Quick study academic calculus 2 600dpi

DEFINITIONS

INTERPRETATIONS

• !leuristies. The defi nite integral captures the idea of
addin g the va lues or a run clion over a continuulll .
• Riemann sum. A su itably we ig hted sum of values. A
de fin ite illlcgra l is the li mi t ing value of sueh sums. A
Ri e man n s um of a funct ion f de f incd on [II ,h] is
de te rmincd by a partitio n , which is a fini te division of
[{I .h·1 into subintervals, ty p ically exprcssed by
{I =xu
• Area under a cun·e. I ('lis nonnegati ve and cont inuous on

fro m each sub intcrval , say c; fro lll [x; " x ;]. The associated

" i)(Xi - X;
Rie mann sum is: ~f(c
j

d.

I

A regula r pa rtitio n has sub intervals all the samc Icngth,
{\. x =( h ~{I)/II , x ; = {I + i {\'x. A parti ti on 's norm is its
max im um subintc rva l length. A left s um takes the left
e nd poi nt Cj=Xj_, o f each , ubinterval ; a r ight su m , thc
ri ght endpoin t. An uppe r sum of a continuous f takes a
po int <"; in each subint c rva l w here thc max imu m \alue off

is ac hieved; a lowe r sum , thc m in imum val ue . E.g., tl e
uppe r Rie ma nn su m of cos x on [0,3] with a reg ul ar
partitio n of II int cr va ls is the left sum (since the cosine is
decreas ing o n the inte rva ll :

f ICOs(U- I):t)+
I]..:i.
i I
11
Il

A '(x )=f(x ) (valid
endpoints).

[a,h], the n ( >tf ed dx !; ivcs the a rea hetwee n the x-a xi s
• I>

a nd the graph. T he area func tion A (x) =

.c'f (n dl gives

--- -- - - - -- - -..-

A(x)

' 1,

a .(/

x

U

A rough estimatc o f a n integ ral may be made by est ima ting
the ave ra ge va lue ( by in spec tin g th e gra ph ) a nd
multip ly ing it by the le ng th o f the illl erva!' (See M el/II
Ii' /l/I! Theorem (MVT) jiJl" illtegra /s. in the Theon ' sccti on. )
• Accumulated Change. The int egra l of a ra te o f c hange of
a q ua ntity over 3 time interva l gives the tOlal change in th e
q uanti ty over the time inte rva l. E.g., if v (l) =s '(I ) is a
ve locity (the rate o f e ha nge of pos it ion), then v (l )1'lt is the
approxim ate displ acement occ urring in th e time inc re me nt
/ to t +{\./ ; addi ng the di sp lace me nts for all time inc rc m c n.ts
gives the approx imate change in pos ition ovcr the enti re
time inte rva l. In the lim it of small time in crcme nts, o ne

."

gc ts the exac t to ta l di splacement: / v(f) dl =s (h)~-'(IIl .

."

• Differentiation of integrals. r unctions a rc oftcn de fined
a" integra ls. E.g., the "si ne in tegral fu nction" is

Si(x)= ./,:"( Si;l I )d l.

To dirfe re n t iatc s uch, u se t he ' ccon d pa rt of the

fundam ent al theo rem: Si '(x)= sin x /.\'. A fu nction

such

as

( 'f (t)(l/

•a

l 'j (x )dx=

lim ~f( C i)Llsj.
11 .lxll .() i
T he lim it is sa id to cx ist ifsome number S (to be called the
integra l) satis fies the roll ow ing: Every £> 0 admits a 6
such tha t all Riem a nn sums on partitions of [tI ,h] with
norm less than 6 dilTer from S by less than £ . If there is
suc h a valuc S. the I'unct ion is sa id to be int egra ble a nd the
value is de noted

l 'j(x )dx or I f .Thc function must be
(I

(l

bounded to be integra bl e. The fu nction f is ca ll ed the
integrand a nd the points {I a nd h arc called the lo,ver lim it
a nd uppe r li mit o f integralion, respectively. T he word
integral rerers to thc rormation of

ja-I>f


from f and [(I .h], as


\Ve il as to the resuiting value ifl here is one .
• ·\ ntiderhative. A n "ntidc riva tivc of a fi.lI1ction

f is ,[
func ti o n A whose de rivative is f : A'(x )=f(x) for all x in
some do ma in (usua lly a n inter va l) . A ny t'.I'O antiderivati ves
of a fu nctio n o n an in terva l differ by a co nstan t (a
consequence of the Mea n Va lu e T henrem ). E.g. , bot h

J1.. j" j

W dl=

dx "

J1.. A (x ')= A ' (x') 2x = 2.xj(x'1.

2I (x -

a )'- a ll(I iIX "., -

(j .lil-'-'
c rln g
de noted

I--.

(IX

.
a re ant 'd
i c n.
vatlvcs
0f

j)·(x){I (x)dx=/(;)j~b(l (x)Clx.
1

jj(x)dx,' ,s

a lll ide rivatives o n a typica l (often
inten a!' E.g. , ( lo r x < ~ I , or for x > I).

Ml~ ~~ ~~~~~~B""~I~

~---------------------b

a

unspecifi ed )

X- dx =! x ' - 1 + C.
j '!x' - i
Th e consta nt C, w hich may have any real valuc, is the
constant of integr a tion . (Com puter prog ra ms, and this

cha rt . may om it the constant, it being understood by the
kn owledgea hle user that the g iven a nl idcrivative is just one
representative of a ra m ily.)

is f ; and

if (x) ldx.

