Calculus Equations and Answers qs

Prévia do material em texto

Essential Tools for Understanding Calculus - Rules, Concepts, Variables, Equations, 
Examples, j) Helpful Hints & Lh Common Pitfalls 
STRATEGY FOR SOLVING 
PROBLEMS EFFECTIVELY 
I. Understand the principle (business or scientific) 
required. 
II.j) Develop a mathematical strategy. 
A. There are eight useful steps that will help you 
develop the correct strategy. 
I. Sketch, diagram or chart the relationships and 
information that is subject of the problem. 
2. Identify 	 all relevant variables, concepts and 
constants. 
3. Describe the problem situations using appropriate 
mathematical relationships, functions, formulas, 
equations or graphs. 
4. Collect all essential information and data. 
S. Lh ~ extra and unnecessarY information 
and data. 
6. Derive a 	mathematical expression or statement 
for the problem, making sure all measurements 
are in' the correct unit. 
7. Complete the appropriate mathematical 
manipulations and solution techniques. 
8. Check the final answer by using the original 
problem and information to make certain that the 
answers, units, signs, magnitudes, etc., all make 
sense and are correct! 
FUNCTIONS 
I. Definitions 
A. A relation is a set oforder pairs; written (x,y) or (x, 
fix»~. 
B. A function is a relation that has x-values that are all 
different for differenty-values . A vertical line test 
can be used to determine a function; every vertical 
line intersects the graph, at most, once. 
C. A 	one-to-one function is a function that has y­
values that are all different for different x-values. 
A horizontal line test can be used to determine a 
one-to-one function; every horizontal line intersects 
the graph, at most, once. 
D. Domain is the set of all x-values of a relation. 
E. Range is the set of all y-values of a relation. 
F. A function is an even function iff(- x) = fix). 
G.A function is an odd function iff(- x) = -f(x). 
H. The one-to-one functions f(x) and g(x) are inverse 
functions iff(g(x» = g(f(x» =x;f '(x) and g-'(x) 
indicate the inverse functions of fix) and g(x), 
respectively. Inverse functions are reflections over 
the line graph ofy = x . 
I. 	 Dependent variable is the output variable in an 
equation and depends on or is determined by the 
input variable. 
1. 	 Independent variable is the input variable in an 
equation. 
II. Common Function Summary 
A. Linear:f(x) = mx + b 
I. m is the slope; 
m =Y2 - y, = y, - Y2 = ~y = rise 
x 2 -x, x, -x2 ~ run 
2. b is the y-intercept. 
3.lt 	is a constant function when m = 0; it is a 
horizontal line. 
B. Absolute value:f(x) = a~ - hi + k 
I. (h, k) is the vertex. 
2. If a> 0, the graph opens up. 
3. If a < 0, the graph opens down. 
4.±aare the slopes of the two sides of the graph. 
C. Square root: f(xl=a.Jx-h +k 
I. (h, k) is the endpoint. 
2. f2 If a> 0, the graph goes to the right. 
3. p 	If a < 0, the graph goes to the left. 
D. Polynomial: f(x) = a"x' + a~,x~' +...+ a,x + ao 
1. ao is the y-intercept. 
2. There are, at most, n zeros or x-intercepts where 
f(x) = o. 
3. There are, at most, n - 1 points ofchange or turns 
in the graph. 
4. j) The extreme left and right sections of the 
graph both go up or both go down if n is even, 
and go in opposite directions if n is odd. 
E. Quadratic:f(x) = a(x - h)2 + k 
I. This is a special case of a polynomial. 
2. (h, k) is the vertex. 
3. If a> 0, the graph opens up. 
4. If a < 0, the graph opens down. 
S. Use quadratic formula to solve for the zeros or 
x-intercepts: X= -b+~ . 
() 2a 
F. Rational: f(x) = p x 
qCx) 
l.p(x) and q(x) are polynomials, and q(x) f. o. 
2. If degree p(x) < degree q(x), the asymptote is the 
x-axis wherey = O. 
3. If degree p(x) = degree q(x), the asymptote is y = 
(lead coefficient ofp(x»/(lead coefficient ofq(x» . 
4. If degree p(x) > degree q(x), the asymptote is y = 
(the quotient ofp(x).,. q(x»; a diagonal asymptote. 
G. Exponential:f(x) = ax 
I a>Oandaf.]. 
