CSSS 505

Calculus Summary Formulas

Differentiation Formulas

1. 1)( −= nn nxxdxd

17. dxdu

dxdy

dxdy ×= Chain Rule

2. fggffgdxd ′+′=)(

3. 2)(g

gffggf

dxd ′−′

=

4. )())(())(( xgxgfxgfdxd ′′=

5. xxdxd cos)(sin =

6. xxdxd sin)(cos −=

7. xxdxd 2sec)(tan =

8. xxdxd 2csc)(cot −=

9. xxxdxd tansec)(sec =

10. xxxdxd cotcsc)(csc −=

11. xx eedxd =)(

12. aaadxd xx ln)( =

13. x

xdxd 1)(ln =

14. 21

1)sin(x

xArcdxd

−=

15. 211)tan(x

xArcdxd

+=

16. 1||

1)sec(2 −

=xx

xArcdxd

Trigonometric Formulas

1. 1cossin 22 =+ θθ2. θθ 22 sectan1 =+3. θθ 22 csccot1 =+4. θθ sin)sin( −=− 5. θθ cos)cos( =− 6. θθ tan)tan( −=− 7. ABBABA cossincossin)sin( +=+8. ABBABA cossincossin)sin( −=−9.

BABABA sinsincoscos)cos( −=+

10.

BABABA sinsincoscos)cos( +=−

11. θθθ cossin22sin =2

12.

θθθθθ 222 sin211cos2sincos2cos −=−=−=

13. θθ

θθcot

1cossintan ==

14. θθ

θθtan

1sincoscot ==

15. θ

θcos

1sec =

16. θ

θsin

1csc =

17. θθπ sin)2

cos( =−

18. θθπ cos)2

sin( =−

Integration Formulas

Definition of a Improper Integral

∫b

adxxf )( is an improper integral if

1. becomes infinite at one or more points of the interval of integration, or f2. one or both of the limits of integration is infinite, or 3. both (1) and (2) hold.

1. Caxdxa +=∫

2. ∫ −≠++

=+

1 ,1

1

nCnxdxx

nn

3. ∫ += Cxdxx

ln1

4. ∫ += Cedxe xx

5. ∫ += Ca

adxax

x

ln

6. ∫ +−= Cxxxdxx ln ln

7. ∫ +−= Cxdxx cos sin

8. ∫ += Cxdxx sin cos

9. ∫ +−+= CxCxdxx coslnor secln tan

10. ∫ += Cxdxx sinln cot

11. ∫ ++= Cxxdxx tansecln sec

12. ∫ +−= Cxxdxx cotcscln csc

13. Cxxdx +=∫ tan sec2

14. ∫ += Cxdxxx sec tansec

15. ∫ +−= Cxdxx cot csc2

16. ∫ +−= Cxdxxx csc cotcsc

17. ∫ +−= Cxxdxx tan tan 2

18. ∫ +

=

+C

axArc

axadx tan1

22

19. ∫ +

=

−C

axArc

xadx sin

22

20. ∫ +=+=−

CxaArc

aC

ax

Arcaaxx

dx cos1sec122

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 means that for each Lxf

ax=

→)(lim 0 >ε there

exists a 0 >δ such that ε )( <− Lxf whenever δ 0 <−< cx .

1b. A function y = is )(xf continuous at ax = if i). f(a) exists ii). exists )(lim xf

ax →iii). )(lim af

ax=

→4. Intermediate-Value Theorem

A function that is continuous on a closed interval [ takes on every value between .

)(xfy = and )( fa

]ba,)(bf

Note: If is continuous on [ and and differ in sign, then the equation f ]ba, )(af )(bf has at least one solution in the open interval ( . 0)( =xf ),ba

5. Limits of Rational Functions as ±∞→x

i). 0)()(lim =

±∞→ xgxf

x if the degree of )( of degree the)( xgxf <

Example: 03322

lim =+

−∞→ x

xxx

ii). )()(

xgxf

x ±∞→lim is infinite if the degrees of )( of degree the)( xgxf >

Example: ∞=−

+∞→ 82

23lim

x

xxx

iii). )()(

xgxf

x ±∞→lim is finite if the degree of )( of degree the)( xgxf =

Example: 52

2510

2322lim −=−

+−∞→ xx

xxx

6. Average and Instantaneous Rate of Change

i). Average Rate of Change: If ( ) ( )1,1 and 0,0 yxyx

y

are points on the graph of

, then the average rate of change of with respect to )(xfy = x over the interval

is [ ]10 , xxxy

xxyy

xxxfxf

∆∆=

−−

=−−

01

01

01

01 )()(.

ii). Instantaneous Rate of Change: If ( is a point on the graph of , then

the instantaneous rate of change of with respect to

)00 , yxy

)(xfy =x at is . 0x )( 0x′f

7. h

xfhxfh

xf )()(0

lim)( −+→

=′

8. The Number e as a limit

i). en

nn=

+

+∞→

11lim

ii). ennn

=

+

→

1

11

0lim

9. Rolle’s Theorem If is continuous on [ and differentiable on ( such that , then there

is at least one number in the open interval such that .

f ] )ba,c

ba,)

)()( bfaf =0=( ba, )(′ cf

10. Mean Value Theorem If is continuous on [ and differentiable on ( , then there is at least one number c

in such that

f

(a] )

)ba, ba,

b, )()( cfa

aff ′=)(b

b−−

.

11. Extreme-Value Theorem If is continuous on a closed interval [ , then has both a maximum and minimum

on [ ] .

fa

]ba, )(xfb,

12. To find the maximum and minimum values of a function , locate )(xfy =1. the points where is zero )(xf ′ or where fails to exist. )(xf ′2. the end points, if any, on the domain of . )(xfNote: These are the only candidates for the value of x where may have a maximum or a )(xf minimum.

13. Let f be differentiable for and continuous for a , bxa << bxa ≤≤1. If ′f for every 0)( >x x in , then is increasing on . ( ba, ) f [ ]ba,2. If ′f for every 0)( <x x in , then is decreasing on [ ] . ( ba, ) f ba,