• Fundamental theorem of calculus. O ne part o f the
theore m is use d to evaluate integrals: Iffis continuo us o n
[(I, h]. a nd A is a n ant idcr iva ti ve off n n th at inte rval , then

j

·"f (x) dx=A(x )I"C= A (b) - A(a).
(,

{/

The other part is used to construct antide rivativcs:

If f is conti nu ous o n [a , h) . th en the fun cti on


A (x)l= ( j U)d l is an a ntiderivative of f on [(I ,h] :
. a

~

b

apprehend or cvaluate. In effect, the "a rea" is smoothly
rcd istributed w it hout changing the integral;' value. !I' g is
a fun ction w ith cont inuous deriva tive andf i, continuous.
then

" lI )dn = ...I'df (.9(f ))g ' (f )dl,
1"f(



where c. d are

points with g (C)= il a nd g(rI )= h.
In pract ice, su bstit !!!r lI = g(l); compu te t11/ = g'(I)dl; and
find what t is when 1/ = 0 and I/ = h. E.g.. I/ =sin t e ffe cts the
transrorm ation

I> ( )
Iff is non negative , then .(".f
x dx is no nnegati ve.

i[ ' j(x )dx i< t

II!I

/ .,'

lnnnunuu : ~ ~

whkh becomes

X - {I ,

an exp ression for the fam ily or

n

integra tion can be cha nged to make an in tegral eas ier to

Use this
in tegra l evalu at ions w ith rough
ove restimates o r u nderes tima tes.

l/.hJ, the n so

,

• Cbange of variable formula. An integrand and limit, of

"
to ch ec k

U

C

is attained

i\1VT for Integra h

L · (b -alS l "f (xldX S, M . (/)- a) .

Iff is int egrablc on

f

somewhere on the inLe rval : -b - -a . a .f(x) dx =.f( ; ).

\ -u- du= I' " "~
! - s in-I co~ I til.
/.",....----,;
of

of

on

·

I

Oil

by 2a-.
I ., T he indefinite in tegra l ora runction!

'

dx

In thc case g - I , the a\eragc value of

• Integrability & inequalities. A co ntinuo us fun cti on o n a
closed inter val is integra bl e. Integ rab ility On [tI .h] imp li es
int egra bi lity on c losed sub inte rva ls of [tI .h]. Ass umin gfi s
in teg rable , if L "'f(x) 5. M for a ll x in [{I ,hJ, th e n

~

IIiII

·Mean value theorem for integrals. Ir f an d ~ a rc
conti nuo us o n [II ,h], then there is a S in [a.h] such Lhat

f«,

L_

in\o l vin g

. a

THEORY

II

c OI11[l osition

A( u )= ( "f (tldt. T o d i frcre nt ia te, u sc the c hai n rule

x~

• Definite integral. The definite inte gra l off frol11 {I to b
may bc described as

is a

and the fundamcn ta l th eorem :

(x )= y ,, + ('fll )d l.
.

-;.o.-.-.--­

::;.
- ;.;.
--~
-

Illay bc defined by average VII[lIe = 'h-=- / f(xl dx.

-Integral curve. Imag ine that a run cti on f determines a
slope fix) lor eac h x . Plac ing linc segments w ith s lo pe
fi x) at po ints (x.y) fo r va ri ousy, and doi ng thi s lor va rio us
x , one gets a slope field . A n antidc ri vati ve o ffi s a fu nc tion
whose graph is tange nt to th e slope field at each po int. T he
gra ph o f the a nt ide ri va ti vc is called a n integ ral curve of
the slope fi e ld.
• Solution to initial value problem . The so lution to the

diiTerential eq umiony ' ",/"( x) with initia l va luey(xnl = y" is

3

F unda m e nta l T h e orem

fl.x)

the arca accu lllul ated up to x. Iffis negati ve, the integral
is the negat ive o f the area .
• Average value. The awragc va llie 011 over a n int erval [a ,h]

i

for o ne-si de d de rivatives

."

(l

2

cos ' l dl , si nce ! 1- s in 'l = cos I

for () < 1< 1[/ 2. T he for mul a is ollen used in revcrse,
staJ1ing w ith ], "1-'(.9 (x ))O' Cddx.

ec Technic/iII!s on pg. 2.

• i'atural logarithm . A rigorous defini tion is In x =

(' Ida
. T he change of variable for mula with 11 = I I I
U

. ,

. , ., I

yiel ds ./ ,

/., - I

/., \

a rill = . , I -cc1- d l = - _, . I dt sho\\ ing that

In ( f Ix) = ~I nx. The other elementary propertie, of the
natural log can likewise be easily derived from thi,
definition. in this approach, a n inverse function is deduced
a nd is derined to be the natural exponent ial function .

•,

C

n


INTEGRATION FORMULAS

Other routine integration-by-parts integrands arc arcsinx,
XCOSX, and xe UX ,
• Rational functions . Every rational function may be
written as a polynomial plus a proper rational function
(degrec of numerator less than degree of denominator). A
proper rational function with real coefficients has a partial
fraction decomposition: It can be written as a sum with
each summand being either a constant over a power of a
lincar polynomial or a linear polynomial over a power ofa
quadratic. A factor (X-C)k in the denominator of the
rational function implies there could be summands
Inx. xnInx. xsinx,

• Basic indefinite integrals. Each lormula gives just one
antidcrivativc (all others dillering by a constant from that
given). and is valid on any open interval where the
integrand is defined:
~

1/ !

1

*
J x"dx=~(n
n+l

f 1dx=lnlxl
x

-1)

fe"xdx=e;x(k * O)
feos x dx=sin x

fsin x dx= -cns x

fl'~:' =arctan x

x
f~=81'Csin
Ii-x'

• Further indefinite integrals. The above conventions hold:

J~()t x dx=ln lsin xl

ftan x dx=lnlsec xl

.A.!...+",+~"
x-c
(x-c)"
A factor (x'+bX+C)k (the quadratic not having real roots)
in the dcnominator implies there could be summands

Isec x dx=lnlsec x + tan x l

x'+bx+c

".

