Calculus Cheat Sheet
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1Taylor and Maclaurin Series
Taylor series of f at a:
Taylor and Maclaurin Series
( )
f
(a)
f (a)
f (x) =
(x
a) = f (a) +
(x
a)
n!
1!
=0
Per-Olof Persson
f (a)
f (a)
2
3
+
(x
a)
+
(x
a)
+
2!
3!
Maclaurin series if a = 0:
Department of Mathematics
University of California, Berkeley
( )
f
(0)
f (0)
f (x) =
x = f (a) +
x
Math 1B Calculus
n!
1!
=0
f (0)
f (0)
2
3
+
x
+
x
+
2!
3!
Remainders
Common Maclaurin Series
Theorem
1
2
3
=
x = 1 + x + x
+ x
+
R = 1
If f (x) = T (x) + R (x), where T is nth partial sum of the
1
x
=0
Taylor polynomial of f at a, and
2
3
x
x
x
x
e =
= 1 +
+
+
+
R =
n!
1!
2!
3!
lim R (x) = 0
=0
2 +1
3
5
x
x
x
sin x =
( 1)
= x
+
R =
for x
a < R, then f = sum of its Taylor series on x
a < R
(2n + 1)!
3!
5!
=0
2
2
4
x
x
x
Theorem (Taylor’s Inequality)
cos x =
( 1)
= 1
+
R =
(2n)!
2!
4!
=0
( +1)
If f
(x)
M for x
a
d, then the remainder R (x) of
2 +1
3
5
x
x
x
1
the Taylor series satisfies
tan
x =
( 1)
= x
+
R = 1
2n + 1
3
5
=0
M
+1
k
k(k
1)
R (x)
x
a
for x
a
d
2
(1 + x) =
x = 1 + kx +
x
+
R = 1
(n + 1)!
n
2!
=0
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