Calculus I
The Learning Center
http://www.rose-hulman.edu/lc
Helpful Calculus I Information
Basic Differentiation Rules
1.
2.
3.
4.
5.
d
cu   cu 
dx
d
uv  uv  vu 
dx
d
c  0
dx
d
x  1
dx
12.
13.
14.
15.

d
ln u   u
dx
u
16.
d
tan u   (sec 2 u)u 
dx
d
sec u   (sec u tan u)u 
8.
dx

d
arcsin u   u 2
9.
dx
1 u

d
arctan u   u 2
10.
dx
1 u
d
u
arc sec u  
11.
dx
| u | u 2 1
7.
18.
19.
20.
21.
22.
Vectors
|| v ||: (v1 ) 2  (v2 ) 2
Length of a scalar multiple:
|| cv ||:| c |  || v ||
Unit vector in the direction of v:
 
 
d
cos u   (sin u )u 
17. dx
d
6.
sin u   cos u u 
dx
Length of a vector:
d
u  v  u   v
dx
d  u  vu   uv 

dx  v 
v2
d n
u  nu n 1u 
dx
d
| u |  u (u ), u  0
dx
|u|
d u
e  eu u
dx
d
cot u   (csc 2 u)u 
dx
d
csc u   (csc u cot u)u 
dx

d
arccos u    u 2
dx
1 u

d
arc cot u    u 2
dx
1 u
d
arc sec u    u2
dx
| u | u 1
u :
1
v
|| v ||
Parallel vectors:
Two nonzero vectors u and v are parallel if there is some scalar c such that u=cv
Dot Product:
u  v : u1v1  u 2 v2
Projection using the Dot Product:
 u v 
projv u :  2   v
 || v || 
Angle between two vectors:
uv
cos   :
|| u ||  || v ||
Limits
Strategies for finding limits:
1. Learn to recognize which limits can be evaluated by direct substitution.
2. If the limit of f(x) as x approaches c cannot be evaluated by direct substitution, try to find
a function of g that agrees with f for all x other than x=c (Choose g such that the limit of
g(x) can be evaluated by direct substitution).
3. Conclude through analysis that limf(x)=limg(x)=g(c)
4. Use a graph or table to reinforce your conclusion.
Basic limits:
lim b : b
lim x : c
lim x n : c n
x
 c
x
 c
x
 c
Limit of a polynomial function:
lim p( x) : p(c)
x
 c
Limit of a rational function:
lim r ( x) : r (c)
x
 c
where, r ( x) :
p( x)
q( x)
Calculus II
The Learning Center
http://www.rose-hulman.edu/lc
Helpful Calculus II Information
  f (u)  g (u)du   f (u)du   g (u)du
 kf (u)du  k  f (u)du
11.
2.
 du  u  C
u n 1
12,  u du 
 C , n  1
n 1
3.
13.
7.
 u  ln | u | C
 sin udu   cos u  C
 tan udu   ln | cos u | C
 sec udu  ln | sec u  tan u | C
 sec udu  tan u  C
8.
 sec u tan udu  sec u  C
18.  csc u cot udu   csc u  C
9.

1.
4.
5.
6.
n
du
2
10. 
du
a2  u2
du
u u2  a2
 arcsin

u
C
a
1
|u|
arc sec
C
a
a
14.
15.
16.
17.
19.
 e du  e  C
 cos udu  sin u  C
 cot udu  ln | sin u | C
 csc udu   ln | csc u  cot u | C
 csc udu   cot u  C
u
u
2
a
2
du
1
u
 arctan  C
2
a
a
u
Calculus III
The Learning Center
http://www.rose-hulman.edu/lc
Helpful Calculus III Information
a  b  a b cos 
f 
K
f
f
f
i
j k
x
y
z
| x (t ) y (t )  x (t ) y (t ) |
( x(t ))
2
 ( y (t ))

2 3/ 2
 a b 
projba=  2  b
 b 


axb  a b sin 
DU f  f  u
f  g
2
d 
A=  v T  K v N
 dt 
V   dV
M    ( x, y, z )dV
 x  r cos 

 y  r sin 
dA  rdrd

 x   sin  cos 
 y   sin  sin 


 z   cos 
dV   2 sin ddd
H
K
A   G dA