J~sc x dx=ln lcsc x+cot xl

Math software can handle the work, but the following case

l x-al
f~=lln
x'-a' 2a x+a

Iix ldx= lxlxl

X

f l x-~-+\: a­.,=lnlx+lx'+a' l=sinh

dx=cosh x

1~+lna

.,=Inlx+lx'a'icosh 1~+lna
fld~
x-a­
(take positive values for cosh· l )

flx' ± a'dx=!xl x'a'

± ~Inlx+ Ix' ± a' i

(Take same sign. + or -. throughout)

fla

2
2
• Common definite integrals:
I
l'

1" x"dx= -n+l-

II

fU - Idll,ju-"du(tI > O,fll(U'+ll "du (handled with
substitution

and fill'+l) "du (handled with

w=1I 2 +1) ,
II

= tan t).

,l

th~ integrand is

),dx= ) , + 2 < 2.4.

GEOMETRY

• Areas of plane regions, Consider a plane region admitting
an axis slIch that sections perpendicular to the a.xis \ur) in
lcngth according to a known function L(p). tl"'I'''' b. The
area of a strip of width I'!.p perpendicular to Ih~ axis at p i,
M = L(pll'!.p, and the total area is

IMPROPER INTEGRALS

Over [1I.bJ is

IIJ
on [II,B] for all B>II, then I" j(x)dx ~ ,!i lll " jlx)dx
provided the limit exists.

• Substitution. Refers to the Change of variable formula
(see the Thmrv section), but ollen the formula is used in
reverse. For an integral recognized to have the form

f."F(y(x»)y' (x)dx (with F and g' continuous), you can put
tllI=g'(x)dx, and modify the limits of integration

[ "F(,q(xl)y' (x)dx= j~(I("1
. F(u)dll.
,1/((1)

In effect, the integral is over a path on the II-axis traced out
by the flJJlction g. (I I' g(b) = g(a) [the path returns to its
start], then the integral is zero.) E.g., u= I+x' yiclds

l_ x _ 1 -1 [I _ l _ ? d -11 (1+ x,

')
-l+x'cx- 2 . o l+x'_x x- 2 n

Substitution may be used lor indefinite integrals.

E.g..

l we

A= .(,L(p )dp. E.g"

/J .

iCJ

I "j(x)dx = lim 1."j(x)c/x.
d,1

;1 •

•t

In each case, if the limit exists, the improper integral
converges. and otherwise it diverges. For f defined on
(_ 00,00) and integrable on every bounded interval,

·
f .~jlx)dx
= ,[i~ l'j(x)dx+ ,li~.,["j(x)dx
deI

(the

.4

choice of c being arbitrary), provided each integral on the
right converges.
• Singular integrands. Iff is defined on (lI ,b] but not at
x=a and is integrable on closed subintervals of(a,b], then

),"" j(x)dx (hI
= }i!~

Ih, j(x)dx

provided the limit exists. A

similar definition holds if the integrand is defined on
[1I,b).

E.g.,

l',,4-x'
".!-.,dx

lim
"- '

is

II

l',,4-x'
~dx =
1J

,lim
-., arcsin(f)=!f,
2
2

' , f~
E.g.
l+x 2 dx-lfdll-llna-lln(l+x')
u - 2
-2
'
' -2

­
f y(x)",1I (x)dx----,z:tl,I I fY'(x)
y(x)dx-ln1y(x)l,

"~

_lI(X)"

I

.

(C)

t

• Volumes of solids
Consider
a
solid
admitting an axis slich
that
cross-sections
perpendicular to the
axis vary in arca
according to a known
function A(p), a"'p",b.


The volume of a slab of'

thickness I'!.p pl'rpcndi"ular to

(l

For indefinite integration, fll dv=uu- fv dll,
The procedure is used in derivations where the iemetions
arc gencral, as well as in explicit integrations. You don't
need to usc "II" and "v." View the integrand as a product
with one factor to be integrated and the other to be
differentiated; the integral is the integrated factor times the

one to be differentiated, minus the integral of the product
of the two new quantities. The t~lctor to be integrated may
be I (giving v=x).
E.g., farctan x dx=x arctan x- f 1':x ,dx

ih~

axis at p is III ' =.1(plt!.{',


and the total I'olumc is V = ."('"i\(p)dp, Lg.. a pyramid

a

('

V=j""/I.(z)dz= ("rrrlz)'dz.
ct

The common formula is j'''1t dU=llul" -j'''u dll.

bp

: IIp

a

·Solids of re\olution. Consider a solid of r~I'()lution
det~rmined by a known radius ilmction r(:) , a"' :'" h, along
it s axis of revolution. The area of th~ cross·,,,ctional

"disk" at : is A (:)=nr(:)l. and the I'olum e is

feu rxly' (x)dx=e yrxl ,
['u(x)u' (X)dX=Il(X)U(x)l~ - ['U(X)Il' Cddx.