2. If a > 1, the function is increasing. 
3. If a < 1, the function is decreasing. 
4. Rules for exponents: 
a. x'" • x" = x"'" 
h. XIII =x",-n 
xn 
c. (x'" r =x",n 
d 	 - ", __1_ 
.X 	 - x'" 
e. _1_= x"' 
x - ", 
f. (xy)"' =x"'y"' 
g'(;T=~: 
h.x~ =~=(!!/Xr 
i . ~=!!/X.~ 
j . ~=~ 
k.~="'!!/X 
I. Ifx' =x', then a = b. 
m. Ifax = bx , then a = b if a f. O. 
H. Logarithmic:f(x) = log.x 
I.x>O 
2. a> 0 and a f. 1. 
3. Lhf(x) = log.x, IF and only IF. aftx )= x. 
4. a is the base. 
S. Logarithms are exponents. 
6. Rules for logarithms: 
a. log.I = 0 
b.log.a = I 
c.log.ax=x 
d. a'og.x = x 
e. Iflog.x = log.y, then x =y . 
f. f(x) = log, x = In x; this is the natural log. 
log. x logx Inx 
g.log. X= 10gb a = loga =III/l; 
r-----------------~ 
this is the change-of-base rule. 
h. log.xy = log.x + log.y 
i.log.y=log.x-log.y 
j. log.xY = ylog.x 
I.Trigonometric 
1. Basics 
a. j) Angles can be measured in degrees and 
radians. 
i. 	 I radian = (] !O) degrees 
ii . I degree = (] ;0) radians 
iii.proportion conversion of angle measurements: 
angle 	in degrees angle in radians 
180 0 It radians 
iv. unit circle 
(a) center at (0, 0) 
(b) radius = one unit 
(c) points on the circle = p(x,y) 
(d) j) Positive angles move counterclockwise ~ 
from P(I, 0). 
(e) j) Negative angles move clockwise from 0 
P(I,O). 
(f) 	j)Angles rotating one or more full times ~ 
require adding ±21t for each rotation. ~ 
v. 	function definitions III 
(a)sinS=y Z 
(b) cos S =x 
(c) tanS=~ 	 ~ 
(d) cscS=.!. y 
(e) secS=.!. 
x 
(f) cotS=,,!. y 
vi . j) useful values 
8 = degrees; t = radians',{) = undefined 
8 0 30 4S 60 90 180 
t 0 .n. 6 
.n. 
4 
.n. 
3 
.n. 
2 
It 
sin 0 l 
2 
J2 
2 
.J3 
2 
1 0 
cos I .J3 
2 
J2 
2 
l 
2 
0 - ] 
.J3tan 0 1 
.J3 {) 0 3 
2. Graphing properties 
a. Amplitude of sine and cosine is half the 
difference between the maximum and tht! 
minimum values, or lal. ~ 
b. Period is the radians needed to complete one 
full cycle of the curve, or 2: . 
c. Horizontal shift or phase shift is c. " 
d. Vertical shift or average value is d. W 
e. Sine:f(x) =aslnb(x- c) + d m 
f. Cosine:f(x) = acosb(x - c) + d 
g. Tangent:f(x) = atanb(x - c) + d ZLh [CAUTION! Tangent has no amplitude. so 
a affects the vertical stretch and shrink only.] ~ 
h. Cosine is even; sine & tangent are odd. 
3. Important identities & formulas 
1 
0 
Functions {Sontinued) 
a. Pythagorean identities 
i. 	 sin' u + cos' U = 1 
ii.l + tan'u = sec'u 
iii.l + cot'u = csc'u 
b. Sum/difference formulas 
i. 	 sin(u ± v) = sinu cosv ± cosu sinv 
ii. cos(u ± v) = cosu COSV 'f sinu sinv 
iii tan(u+v)= tanu+tanv 
. 	 - 1 'f tan u tanv 
c. Half-angle formulas 
1. sini=±p-~osu 
ii . cosi=±~I+~OSU 
iii. tan.!!. = 	I-cosu =~ 
2 sinu l+cosu 
d. Double-angle formulas 
i. 	 sin2u = 2sinu cosu 
ii. cos2u = cos' u - sin'u = 1-2sin'u = 2cos' u - 1 
iii.tan2u= 2tanu 
I-tan' u 
e. Power-reducing formulas 
1. 	 sin' u l-cos2u 
2 
ii. 	cos' U= l+cos2u 
2 
iii. tan' U= l-cos2u 
1+cos2u 
Ill. Basic Common Function Graphs 
A.Cons*<lnt:j(x) = c F. Cubic:f(x) =x' 
B. Linear identity:j(x) =x G.Rational: j(x)=~ 
C. Absolute value: H. Exponential: 
j(x) = Ix! j(x) =eX 
D.Square root: I. Logarithmic: 
j(xb-.JX j(x)=lnx 
E. Quadratic function: J. Sine: 
f(x) =x' j(x) = sinx 
y 
f-- 2+--+-+-f-IH 
1'\--­ I t-:r ...... "+-t---+:v.:,,,x 
K.Cosine:f(x) = cosx 
yI I I 
I I I 
I r If j x
" j 2" 
-	
-I 
=+ 
"'~ 
-
-2 I l 
I I 
M.General Transformations 
1. When given the function f(x) and the number a, 
then the function: 
f(x) ± a has a vertical shift up for +a; down for-a. 
