards
horizontal asymptotes occur when
lim f(x) = L ; the asymptote is y=L
x→±∞
42
New cards
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
New cards
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
44
New cards
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.
45
New cards
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