Calculus Cheat Sheets

CSSS 505 Calculus Summary Formulas Differentiation Formulas 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

d n ( x ) = nx n −1 dx d ( fg ) = fg ′ + gf ′ dx d f gf ′ − fg ′ ( )= dx g g2 d f ( g ( x)) = f ′( g ( x)) g ′( x) dx d (sin x) = cos x dx d (cos x) = − sin x dx d (tan x) = sec 2 x dx d (cot x) = − csc 2 x dx d (sec x) = sec x tan x dx d (csc x) = − csc x cot x dx d x (e ) = e x dx d x (a ) = a x ln a dx d 1 (ln x) = dx x 1 d ( Arc sin x) = dx 1− x2

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

12. cos 2θ = cos

2

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(

θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2 sin 2 θ

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 =

3. 4. 5. 6. 7. 8. 9. 10. 11.

12.

x n +1 + C , n ≠ −1 n +1

1 ∫ x dx = ln x + C x x ∫ e dx = e + C

14. 15. 16.

ax +C ln a ∫ ln x dx = x ln x − x + C

17.

x ∫ a dx =

∫ sin x dx = − cos x + C ∫ cos x dx = sin x + C ∫ tan x dx = ln sec x + C

13.

or − ln cos x + C

∫ cot x dx = ln sin x + C ∫ sec x dx = ln sec x + tan x + C

∫ 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 2

2

2

18.

∫a

19.

∫

20.

∫x

dx 1  x = Arc tan  + C 2 a +x a dx  x = Arc sin   + C a a2 − x2

2

dx x2 − a2

=

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.

Rolle’s Theorem If f is continuous on a, b and differentiable on

(a, b ) such that

Mean Value Theorem If f is continuous on a, b and differentiable on

(a, b ) , then there is at least one number

[ ]

f (a) = f (b) , then there is at least one number c in the open interval (a, b ) such that f ′(c) = 0 .

[ ]

in 11.

n  1 lim 1 +  = e n → +∞ n  1 n   lim 1 +  n = e n → 0 1 

(a, b ) such that

f (b) − f (a) = f ′(c) . b−a

c

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.

If f ′( x ) > 0 for every x in

2.

If f ′( x ) < 0 for every x in

(a, b ) , (a, b ) ,

[ ] is decreasing on [a, b ] .

then f is increasing on a, b . then f