j(x ± a) has a horizontal shift right for -a; left for +a. 
aj(x) has verticalstretch if a> 1; vertical shrink if 
0< a < 1; x-axis reflection if a is negative. 
j(ax) has horizontal shrink ifa> 1; horizontal stretch 
ifO < a < 1; y-axis reflection if a is negative. 
LIMITS & CONTINUITY 
I. Definitions 
A. A is the limit of f(x) as x approaches a, written 
~!T. j(x) = A; means that for each neighborhood 
~A (neighborhood of A with radius p), there exists 
a punctured neighborhood N,a such that f(N,a) is a 
subset of~A when N,a is a subset ofthe domain off. 
B. ~!T. j(x) = A if for every E > 0, there exists Ii > ° 
such that If(x) - AI < E when Ix-aI<Ii, assumingj(x) 
is defined for all x in an open interval containing a. 
C. Geometrically, ~i,"! j (x) is the y-value of the point in 
which the graph ofj should intersect the line x = a. 
D. (l Generally, lim j(x) = A implies thatf(x) comes 
/-' 	 X-HI 
infinitely close to A as x gets infinitely close to a. 
E. Limits 	 can equal +00 or -00, usually as j(x) 
approaches an asymptote. 
F. (l Informally consider that the number limj(x)~ 	 x~a 
is the number that is approximated by j(x) when x 
is close to, but not equal to, a. 
G.One-sided limits 
1. A left-sided limit equals L, written 
lim j(x) = lim j(xl= L, ifj(x) gets close to L 
X~fI- xi" 
as x gets close to a from the left; that is, x gets 
close to a but remains less than a. 
2. A right-sided limit equals R, written 
lim j(x)=limj(x)=R, ifj(x) gets close to R 
X-)Q+ xJ,Q 
as x gets close to a from the right; that is, x gets 
close to a but remains greater than a. 
3. l' limj(x) exists and limj(x)=A, lE..AWl ~ x-). X-)(I 
H. The functionjis continuous at the point a ifj(a) 
is defined and t!T. j(xl= j(a). 
1. 	 The functionjis continuous iffor every E > 0, there 
exists Ii > °such that for x andy in the domain ofj 
when Ix - yl < Ii, then If(x) - j(Y)1 < E. 
II. Theorems 
A.lim[j(xl±g(x)]=limj(xl±limg(x) 
X-HI X-)Q X-)Q 
B. If function g is continuous at point A and 
limj(xl=A, then limg{J(x») = g(limj(x»).
X-)(I X-)Q X-)Q 
C. lim[j(x). g(x)]=(limj(x»)(limg(x»)
X_CHI X----)(I X-HI 
j(x) lim j(x) 
D lim--=~ provided g(x "# 0) and 
.X-->. g(x) limg(x) ' 
X-->. 
limg(x) "" 0. 
E. ;:{J(x»)" =(limj(x)"), provided n is a positive 
X----)Q X-)Q 
integer. 
F. Iimj(x)=A is equivalent to lim[j(x)-A]=O.
x----). 	 X----)Q 
G.lf j(x) < g(x) < hex) for every x in a punctured 
neighborhood of a (that is, x near a), and 
limj(x)= IimhCxl= A, then limg(x)=A.
X-HI x--+a X.....,IIf 
2 
H.~Forexample, when finding lim x 3 -28 such thaI 0/' 	 x-+l x­
x 3 
, x-2 x-2' 
x"# 2 - 8 becomes (x - 2)(x2 + 2x + 4) and then 
(x' + 2x + 4); consequently, when x is close to 2, 
(xl + 2x + 4) is close to 12; therefore, lim xl-: = 12. 
