Cheat Sheet – Exam 3
Derivatives
1.
2.
3.
4.
5.
6.
d
(tan x) sec 2 x
dx
d
(cot x)
csc 2 x
dx
d
(sec x) sec x tan x
dx
d
(csc x)
csc x cot x
dx
d
1
(sin 1 x)
dx
1 x2
d
1
(cos 1 x)
dx
1 x2
14.
a2 x2
1
dx
2
a x2
1
15.
x x
2
a
2
1
tan
a
dx
cos2 x sin 2 x
27. sin A cos B 12 [sin( A B) sin( A B )]
[cos( A B) cos( A B)]
[cos( A B) cos( A B)]
Right Angle Trigonometry
30. sin
opp
hyp
csc
hyp
opp
31. cos
adj
hyp
sec
hyp
adj
32. tan
opp
adj
cot
adj
opp
Half-Angle Formulas
C
33. sin 2 x
1
2
(1 cos 2 x)
34. cos 2 x
1
2
(1 cos 2 x)
0
0
π/6
sin x
cos x
1
3
tan x
0
1
2
2
x
a
C
ln x dx
17.
tan x dx ln | sec x | C
18.
sec x dx ln | sec x tan x | C
19.
cot x dx
x ln x x C
π/4
π/3
2
3
2
1
1
3
16.
ln | csc x | C
1
2
29. sin A sin B
x
a
1
1
2
28. cos A cos B
C
1
sec
a
ln | sec x tan x |
C
2
26. cos 2 x
x
a
1
sec x tan x
2
sec2 x
24. 1 cot 2 x csc2 x
25. sin 2 x 2sin x cos x
Integrals
1
sec3 x dx
23. 1 tan 2 x
d
(sinh x) cosh x
11.
dx
d
(cosh x) sinh x
12.
dx
dx sin
21.
22. sin 2 x cos2 x 1
d
1
(tan 1 x)
dx
1 x2
d
1
(cot 1 x)
8.
dx
1 x2
d
1
9.
(sec 1 x)
dx
x x2 1
d
1
10.
(csc 1 x)
dx
x x2 1
1
csc x dx ln | csc x cot x | C
Trig Identities
7.
13.
20.
2
1
2
π/2
1
2
2
3
0
Undef.
7.1 Integration by Parts
udv uv
vdu
L – Logs
I – Inverse Trig
A – Algebraic
T – Trig
E – Exponential
7.2
7.4 Partial Fractions
Higher on the list = u
Case 2:
p( x)
( x 2)( x 4) 2
Case 3:
p( x)
( x 2)( x 2 4)
Lower on the list = dv
Mn
x[ f ( x1 )
x
3 [ f ( x0 )
Sn
a) If sec x has an even power, pull a sec x aside
and convert the rest to tan x. Substitute u =
tan x
b) If tan x has an odd power, pull a
tan x sec x aside and convert the rest to
sec x. Substitute u = sec x
c) If sec x has an odd power and tan x has an even
power, convert all tan x into sec x.
x 2
f ( x2 ) ...
4 f ( x1 )
2 f ( x2 )
...
f ( xn )]
f ( xn 1 )
2 f ( xn
2
)
f ( xn )]
4 f ( xn 1 )
K (b a )3
where | f ''( x) | K
12n 2
| EM |
K (b a )3
where | f ''( x) | K
24n 2
| ES |
K (b a )5
where | f (4) ( x) | K
4
180n
Type I:
a
f ( x)dx
f ( x)dx
f ( x)dx
lim
t
b
lim
a
lim
t
x
a
f ( x)dx
f ( x)dx
t
t
t
f ( x)dx lim
t
t
a
f ( x)dx
Type II: If f(x) is discontinuous at a.
b
a
f ( x)dx
lim
t
a
b
t
f ( x)dx
If f(x) is discontinuous at b.
b
a
f ( x)dx
lim
t
b
t
a
f ( x)dx
If f(x) is discontinuous at c for some a
7.3 Trig Substitution
b
a2
x2
x
a sin
1 sin 2
cos 2
a2
x2
x
a tan
1 tan 2
sec2
x2 a2
x
a sec
sec2
1 tan 2
f ( x n )]
7.8 Improper Integrals
csc x cot x dx
a) If csc x has an even power, pull a csc x aside and
convert the rest to cot x. Substitute u = cot x
b) If cot x has an odd power, pull a
cot x csc x aside and convert the rest to
csc x. Substitute u = csc x
c) If csc x has an odd power and cot x has an even
power, convert all cot x into csc x.