~3"~

V= f~'( l - ~r dz = S~lh ,

n

'/4-x' dx=arcsm 2-2

• Integration by parts. Explicitly,

[a,b]. Sometimcs it i. simpler to I' iew a region as bounded
by two graphs "over" the y·axis, in which case the
integration variabk isy.

having square horiLollwi Cfllss-scctions. with bottom ~ide
length !\ and height Jr. has cross-st:L:tional area
A(:)=[.\'( I-://r)]' at height ... Ih ,,)!tUllC is thu>

Singular Integrand
.,

.I"." if/(x) - jlx)idx. prm i(kd K(X) "-f(x) on

~

xdx= Iim(1-e 1J)=1

u

Some general formulas are:

(J

convergcs since

l-In lx-hi).
Ji x - a i( x-~ndx=....l....b(lnlx-a
a-

def

TECHNIQUES

a

0'


)

APPLICATIONS

In gencral, the indefinite integral of a proper rational function
can be broken down via partial Ihlction decomposition and
linear substitutions (of lorm II =ax+b) into the integrals

-(X)

IJ

(

Thus r(

Likewise, lor appropriate}:

l

1 l':x' . -dx

bounded by 1/2.112: on [fI , l.! and is always less Ihan 1/.\"'12.
It
convcrges
to
a
numhcr
lcss
than

a-1

• Unbounded limits. III is defined on [11,00] and integrable

cos 26) equals cos'6 or

-r/

-II

x~ a + x -Dh

IJ

II

appropriately:

1 x-

1

where C. D are seen to be C= - D= ....l....
l .

l "sin xdx=2

d8= R

l ""sin' (1d8=1";'1-cos28
,,2
4

II =g (x),

1 x-

The above integrals arc llseful in comparisons to

establish convergence (or divergence) and to get bOllnJs.


the arca of the region bounded by thc graphs or!, and K

1+cos20 d8= TI

j"'rr"cos'8d8=1"n
,,2
4


±

x=oo, p=O or < 1 div~rgL's at x=oo.

1
'11 .
.

-;j dx converges to I and
~ dx dl\crg~s.
E.g..

2:~

(l

.,
j 'Ir'-x'dx= IT"4

To remember which of II, (I
sin'6, recall the value at zero.

A"+B,,x
(x'+hx+(-)'"

shoulcJbel'lmiliar.lfa"b,( x-a ~ X-1
l)

substitution

lxla' x' +~arcsin'!

2 -x'dx=

N
tfdXI
I
oe: x(lnx)" - - (1I - 1l0nx)" I'P> cOJ1\crges at

'1

+ +

fSinh

I·

)/dx convcn.!.cs
for p> I. divcr!.!cs otherwi se.
.; x ( 'I~
nx
~
....

E.g. ,

Al +Blx

fcosh x dx=sinh x

1

1d
"lor p < I ,(I Iverge s 01I
'
-u'1 x"
x converg~s
lcrWlse.

rtf is not defined at a finite number of points in an interval

)a

intervals between such points, the integral f.'1' is defined

Ifth" solid lies betw"~n two radii rl(:) and r2(:) at each
point: along Ih" axis of revolution. the cross-sections
are "washers," and the volume is th~ ob\ iOllS

as a sum of lell and right-hand limits of integrals over
appropriate closed subintervals, provided all the limits exist.

differcnce or volumes like that above. Sometim es
a radial coordinate r, a", r'" b, along an a~i s

[II,b], and is integrable on closed subintervals of open

E.g.,

fl ~dx= lim
IX'

a -' II

1" ~dx+ lim [I ~dx
- IX'

b ' 0 ·1,

X'

if the

limits on the right werc to exist. They don't, so the integral
diverges.

• Examples & bounds .

1 ~dx
x
I

converges for p > 1, diverges otherwise.

2

perpendicular to the axis of revolution .
paramctri/es the heights /r (r) of cylindrical
sections (shells) or the solid parallel to ihc ax is or
revolution. In this case, the area of the shell at r is
A(r)=2nr/r(r), and the volum e of the solid is

V=

["A (r)dr = ~-("2ITrh(r
)dr.
,

~o


• \rc lenl1;th A graphY=I(x) between x=a andx=h has length

V= I"/1

+f' (x)"dx.

f(a) " I
.
f ib»)
.'
T,,= ( --:r-+ ;~f(a-Ih ) + --:r- h. T im IS al so the

A curve C parametrized by «x(1).

1 I"

I"
,
y(t i), aSlsh. has length c ds= " "X'
(t)- (- y' d( t)- dt .

• \rea of a surface of re".lution The surface generated by
revolving a graph y=I(x) between x=a and x=h about the
x-axis has area 1"2rr.f(x)/I+f'(xl'dx. If the

TAYLOR'S FORMULA

'Ta~lor pol~nomials .

The nth degree Taylo r polynomial of

average of the left sum and right sum fo r the g il'en partition.
T he approx imation re mains val id iff is not positive.
• \lidpoint rul~ T his e va luates the Riemann sum on a
regu lar parti tion with the sampling g ive n by the midpoints

1c2rr.yds= I"

I,

I

.,

.,

2rr.yU) x' (t)- + y' (t)- dl.

PHYSICS

j 'a(u)du, X(t)=X(tIl) + .,"['v(u)du. E.g .. the height X(I)
~

of an object thrown at time 1,,=0 li'om a height x(O) =x"
with a vertical velocity 11(O)=VO undergoes the acceleration
-g due to gravity. Thus v(l) = I'(VII) +
and x(l) = x" +

1(

l' (- u)du

= v,,-I:t

II

VII

-,qu)du = x" + I,,,t-i,ql'.

• \\orL If F(x) is a variable force acting along a line
parametrized by x. the approximate work done over a sIIlali
displacement ~x at x is ~W=F(x)~" (force times
displacement), and the work done over an interval [a.h] is

In a nuid lifting problem, often ~W=~F'''(Y), where

h(y) is the lifting height for the "slab" of fluid at y with
cross-sectional arca A(y) and width ~y. and the slab's
weight is ~F=pA(y)~y , p being the fluid's weight-density.
T hen W= tpA(y)h(y)dy.
-0

1..,
f U' I(c )(x - c)"
n.·

(provided the derivati ves ex ist). When

c=O, it's also called a M a cLaurin polynomial.
'Ta~lor 's I'ormula
Assume I has 11+1 co ntinuous
derivatives on open interval and that c is a point in the
interval. Then lor any x in the interval.f(x)= P,,(x) +R,,(x).