III. Rules 	 x----)2 x-
A. For polynomial 	p(x) to the n'h power with 
the lead term of ax" and polynomial Q(x) to 
the m'h power with the lead term of bX"', if 
p(x) 
x =Q(x) and Q(x) "# 0, then when: j() 
l.n=m, the limj(xl=![ 
x----)oo b 
2. n > m, the lim j(x) = lim P «x» 
X--,)ooo X----) <XI Q x 
3.n<m,the limj(x)=O 
X-->~ 
B. lim c = c, when c is a constant. x-.. 
cr 	eX_I=1 ·x~ 	 x 
D.lim x: =0 
X----)<><> e 
E. lim(X+l)X=e 
X----) OO 	 X 
F 	 lim aX -I = Ina 
• X-+O X 
G. lim sinx = I 
x--+o x 
H.limcosx-l =0 
x----)o 	 x 
I. 	 lim tanx =1 
X~O x 
DERIVATIVES 
1. Definitions 
A. If x is a number in the domain of a function, /. 
j(x+h)- j(x)( ) then the derivative f' x = lim , 
provided the limit exists. h-.O h 
I. If the limit exists at x; thenjis differentiable at x; 
2. j) Not all continuous functions are 
differentiable. 
B. If a function,/. is differentiable at point a, then the 
tangent line to the graph ofj at the point (a, /{a» 
j '() 'th j'() I' j(a + h)- j(a)has sIope a WI II = 1m . 
X~. h 
I. This tangent line is unique. 
2. This derivative is the instantaneous rate of 
change of j; it tells how fast j is increasing or 
decreasing with respect to x as x approaches a; 
that is, near x =a. 
3. rP For example, when y==.J4-x2 , the 
slope of the tangent line to the curve at point 
(x,y) is -;; therefore, the derivative of y is 
y'=f'Cx)=-~=~ . 
y .J4-x' 
y 
-2 -\ 
C. The first derivative function 	notations include 
f'(x), y', dy, DxY, and ..!L j(x); second 
dx . fix. • d1y 
derivative notations mclude j (x), y, dX' 
and D;y. 
D. The derivative atx = a is usually written as: 
/'(11), D(f)(II), or 1;L. 
2 
II. RuleslFormulas 
A. Assume fix) and g(x) are differentiable functions, 
D,f(x) = :x lex) = f'(x), and D.g(x) = ! g(x) = 
g'(x) for the following statements: 
1. L'Hopital's Rule: If/and g are differentiable 
for x near a and Iim/(x)=limg(x)=O; or, 
X~1l X--+Q 
lim/(x)=limg(x):±oo and g'(x) t- 0, then 
X--+d' x--+« 
lim lex) = lim f'(x) . 
x~. g(x) x~. g'(x) 
2. Chain Rule: Ifll =g(x), thenD,f(II) =DJ(II)D.II; 
dy dy dll 
or, D.y=D.yD.u; or, dx = dll dx wheny=f(II). 
3. RoUe's Theorem: If/is continuous in the closed 
interval [a, b] and iflea) = feb), then there is at 
least one point m in the open interval (a, b) such 
thatf'(m) = O. 
4. Mean Value Theorem: If/is continuous in the 
closed interval [a, b], then there is a point m in the 
. /(b)- lea) ()
open mterval (a, b) such that b-a f' m . 
5. Lmg(x)=mg'(x) for all real numbers m. 
dx 
6. L{J(xhg(x») = f'(xhg'(x) 
dx 
7. L[/(x)g(x)j =lex) g'(x)+g(x) f'(x) 
dx 
8.L[/(X)]= g(x) f'(x)- /(x)g'(x) for g(x}t-O. 
dx g(x) [g(x)jl 
9. Lc=O, when c is a constant. 
dx . 
10.Lx=1 
dx 
II. L(l)=_...Ldx x Xl 
12.L(mx+b)=m' for all real numbers m. 
dx 
13. L(xt'= nx"-I, when n is a real number; n t- 0, 
dx 
x..-1 is defined. 
14. L £ = I, 
dx 2vx 
15. L JI (x) I ; derivative of an 
dx f'{j-I (x») 
inverse function whenf'if-l(x» t- o. 
16. L r=r 
dx 
17.L~ =~Ina 
dx 
18. L lnx=ldx x 
19.A... log x=_I_ 
dx • xlna 
20. L(sinx) = cosx 
dx 
21. L(cosx)=-sinx
dx 
22. L (tanx)=secl x 
dx 
23. L (cotx)=-cscl x 
dx 
24. L(secx) = secx' tanx 
dx 
25. L(cscx)= -cscX'cotx 
dx 
26. L(arcsinx) = ,....!-.,
dx vI_Xl 
27.L(arccosx)=- ,....!-.,
dx vI-xl 
28. L(arctanx) = _1_ 
dx l+xl 
29. L (arccotx)= __I-
dx I+x 
30. L (arcsecx)= ~ 
dx x xl-l 
31. L(arccscx) 1 
dx x.Jxl-l 
III. Applications 
A.lmplicit Differentiation 
1. Used when it is difficult or undesirable to solve 
an equation for y, such as x' + y' = 1. 
