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CSSS 505 Calculus Summary Formulas

Differentiation Formulas 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

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

17.

dy dy du Chain Rule = × dx dx dx

d 1 ( Arc tan x) = dx 1+ x2 d 1 16. ( Arc sec x) = dx | x | x2 -1

Trigonometric Formulas 1. 2. 3. 4. 5. 6. 7. 8. 9.

sin 2 + cos 2 = 1 1 + tan 2 = sec 2 1 + cot 2 = csc 2 sin(- ) = - sin cos(- ) = cos tan(- ) = - tan 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

13. tan =

sin 1 = cos cot cos 1 14. cot = = sin tan 1 15. sec = cos 1 16. csc = sin

17. cos( 18. sin(

10. cos( A - B) = cos A cos B + sin A sin B 11. sin 2 = 2 sin cos

2

2

- ) = sin - ) = cos

2

12. cos 2 = cos

- sin 2 = 2 cos 2 - 1 = 1 - 2 sin 2

Integration Formulas Definition of a Improper Integral

b

1. 2. 3.

a

f ( x) dx is an improper integral if

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. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

a dx = ax + C

n x dx =

12. 13. 14. 15. 16. 17. 18. 19.

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

x a dx =

x n +1 + C , n -1 n +1

ax +C ln a ln x dx = x ln x - 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

sin x dx = - cos x + C cos x dx = sin x + C tan x dx = ln sec x + C

a

x

2

dx 1 x = Arc tan + C 2 a +x a dx x = Arc sin + C a a2 - x2

dx x2 - a2 = x 1 1 a Arc sec + C = Arc cos + C a a a x

or - ln cos x + C

20.

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

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 > 0 there

xa exists a > 0 such that f ( x ) - L < whenever 0 < x - c < .

A function y = f (x ) is continuous at x = a if i). f(a) exists ii). lim f ( x) exists iii).

1b.

4.

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) .

xa lim = f (a) xa

[ ]

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.

Average and Instantaneous Rate of Change i). Average Rate of Change: If x , y

( 0 0 ) and (x1, y1 ) are points on the graph of

7.

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 h0

8.

The Number e as a limit i).

ii). 9.

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

n 1 lim 1 + = e n + n 1 nn lim 1 + = e n 0 1

10.

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

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

[ ] [ ]

(a, b ) such that

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

c

(a, b ) such that

11.

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 . To find the maximum and minimum values of a function y = f (x ) , locate 1. 2. the points where f (x ) is zero or where f (x ) fails to exist. the end points, if any, on the domain of f (x ) .

[ ]

[ ]

12.

13.

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 If f ( x ) < 0 for every x in

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

then f is increasing on a, b . then f

[ ] is decreasing on [a, b ] .

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