(n ~l!!.1"''' I II(q).

(X- C),,+I for somc

ite m Solution to initial value problem; an example of that
type is in Motion in one dimension. In those, the expression
for the derivative involved only the independent variable. A
basic DE involving the dcpendent variable is y'=ky. A
gcneral DE where only the first-order derivative appears
and is linear in the dcpendent variable is y'+p(l)y =q(t).
Generally more difTicult arc equations in which the
independent variable appears in a Ihlt{ nonlinear} way;
c.g.. .1" = y2 - x . Common in applications are second-order
DEs that are linear in the depcndent variabJe; c.g ..

y"=-ky. x2y"+xy'+x2y=O.
·Solutions. A solution of a DE on an interval is a function
that is dillcrentiable to the order of the DE and satisfies the
equation on the interval. It is a general solution if it
describes virtually all solutions. if not all. A general
solution to an 11th order DE generally involves II constants.
each admitting a range of real values. An initial value
problem (IVP) for anlllh order DE includes 11 specification
of the solution's valuc and II-I (krivativcs at some point.
Generally in applications, an IVP has a unique solution on

the remainde rs fo r the Maclaurin polynomial s of I(x) =
(-1)"
I n (l +x), -1I • X"+I.

(n + 1)(1+;)"

T he re is a

S between

0 a nd x such that In( I + x)

first-ord~r

.

=

x- ,£ + _ _1_ _

111" =

interval :

linl' ar DE . Thc equation y'=ky,

dy

I

(

1-

Eac h

• Slmp,on s rule The weighted sum:{1'1 ,+
, :1211
' I (In the
int erval

[a , h ]

yield s

rule

S impso n's

s= b-;;a (f(a)+4f(a ! h)+.f(b»).

A

T his is also the integral of the Simpson' s ~ ;?
quadra tic that interpo la tcs the
fun ction at the three points. For
a reg ul ar pa rtitio n of [a, h] into

a n even num be r /1= 2", of ~-'---::-:;:""L--'--\:


i!f1'

U
b
interval s, a formula is:
h
til
I
III
1
S2m=::r U (a)+4 ;~/(a )+[ 2i+ 1 ]" + 2 ;~/(a+ 2i'/I )+ fib)}

whe re " = (h - a)llI.

2 3(1+;)'"

"

impson 's r ul e is e\ act 011 cubics.

• . ... ror bounds As x approaches c. the remainder ge ne rall y
becomes smallcr, and a given Taylor po lyno mial provides a
bettcr approximation of the fun ct io n value. With the
assump tio ns and notation above, if !f'''
by

M

on

th e

"(fl~l!! Ix-c l" i I

interval,

II

then

(x)1 is bOlmded
If(x) - P,. (x )1

tor all x in the interva l. E.g .. fo r

~"I<\.

because the thi rd derivative o f eX is bounded by 3 on (-1.1).
• Big 0 notation T he statement f (x)=p(x)+O(x m )
f(x) - p ix) .
(as x .....O) me ans tha t
x '"
IS bounded near x=o.
(Some authors require that the limit of thi s ra tio as x
approac hes 0 ex is!.) That is, f(x)-p (x) approaches 0 at
essentially the same rate as x'" E.g.. Taylor's for mula
has

continuous thi rd derivative on an open interva l containing
O. E.g.. sinx=x +O(x J). [Si m ilar relations can be infe rred
fro m the ide ntit ies in the item Basic MacLaurin Series.]
'I'llupital's rule. This resol ves indeterminate ratio s or

(H or ~). IfFI~f(x)=

0 =

F~g(x)

and if IJt.nJ(x) = 0

= IJ'~: g(x) are defined and glx)"O. ror x ncar a (but not

'1

I

I'

fix)

nccessan y at a), t 1en }~ g(x) =

()

I'

=}~

F( x )
'd d
g' (x) proVI e

the latter li mi t exi sts, or is infin ite . The rule also holds
whe n the limits ofIand g are infinite. No te that/'(a) and
g'(ll ) are not required to exist. To resol ve an in deter m inate
dill"e re nce (00 _ 00) . try to rewrite it as an indcterminate
ratio and apply l ' Hli p ita l's ru le . To resol ve a n
indeterminate exponential (O".loo. oroo") , take its logarithm
to get a product and rewrite this as a suitable indeterminate
ra tio: apply L: II"pita l 's rul e; the expone ntial of the result
resol ves the original indeterminate exponential.
F'or}I~~
r II"
Inlx
l/x
0
x yougctan dfdr
111 Imx-01
/ x L rml.\--«)_l
/x~'
where lim

some interval containing the initial value point.
• Basic

S

(s

between c and x
varying with x). The expression fo r

R ,,(x ) is called the Lagrange form ofthe remainder. E.g ..

implies I (x )=f (O) of'(0)x+!f"(0)X 2+O(X3 ) if I

• F'.. mples. A ditkrcntial equation (DE) was solved in the

I' 2"IIh)h.

summand is the area o f a trapc/oid \vh o~l.· top is the
tangent line segment th ro ugh the midpoin t.

