CSSS 505
Calculus Summary Formulas
Differentiation Formulas
d n
( x ) = nx n −1
dx
d
2.
( fg ) = fg ′ + gf ′
dx
d f
gf ′ − fg ′
3.
( )=
dx g
g2
d
4.
f ( g ( x)) = f ′( g ( x)) g ′( x)
dx
d
(sin x) = cos x
5.
dx
d
(cos x) = − sin x
6.
dx
d
7.
(tan x) = sec 2 x
dx
d
8.
(cot x) = − csc 2 x
dx
d
(sec x) = sec x tan x
9.
dx
d
(csc x) = − csc x cot x
10.
dx
d x
(e ) = e x
11.
dx
d x
12.
(a ) = a x ln a
dx
d
1
(ln x) =
13.
dx
x
1
d
14.
( Arc sin x) =
dx
1− x2
1.
d
1
( Arc tan x) =
dx
1+ x2
d
1
16.
( Arc sec x) =
dx
| x | x2 −1
15.
17.
dy dy du
Chain Rule
=
×
dx dx dx
Trigonometric Formulas
1.
2.
3.
4.
5.
6.
7.
8.
9.
sin 2 θ + cos 2 θ = 1
1 + tan 2 θ = sec 2 θ
1 + cot 2 θ = csc 2 θ
sin(−θ ) = − sin θ
cos(−θ ) = cosθ
tan(−θ ) = − tan θ
sin( A + B ) = sin A cos B + sin B cos A
sin( A − B) = sin A cos B − sin B cos A
cos( A + B) = cos A cos B − sin A sin B
10. cos( A − B) = cos A cos B + sin A sin B
11. sin 2θ = 2 sin θ cos θ
sin θ
1
=
cosθ cot θ
cosθ
1
14. cot θ =
=
sin θ tan θ
1
15. secθ =
cosθ
1
16. cscθ =
sin θ
13. tan θ =
17. cos(
18. sin(
12. cos 2θ = cos θ − sin θ = 2 cos θ − 1 = 1 − 2 sin θ
2
2
2
2
π
2
π
2
− θ ) = sin θ
− θ ) = cosθ
Integration Formulas
Definition of a Improper Integral
b
∫ f ( x) dx is an improper integral if
a
1.
2.
3.
f becomes infinite at one or more points of the interval of integration, or
one or both of the limits of integration is infinite, or
both (1) and (2) hold.
1.
∫ a dx = ax + C
2.
n
∫ x dx =
1
x n +1
+ C , n ≠ −1
n +1
∫ x dx = ln x + C
4. ∫ e dx = e + C
3.
x
5.
6.
x
ax
+C
ln a
∫ ln x dx = x ln x − x + C
x
∫ a dx =
∫ sin x dx = − cos x + C
8. ∫ cos x dx = sin x + C
9. ∫ tan x dx = ln sec x + C or − ln cos x + C
7.
∫ cot x dx = ln sin x + C
11. ∫ sec x dx = ln sec x + tan x + C
10.
∫ csc x dx = ln csc x − cot x + C
13. ∫ sec x d x = tan x + C
14. ∫ sec x tan x dx = sec x + C
15. ∫ csc x dx = − cot x + C
16. ∫ csc x cot x dx = − csc x + C
17. ∫ tan x dx = tan x − x + C
12.
2
2
2
dx
1
 x
18.
∫ a + x = a Arc tan a  + C
19.
∫ a − x = Arc sin a  + C
20.
2
2
 x
dx
2
2
dx
∫ x x −a
2
2
=
x
1
1
a
Arc sec + C = Arc cos + C
a
a
a
x
Formulas and Theorems
1a.
Definition of Limit: Let f be a function defined on an open interval containing c (except
possibly at c ) and let L be a real number. Then lim f ( x ) = L means that for each ε > 0 there
x→a
exists a δ > 0 such that f ( x ) − L < ε whenever 0 < x − c < δ .
1b.
A function y = f (x ) is continuous at x = a if
i).
f(a) exists
ii).
lim f ( x) exists
iii).
4.
x→a
lim = f (a)
x→a
Intermediate-Value Theorem
A function y = f (x ) that is continuous on a closed interval a, b takes on every value
between f ( a ) and f (b) .
[ ]
[ ]
Note: If f is continuous on a, b and f (a ) and f (b) differ in sign, then the equation
5.
f ( x) = 0 has at least one solution in the open interval (a,b) .
Limits of Rational Functions as x → ±∞
f ( x)
lim
i).
= 0 if the degree of f ( x) < the degree of g ( x)
x → ±∞ g ( x)
x 2 − 2x
Example: lim
=0
x → ∞ x3 + 3
f ( x)
ii).
lim
is infinite if the degrees of f ( x ) > the degree of g ( x )
x → ±∞ g ( x )
x3 + 2x
Example: lim
=∞
x → ∞ x2 − 8
f ( x)
lim
iii).
is finite if the degree of f ( x ) = the degree of g ( x )
x → ±∞ g ( x )
2 x 2 − 3x + 2
2
=−
Example: lim
5
x → ∞ 10 x − 5 x 2
6.
7.
( 0 0 ) and (x1, y1 ) are points on the graph of
Average and Instantaneous Rate of Change
i).
Average Rate of Change: If x , y
y = f ( x) , then the average rate of change of y with respect to x over the interval
[x0 , x1 ] is f ( x1 ) − f ( x0 ) = y1 − y 0 = ∆y .
x1 − x0
x1 − x0 ∆x
ii).
Instantaneous Rate of Change: If ( x 0 , y 0 ) is a point on the graph of y = f ( x ) , then
the instantaneous rate of change of y with respect to x at x 0 is f ′( x 0 ) .
f ( x + h) − f ( x )
f ′( x) = lim
h
h→0
8.
The Number e as a limit
i).
ii).
9.
10.
n
 1
lim 1 +  = e
n → +∞ n 
1
n


lim 1 +  n = e
n → 0 1 
Rolle’s Theorem
If f is continuous on a, b and differentiable on (a, b ) such that f ( a ) = f (b) , then there
[ ]
is at least one number c in the open interval (a, b ) such that f ′(c) = 0 .
Mean Value Theorem
If f is continuous on a, b and differentiable on (a, b ) , then there is at least one number c
[ ]
in (a, b ) such that
11.
f (b) − f (a)
= f ′(c) .
b−a
Extreme-Value Theorem
If f is continuous on a closed interval a, b , then f (x ) has both a maximum and minimum
[ ]
[ ]
on a, b .
12.
13.
To find the maximum and minimum values of a function y = f (x ) , locate
1.
the points where f ′(x ) is zero or where f ′(x ) fails to exist.
2.
the end points, if any, on the domain of f (x ) .
Note: These are the only candidates for the value of x where f (x ) may have a maximum or a
minimum.
Let f be differentiable for a < x < b and continuous for a a ≤ x ≤ b ,
1.
2.
[ ]
If f ′( x ) < 0 for every x in (a, b ) , then f is decreasing on [a, b ] .
If f ′( x ) > 0 for every x in (a, b ) , then f is increasing on a, b .