Bx C
x2 4
| ET |
m
2
A
Error Bounds:
b
n
x 4
Pattern for Simpson’s Rule Coefficients:
1,4,2,4,2,4,…,4,2,4,2,4,1
secn x tan m x dx
2
x 2
C
( x 4) 2
x
Tn
a) If sin x has an odd power, pull a sin x aside and
convert the rest to cos x. Substitute u = cos x
b) If cos x has an odd power, pull a cos x aside and
convert the rest to sin x. Substitute u = sin x
c) If both have even powers, use the half-angle
formulas.
B
b a
n
x
2 f ( x1 ) ... 2 f ( xn 1 )
2 [ f ( x0 )
7.7
sin n x cos m x dx
A
a
f ( x)dx
lim
x
c
t
a
f ( x)dx lim
Don’t Forget +C
t
c
c b
b
t
f ( x)dx
Parametric Equations x
f (t ), y
g (t )
Eliminate the parameter: Solve for t and
substitute.
dy
dy
dx
Slope of the tangent line: m
dt
dx
dt
Parabola
Center = Vertex = (h, k)
The equation of the parabola with focal point
(h, k p) and directrix y k p is
y
ax 2
1
4p
a
dy
2
Concavity:
dt
d
dt
d y
dx 2
dx
dx
The equation of the parabola with focal point
(h p, k ) and directrix x h p is
dt
dt
Equation of the tangent line: y
dy
dt
Horizontal when
Vertical when
dx
dt
y1
m( x x1 )
Center = (h, k) = midpoint between foci.
Arc Length: L
dx
The equation of the ellipse with focal points
(h c, k ) and (h c, k ) , with major axis length of
2a is
2
2
dy
dt
Polar Equations r
dt
r cos
r2
y
r sin
tan
x2
dr
d
dr
d
sin
cos
r sin
r cos
r sin
y1
m( x x1 )
0
0
Undefined when both are 0.
Area inside the curve:
1
2
Area between curves:
1
2
dr
d
Arc Length: L
Function y
Arc Length: L
r 2d
r12 r22 d
2
2
r d
a
1
a2 c2
( x h) 2
a2
( y k )2
b2
a2
1
b2 c2
Hyperbola
Center = (h, k) = midpoint between foci.
The equation of the hyperbola with
focal points (h c, k ) and (h c, k ) and
vertices (h a, k ) and (h a, k ) is
( x h) 2
a2
( y k )2
b2
Asymptotes: y k
a 2 b2
1
b
a
c2
( x h)
The equation of the hyperbola with
focal points (h, k c) and (h, k c) and
f ( x)
b
b2
1
2b is
sin
cos
r cos
( y k )2
b2
The equation of the ellipse with focal points
(h, k c) and (h, k c) , with major axis length of
y2
Equation of the tangent line: y
Vertical when
dt
y
x
dy
Slope of tangent line:
dx
dr
d
( x h) 2
a2
f( )
x
dr
d
1
4p
a
Ellipse
0
g (t ) f '(t )dt
Horizontal when
ay 2
P = Focus - Center
0
Undefined when both are 0.
Area under the curve (left to right):
ydx
x
dy
dx
2
vertices (h, k b) and (h, k b) is
dx
( x h) 2 ( y k ) 2
1
a2
b2
b
h)
Asymptotes: y k
a (x
a 2 b2
c2
11.1 Sequences
11.3 Error bounds for Integral Test
If f is continuous, positive and decreasing,
then the error is bounded by:
Finding the general term:
Alternating: ( 1) n or ( 1) n
1
Common difference of a: an b
Common ratio of b: abn
Use n 1 to find missing variable.
n 1
f ( x)dx
Sn
n 1
Rn
f ( x)dx and
n
f ( x)dx
S
Sn
n
f ( x)dx
11.5 Error bounds for Alternating Series Test
Limit Laws:
lim(an bn ) lim(an ) lim(bn )
lim(an bn ) lim(an ) lim(bn )
( 1)n bn is an alternating series, then
If
n 1
| Rn | bn 1 and S n
lim(can ) c lim(an )
S
S n 1 or S n
1
S
Sn
lim(anbn ) lim(an ) lim(bn )
lim(an / bn ) lim(an ) / lim(bn )
p
lim[(an ) ] [lim(an )]
p( x)
lim
n
q( x)
Absolute Convergence
p
0
if deg(p) < deg(q)
c
if deg(p) > deg(q)
if deg(p) = deg(q)
where c is the ratio of the coefficients of the
highest degree terms in p and q.
an
n 1
Convergent
Divergent
L’Hospital’s Rule:
If
f ( x)
g ( x)
lim
x
f ( x)
g ( x)
0
f ( x)
or
0
g ( x)
lim
x
, then
f '( x)
g '( x)
11.2 Series
If I ask for what a series converges to, it must
be either geometric or telescoping.
| an |
n 1
Absolutely
Conditionally
Do Not Apply Series Convergence Tests
to Sequences!!!