<,x=l+x+x2/ 2, w ith error no more than ftlxl" = 0 .5 Ix !",

W= t'f'(x)dx.

a+

o f ea ch

w he re R,,(x) =

• '\Iotion in one dimen.ion . Suppose a variable
displacement x(1) along a line has velocity v(t)=x'(t) and
acceleration a(l)=v'(t). Since v is 3n antidcrivative of a,
the fundamental theorem implies: v(t) = V(III) +

M, f

Iat c is P,,(x ) = I(c) + f'(c)( x -c) + !!f"(c)(x-C)2 + ... +

I,

../ g enerating curve C is parametri7cd by «x(1), .1'(1».
u:stsb, anti is revolved about the x axis, the area is

x -0

Ixlx =

ell = I.

SEQUENCES
·S~qu~ncc

SequC'llces an: rllnction~ whose domain~
consist of all integers greater than or equal to somc initial
integer, usually 0 or I. The integer in a sequelll"e at /I i<
usually denoted with a subscripted symbol like a" (rather
than with a functional notation a(/I» and is ealkd a term
of the sequence. A sequence is olien referred to with an
expression for its terms. e.g. , 1//1 (with the domain
understood), in lieu of
fuller notation like :

{l / n},;"" ,orn l , l / n(n = l, 2, ... ).
.) I

I~nt.

SCllu,

An arithmetic sequence

un has a

difference d bdwccn slIccc"s ih! \alll e~ :
u,,=a,,_I+d=Uo+d·/I. It is a scqucllIial \(,fsion of n linear

C0l111110n

l[ll1ction. the common diflerence in the mil' of slope. A

geometric sequence, with terms Nfl" has a common ratio r
hetwcen successive values: gn=I:",lr=g"r". e.g.. 5.0. 2.5.
1.25, 0.625. 0.3125 ..... It is a sequential "'r,ion of an
exponential function. the common ratio in the rok of base.
·ConH~rl!: 'nee A sequcnce (un] Co"\'erges ifsuml.! number

L (called the limit) satisfies the j(,lIolVing: l::.Iery £ >0
admits all N such that la,,-LI < £ I'"·ullll " .v. Ifa limit L
e xists, the re is only one: on(' says i u,,: t..:onn~rgc~ to L. and
writes a,,-L, or ,!i!" all = L. If a sequence does not
converge, it diverges. If a seqllclll'L' a" di"'~rgcs in 'lh::h a
way that every M>O admits all N suth that a,, >Jl h,r all
"" N. then one writes a,, -+ oo. E.g .. if Irl < I then r"..... O: if
r= I then rll-+I; otherwise r" diverges. and ifr> 1. r" -x ,

• Boun 'd n unotone

Sl ph Il

;\11 iilcr~a:-.ing

sequence'

that is bounded above converges (to OJ limit less thall or
equal to any hound). This is a fundamental "let about the
real numhcrs, and is basi~ to series convcrgt.:llcc tL·~t S .

dy

rewntten lit =ky suggests y =kdt where lyFkt+c. In
this way, one finds a solution y=CeAt. On any open
interval. every solution must have that form, because

y'=ky implies

1ft (ye

M),

where yr kl is constant on the

interval. Thusy=Ce kl (C real) is the general solution. The
unique solution with y(a)=y" is y=y"ek(l "J. The trivial
solution isy .. O. solving any IVP y(a)=O.
• (.eneral
first-order
line r
DE

Consider
y'+p(t).I' =q(l). The solution to the associated
homogeneous equation h'+p(t)1r =0 (dhlh =-p(t)dt)
with h(a)=\ is hU)=expl- I'p(u)dul.
If .I' is a solution to the original DE. then (ylh)'=qlh.
where

y(t)=Y

y=h fq / h. The

solution

- [q(u)h(u) 'dul.

with

y(<<)=y"

is

NUMERICAL INTEGRATION
• General notes . Solutions to app lied p ro blems often
involve definite integral s tha t cannot be evaluated easily. if
at all. by finding antiderivativcs. Readily available
so ftware using refined algorithms can evaluate many
integrals to lleeded preci sio n. T he fo ll owi ng methods for
approximating l"f(x)dx are elementary. Thro ughout.

II

is

the number of intervals in the underlying regular partition
and Ir=(h-a )/II.
·Trapezoid rule. The line connecting two points on the
graph o f a positive func tion together with the underlying
illterval on the x axis l'lrl11 a trapezoid whose area is the
average of the two func tion values times the length of the
interval. Adding these areas up over a regular partition
gives the trapezoid ru le approximati on

3

SERIES OF REAL NUMBERS
A ~cri cs is a scqucn..:c ohtained by adding the

." 'r

\

values of another sequence L;a" = ao+...+a,. The Hduc
" u

of the serics at N is the sum ofvalucs lip to a,· and is ~3lkd
\.

La" "n+...+a\. The scries itself is
La". The an arC called the terms of the series.
,.

t' A series L:a" converges if the sc:tJucnce or

a partial sum:

II

II

denoted

II

-(00\

II

"

II

partial slims converges. in which case the limit o f the
sequence of partial SUI11S is called the sum "I' the If the series converges. the notation I'.)r the ~eric s it.. e lf
stands also for its sum:

La" =

1/

II

\

lim
.\ .

"

Lna"
.



Series continued

La

An equation suc h as

~

Z

fI

a nd it s

III
.01I1III

"11l1lI

La" may stand for

looks like a p-series. but is not directly comparable to it.

II

form L;a,.". where r is a real number and a" O. A key identity
I)

1/

\
1- .S ,
L;,," = I+r+r2+ +r;\=-_I__ (,.* 1) It implie s
" "
...
1- ,.
.

is

L;,." = _ I _ (ifl ,.l< ll( a lsoL; a l"" =a(_ L - 1)). and

I- r
" ,
I'-I"
that the scries diverges if Irl > I. The serics diverges if
r =± 1. T he l'~Hlvcrge ncc and ross iblc Sum l)f any geometric

series can be determined lIsing thl.' pl"cccding k1J"l11ll1a.
""

r~a l

L:

num ber.

