Formula sheet
Some important derivatives
f (x)
f 0 (x)
remark
c2R
0
xm
m xm 1
m2R
x
|x|
|x| = sgn x
sin x
cos x
cos x
sin x
tan x
sec2 x
sec x
sec x tan x
cot x
csc2 x
csc x
csc x cot x
ex
ex
1
ln x
x>0
x
1
p
arcsin x
1 x2
p 1
arccos x
1 x2
1
arctan x
1+x2
tan x + c
sec2 x
cot x + c
csc2 x
sec x + c
sec x tan x
csc x + c
csc x cot x
Some important
R indefinite integrals (with c 2 R)
f (x)
f (x) dx
remark
R
R
1
x+c
dx = 1 dx = x + c
xm+1
xm
m 2 R \ { 1}
m+1 + c
p
2 3/2
x
+c
x>0
3x
1
ln
|x|
+
c
x 6= 0
x
ex
ex + c
sin x
cos x + c
cos x
sin x + c
sec2 x
tan x + c
x 6= (2 k + 1) ⇡2
csc2 x
cot x + c
x 6= k ⇡
sec x tan x
sec x + c
x 6= (2 k + 1) ⇡2
csc x cot x
csc x + c
x 6= k ⇡
p 1
arcsin
x
+
c
x
2 ( 1, 1)
1 x2
1
p
arccos x + c
x 2 ( 1, 1)
1 x2
1
arctan x + c
1+x2
Some important trigonometric identities
sin2 x + cos2 x = 1
1 + tan2 x = sec2 x
1 + cot2 x = csc2 x
sin(2 x) = 2 sin x cos x
cos(2 x) = 2 cos2 x
1=1
2 sin2 x
sin2 x =
1
cos(2 x)
2
1 + cos(2 x)
2
sin(x ± y) = sin x cos y ± cos x sin y
cos2 x =
Tangent line & plane
Definition (nonvertical tangent line): Let f : R ! R be continuous and let f 0 (a) exists. Then y = f 0 (a)(x
a) + f (a) is an equation of the nonvertical line tangent to the graph of f at point (a, f (a)).
@f
Definition (tangent plane): Let f : R2 ! R be continuous and let @f
@x (a, b) and @y (a, b) exist. Then z =
@f
@x (a, b)(x
a) + @f
@y (a, b)(y
b) + f (a, b) is an equation of the plane tangent to the graph of f at point (a, b, f (a, b)).
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Convergence tests for positive series
• The integral test: Suppose that an = f (n), where f is positive, continuous, and nonincreasing on an
1
1
R
P
interval [N, 1) for some positive integer N. Then
an and f (t) dt either both converge or both diverge
n=1
to infinity.
N
• A comparison test: Let {an } and {bn } be positive infinite sequences for which there exists a positive
constant K such that, ultimately, 0  an  K bn .
P
P1
(a) If the series 1
n=1 bn converges, then so does the series
n=1 an
P1
P
(b) If the series n=1 an diverges to infinity, then so does the series 1
n=1 bn
• A limit comparison test: Let {an } and {bn } be positive infinite sequences and let lim abnn = L, where L
n!1
is a nonnegative finite number or +1.
P
P1
(a) If L < 1 and 1
n=1 bn converges, then
n=1 an also converges
P1
P
(b) If L > 0 and n=1 bn diverges to infinity, then 1
n=1 an also diverges to infinity
• The ratio test: Suppose that an > 0 (ultimately) and that ⇢ = lim an+1
is a nonnegative finite number
n!1 an
or +1.
P
(a) If 0  ⇢ < 1, then 1
n=1 an converges
P
(b) If 1 < ⇢  1, then lim an = 1 and 1
n=1 an diverges to infinity
n!1
(c) If ⇢ = 1, this test gives no information
• The root test: Suppose that an > 0 (ultimately) and that L = lim
n!1
(a) If L < 1 then
(b) If L > 1 the
P1
n=1 an converges.
P
series 1
n=1 an diverges
p
1
n
|an | = lim |an | n
n!1
to infinity.
(c) If L = 1, this test gives no information.
The alternating series test
Suppose {an } is a sequence whose terms satisfy, for some positive integer N, the following three conditions:
(a) an an+1 < 0 for n
(b) |an+1 |  |an | for n
N,
N,
(c) lim an = 0 (this rule can be also replaced by lim |an | = 0),
n!1
n!1
that is, the terms are ultimately alternating in sign and decreasing in size, and the sequence has limit zero. Then
1
P
the series
an converges.
n=1
If the alternating series
convergent.
If the alternating series
gent.
1
P
an converges and
1
P
n=1
n=1
1
P
1
P
n=1
an converges and
n=1
|an | converges as well, we call the series
|an | diverges, we call the series
1
P
1
P
an absolutely
n=1
an conditionally conver-
n=1
Integrating factor
dy
µ(x) with µ(x) =
For
R di↵erential equation of the form dx + p(x) y = q(x) the integrating factor is defined as e
p(x) dx (where we do not include the constant of integration).
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