CSSS CS SS 50 505 5 Calcul Cal culus us Su Summa mmary ry For Formul mulas as Differentiation Formulas
1. 2. 3.
4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
15. 16.
d dx d
( x n ) = nx n −1
17.
( fg ) = f g ′ + g f ′ dx d f g f ′ − f g ′ ( )= dx g g 2 d dx d dx d dx d dx d dx d dx d dx d dx d dx d dx d dx d dx d dx
f ( g ( x )) = f ′( g ( x)) g ′( x) (sin x) = cos x (cos x) = − sin x (tan x) = sec 2 x (cot x) = − csc 2 x (sec x) = sec x tan x (csc x) = − csc x cot x x
x
(e ) = e
(a x ) = a x ln a (ln x) =
1 x
( Arc sin x ) = ( Arc tan x) = ( Arc sec x) =
1 1 − x 2 1 1 + x 2 1 | x | x 2 − 1
dy dx
=
dy dx
×
du dx
Chain Rule
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
13. tan θ = 14. cot θ = 15. secθ = 16. cscθ = 17. cos( 18. sin(
11. sin 2θ = 2 sin θ cosθ 2
2
2
2
12. cos 2θ = cos θ − sin θ = 2 cos θ − 1 = 1 − 2 sin θ
π
2
π
2
sin θ cosθ cosθ sin θ 1
= =
1 cot θ 1 tan θ
cosθ 1 sin θ
− θ ) = sin θ − θ ) = cosθ
Integration Formulas
Definition of a Improper Integral
b
∫a f ( x) dx is an improper integral if 1.
f becomes infinite at one or more points of the interval of integration, or
2. 3.
one or both of the limits of integration is infinite, or both (1) and (2) hold.
1.
∫ a dx = ax + C
2.
∫ x
3. 4. 5. 6. 7. 8. 9.
10. 11.
n
dx =
x n +1 n +1
12. 13.
+ C , n ≠ −1
14.
1
∫ x dx = ln x + C ∫ e dx = e + C x
x
a x
∫ ln a ∫ ln x dx = x ln x − x + C ∫ sin x dx = − cos x + C ∫ cos x dx = sin x + C ∫ tan x dx = ln sec x + C or x
a dx =
15. 16. 17.
+ C
18.
19.
− ln cos x + C
∫ cot x dx = ln sin x + C ∫ sec x dx = ln sec x + tan x + C
20.
∫ csc x dx = ln csc x − cot x + C ∫ sec x d x = tan x + C ∫ sec x tan x dx = sec x + C ∫ csc x dx = − cot x + C ∫ csc x cot x dx = − csc x + C ∫ tan x dx = tan x − x + C dx 1 x Arc tan = + C ∫ a + x a a 2
2
2
2
∫ ∫ x
2
x + C a
dx
= Arc sin
a 2 − x 2 dx 2
x − a
= 2
1 a
Arc sec
x a
+ C =
1 a
Arc cos
a x
+ C
1a.
Formulas and Theorems Definition of Limit: Let f be a function defined on an open interval containing c (except
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 .
possibly at c ) and let L be a real number. Then
1b.
A function y = f ( x) is continuous at i).
f(a) exists
ii).
lim f ( x) exists x → a lim = f ( a) x → a
iii). 4.
= a if
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
f ( x) = 0 has at least one solution in the open interval ( a, b) . 5.
Limits of Rational Functions as i).
f ( x)
lim
x → ±∞ g ( x)
→ ±∞ = 0 if the degree of f ( x ) < the degree of g ( x)
x 2 − 2 x Example: lim =0 3 x → ∞ +3 ii).
lim
x →±∞
f ( x) g ( x)
is infinite if the degrees of f ( x ) > the degree of g ( x )
x 3 + 2 x Example: lim =∞ 2 x → ∞ −8 iii).
lim
x →±∞
f ( x) g ( x)
is finite if the degree of f ( x) = the degree of g ( x)
2 x 2 − 3 x + 2 2 Example: lim =− 5 x → ∞ 10 x − 5 x 2 6.
Average and Instantaneous Rate of Change i).
Average Rate of Change: If x , y
and x , y are points on the graph of 1 1 y = f ( x) , then the average rate of change of y with respect to x over the interval 0
[ x0 , x1 ] is ii).
f ( x1 ) − f ( x0 ) x1 − x0
=
0
y1 − y 0 x1 − x0
=
∆ y ∆ x
.
Instantaneous Rate of Change: If ( x 0 , y0 ) 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 ′( x0 ) .
7.
f ′( x) = lim h→0
f ( x + h) − f ( x) h
8.
The Number e as a limit i).
1 lim 1 + n → +∞ n
n =e
1 ii). 9.
n lim 1 + n = e n → 0 1
Rolle’s Theorem
[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 . If f is continuous on 10.
Mean Value Theorem If f is continuous on in
11.
(a, b ) such that
[a, b] and differentiable on (a, b) , then there is at least one number
f (b) − f ( a ) b−a
= f ′(c) .
Extreme-Value Theorem If f is continuous on a closed interval on
12.
c
[a, b] ,
then f ( x ) has both a maximum and minimum
[a, b] .
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 13.
minimum. Let f be differentiable for a < x < b and continuous for a a ≤ x ≤ b , 1.
If f ′( x ) > 0 for every
in
2.
If f ′( x ) < 0 for every
in
(a, b) , (a, b) ,
then f is increasing on then f is decreasing on
[a, b] . [a, b] .