,

"

" 1
IV
are unbollnded: L: Ii ~ 1 + ~ .
1

1/

...


strictly decrease in ahsolutc valuc and approach a limit of
zero, then the series (:onvcrgcs. Moreover. the truncation

error is less than the absolute value of the first omitted

I

L:( - I Y1 a ll - ±( - 1)fl a" I'< a\ f I ,

term :

"

I

I

If

(assuming

,

It:.

• Basic consideration" For any

if

L:a"

converges, then
" A
co nverges. and conversely. If a" 7t H, then
ll

L:a

di verges. (Equivakntly. irL:u lI co nvl'rgcs, then

says nothing about. e.g., L;
"

1 , ;\

(In .....

O). Thi s

series of positive terms is

h II

un inCfl!as ing sequence of parti a l sums; if the sequence of
partial sum s i ~ hounded the ;;eli e~ COI1\crges. This is the
fo undation of all the follow ing criteria for co nvergence.
• Integral te,t & e~timat< Assume I is continuou s,

positive. and decreas ing on (K.oo). Then L;/(II) converges
if and o nl y if

1 /Ix)dx .

" A

converge s. If th e se rie s

J.­ /(x)dx .

\

L; /(II )S L; f(n) +

co nverges, then

1I

ri gh t s ide

A.

A

II

ovcr~stimating

.,

,,\11

II

l..

III

the

-'I., X

len

s ide

underestimating the sum with error less thanfl N+ I).

Integr al test

,
"

".

"

".

f( ·\'+I)

~(II!)"" = ~,

1
"
---;-\ +
1~ln'
12

• Po\\er series A power series in x is a sequence of
\

polynomial s inxofthc Ilmll L;a"x" (N=O.I. 2, ",J,

-

1

1:\

1.2018 .. " an

I,,1
L; a ni S ,,1
L;l a" l.

ll

n

II

A power series inx- c(or "centered at cO' or "about c") is written

()

Replacing x with a real number q in a power series yields
a series of real numbers. A power series converges at q if

Ixl" l \

x",

co nverges, and

2"n'!. _ lxl

,

'~ ) .,. Iun - -, - --" -1-1" = -2 <1 =>l x l<2.
/~I:'''lr /I . 2 n ! (n+])x
which. with the ratio test. shows that the radius of
conve rgence is 2 .
-Geometric po\\"er \eries , A power series determines a
function on its interval of convergence:
x l • f(x)= L;a" (x-c)", One says the series converges

IR,,(l)I= (11+ 1)(11+ ;)"

1 1 . ().
' <; n+

( - I)" ,

so In2= L; - -n- - '

,

11~I .. r

and

R>O

If

f(x)= L;a,,(x- c)"(lx - clII

necessarily the Taylor coetficients: Q,, = f''''(l')/n!. This
means Taylor series may be found other than by direetlv
computing coctficients. Diffe rentiating. the geometri c
(I - x)·

"

'= L;(// + llx"

I

"

(Ix i< n

()

• Ba ic I\l:tcLaurin wri,
_ 1_= I+x+\,2+ = L;x" (lxl

I- x
. . ., 11 II

arctanx=x~ ~+~ - "' = L: ( - J)1/ ~ ~"

,{
~
" "
The li)lIowing hold tor all real x:

(l xj '5 ])


I

2// + 1


.
x~ x: l
" " x"
e" = I + x + ?T+ ",+"'= L.. i
....
oJ,
II
un.
x:!

( - 1)"x:!rI

Xl

cos x = l-x+?T+-4t - " ' = L; -(-,)-),­
......
II II
_n.
•

x: i

x!i

SInx=x - -;l,+~,-"' = "L;
• ,.),
II

(-1)"x:.!"

I


_n + Jl'
,

(?



I)

• Binomial \crie. For P" O. and ti". Ixl < I .
to the tl1l1et,on. The series L;x", i.e., the sequence of
"

u

p(p
(l+x)I' =I+px + -?-,- - I) x'+ .. , = L; (") x·,

\

1- V i '
L;x" =1 +X+X2+ ... +X\= ·~(x * 1).

polynomials

1/

X

()

converges for x in the interval (-1, I) to 1/( I-x) and
II

II

~X

L;2·;{ "x" = 2~i=(~)"=2x. _ ~,_.

"
II
3" t) 3
3 1- x / .{

for

f (x) = L;(I" t.t:-c)" ,

/I

J;.",

The binomial coefficients are

P(~- l),

(")=
k

()

II

(]=l. C)=p,

(J =


and I"p choose k ")

P(p-1)(p-2!'" (p-k+ 1)


k.

Ifp is a positive integer,

C~O Illr k >p.
A ll rig ht!l "'·_""rI CII. 'f' part 01 till'
rn.l~ b..: rerro.lu('cu or
l(':1n~l1l1t\o:dl!\nn ... fom l ,orInJn)~,
c lcdr'(lI1IC or II1cchalm:al. n1<:ILlJIn~
[lhuIOCOP).rcc:orJUll:,l.oran\ InfOlll\lI1tl~tOr.lgC

(I

a,,£b,,(II ~ N)

derivativc there is f' (x)= L;na" (x-c)" "

"

,

The differentiated series has radius of convergence R, but
may diverge at a n endpoint where the original converged.
Such a tl111etion is integrable on (e - R. e + R). and its

integral vani shing at cis:

converges. but not abso lt1!cly.
·(ompari ~m te\1 Assll m~ u".b,,>O.
has a limit, thell L;a" converges.

•

If e=O. it is also ca lled a Maclaurin series. The Ta)lor
series at x may cOJl\'ergc without converging to f(x). It
converges to fix) if the remainder in Tay lor's f(mnu la.

series g ives __1_ ., = L;nx"

·Intenal ofcomerJ.:ence The set ufrealnu1l1bcrs at which
a power series converges is an interval , called the interval
of convergence. or a point. If the power series is centered
at c. this set is either (i) (-00,00); (ii) (c-R ,e+R) lor some
R>O. possibly together with one or both endpoints; ur (iii)
the point (' alonc. In case (ii), R is called the radius of
convergence 01' the power series, which may be 00 and 0
t(, r cases (i) and (ii i). respectively. Convergence is absolute
for Ix- cl < R. You can often determine a radius of
convergence by solving the inequality that puts the ratio
(or root) test limit les s than
I. E.g. , for
~

1"( ,)

,)

,

11

L;a" (x-c)" =all +a, (x - c) +a, (x - c)' + "'.

"

A serics converges conditionally if it

-If L;b" converges and either

I)

·Computinl!

Such a function ·is differentiahle on (e-R ,c+R). and its

comcrgence I r L;ia" i converges, that is, if

L:a

.. .f k '(

L; T(x-cl h = f(c)+f'(c)(x- e )+ T ( x - c)'+ .. ·.
It

Jlublica ri o!l

4.

tcol)vergcs absolutciy: , then

argument (see below) implies equality for x=1.
• Taylor and \lac! durin 'ric The Taylor series
about e of an infinitely diiTercntiable functionIis

"

Lanx".

The power series is denoted

n

'~' for Ixl<1; a remainder

,

remainders at x=1 tllr the Maelaurin polynomial s ot
Taylor's
formula
above) satisfy
In( I +x) (in

Ix/31< I. The interval of convergence is (- 3,3).
·(alculus or po\\er series. Consider a function given by a
power series centered at c with radius of convergence R:

X>

The ini tial (geometric) series converges on (-1.1). and the
integrated serics converges on (1,-1). The integration says

R (x)= __I- {'" I II('')·(X - c)''' 11 (I: between c and \'
"
(//+1l!'"
.,
.,
1; varying with x and II), approaches 0 as " .... 00. E.g.. the

POWER SERIES

E.g.,

.'\'+1

.'V

' 1
"
---;-\
1~l n '

iU •Absolute

"11l1lI

root test: ,!imn"' " = I (any p) and ,!im(II!)'" = = , More

diverges otherwise. That is. L; x" = -1 1 (lx l
1 -X"-;-\dx
. "..
Z underestimate
with error < 13.1<5.10

~

then

geometric series may be identified through this basic one.

K

III

,Jiml.~~ >I()r ,!i!" lan l' /" >1.

....

ll

converges

geometric serics. The following are useful in applying the

the slim with error less than

J _ "1
. 1
_
L;" - L; ,, +
" dx - 1.2011l .. "

Eg

L:u"

L;a" diverges. These tests arc derived by comparison with

"

the

2
E"
."'., _1_
l+x = I-x+x 2 "'illll,li"c .s_1_
l+x =I-l:+t·
. . .. , ,

"

the resulting series of real numbers converges.

0 ill a s trictly Llecreasing manner).

CONVERGENCE TESTS

D.

If




, Thcse are series \\!hosc terms alternate
in (nonzero) sign. If the term s of an alternating series

La

(ahsolutely).

T he integrated series has radius o f convcrgence R. and may
converge at an endpoint where the original diverged.

In(I+x)=L;(-J)"

M

,, "'· 0

'\eri~

• \lternafin

tl

11n

is ca lled the I)-series.

II)

I~ diverges. li,r the partial

be low ). T he harmonic se ries L;

La
,

1'1 a tl 1' /" < 1 , then
If }!"
I· I~an

I II

The /I-se ries diverges if ps i and converges if p>1 (hy
comparison with harmonic sl'ri~s and the inh:graJ lest,

"

n - ~'

h

precisely. ,!im

If

(11/ .....

"

• Ratio & 1'00t tests. Assume an" O.

1.1

• p-scn', For P , a

SUIllS

V,.
I "
.
. .
'
sin(I / II')
E.g.. L;SlIl (1 / n-) converges smee Hm
.., = 1.

A (numerical) geometric series has the

II

__

The p-series and geometric scries are otien used tl".
comparisons. Try a "limit" comparison when a series

E.~ .. L;I :f,, =4(1 .! ,<,- 1)=2.

~

-lfL;b" diverges and either b,,£an (1I ~ N) or a,,Ib n has
a 110112('/'0 limit (or approaches 00), then L;a" diverges.

L;a ,, =S.
/I

...

..-

== S means the series conve rges

isS. In general statemcnts,

SlIlll

iU •Geometric series
A

/I

4)

or a"lb"

f /W dt= L; _a+
-"--I (x - cY' ' (ix-c l< R).
C

1/

lin

4

and

rClnl"\'31 s~)I("m ..... Uhvul

\HllIen rcrnli"'10n frolll the rubh.JlcT
f\ 2Ui.1I-Z007 UnCh ll r U.lnc. 0 108
~u lt, : Due IU lIS condensed
fqfTnQI,
plc,uc II!>C thl

\)uid.)tudv

nOI h

a~ II

111,111.1.... bUl

a rep lacement for

"" lgncJ ,· l.t$S\\ ' Ir\;

U,S, $4,95 CAN. $7.50
Author: Gerald Harnet. PhD
Customer Hotline # 1,800,230,9522

ISBN-13' 978-157222475-9
ISBN - 1D: 157222475-4

9 1 ~ lll)llli ~1I1!1!1!IJIJ~l l l rl l lil l l l