2. Differentiate 	both sides of the equation with 
respect to x. 
3. Apply the Chain Rule. 
4. Substitute y' for : and 1 for ::; . 
5. Solve for y '. 
6. ~ For example, when finding the derivative 
of y= Jx~\ f(;)[;(X] - 1)with 	 and 
g(x) .Jx+l' dx g(x) becomes 
L[ (x-I)]= .Jx+IDx (x-l)-(x-I) Dx .Jx+I ; 
dx .Jx+I [.Jx+lf 
then, using the powerformula to find Dx .Jx + I , 
.Jx+l (1)- (x-I) 
2.Jx+lthe statement becomes 
x+1 
finally, using algebra to simplify the expression, the 
derivative becomes x+3 or (x+3),Jx+t 
B. Graphs 2(x+ If. 2(x+ 1)1 . 
I. Increasing/decreasing 
a. A function/ is increasing in an interval (a, b) 
iff(a) <feb) whenever a < b. 
b. A function/ is decreasing in an interval (a, b) 
if/Cal >feb) whenever a < b. 
c. If/ is continuous and/'(x) > 0 at every point 
of an open interval (a, b), then/ is increasing 
in this interval. 
d. If/ is continuous andf' (x) < 0 at every point 
of an open interval (a, b), then/ is decreasing 
in this interval. 
e. Considering a point traveling left to right along a 
curve off, ifthe point goes up in any interval of 
the curve, then/ is increasing in that interval; if 
the point goes down in any interval ofthe curve, 
then / is decreasing in that interval. 
2. Concavity 
a. A curve or part of a curve is concave up if the 
curve lies above the lines thatare tangent to the 
points on the curve. 
b. A curve or part of a curve is concave down if 
the curve lies below the lines that are tangent 
to the points on the curve. 
c. Ifj" (x) > 0 at every point in an interval, then 
the graph off{x) is concave up in this interval. 
d.lfj"(x) < 0 at every point in an interval, then the 
graph of/ex) is concave down in this interval. 
3. Inflection point 
a. If / is differentiable in a right and in a left 
interval or neighborhood ofany point a at which 
the graph of/is continuous, and ifj" is positive 
for all values in one ofthe intervals but negative 
for all values in the other interval, then (a,f(a» 
is a point of inflection of the graph off. 
4. Maximum/minimum 
a. Point (a,f{a» is a relative or local minimum 
pOint of any interval of the graph of/if/Cal < 
fix) for any x in this interval; the number lea) 
is the minimum value. 
b. Point (a,f(a» is a relative or local maximum 
point of any interval of the graph of/if/Cal > 
lex) for any x in this interval; the number lea) 
is the maximum value. 
c. The global or absolute minimum is the point 
that has the leastf{x) value in the domain. 
d. The global or absolute maximum is the point 
that has the greatest/ex) value in the domain. 
e. Extreme Value Theorem: If/is a continuous 
function on a closed interval [a, b], then/ has 
a maximum and a minimum; and, the global or 
absolute maximum and minimum occur only 
at critical points or endpoints. 
f. 	A critical point, (x,f(x», is a point ofthe graph 
of/ that satisfies one of these conditions: 
i. f'(x) = 0 
ii.f'(x) does not exist; OR 
iii.(x,f{x» is an endpoint of the graph. 
iv.~YFor example, on the graph ofy = lex), 
at the relative maximum point P and the 
relative minimum point Q, the curve has a 
horizontal tangent as it also does at point R, 
which is neither a maximum nor a minimum 
point; additionally, if the search for 
maximum and minimum points is limited 
to those points whose x-coordinates satisfy 
rex) = 0, then the maximum point S and the 
minimum point T. which is an endpoint, will 
be missed, and these are all critical poipts. 
y s 
-L 
( 
I 
p 
"\. 
R 
./f" 
./ 
/ 
- I- - ­
Q \ 
N \ T 
x 
I 
g. P A maximum point or a minimum point 
must be a critical point, but critical points need 
not be maximum points or minimum points. 
h. If / is differentiable in an open interval that 
contains point a. such thatr(a) = 0, then: 
i. f{a) is a maximum value off, ifj"(a) < 0; 
AND 
ii.f(a) is a minimum value of/ ifj"(a) > O. 
6 [CAUTION! This test does not apply if 
f"(a) = 0.] 
C. P Helpful Hints for Sketching a Curve 
I . Determine the domain for the function,f{x). 
2. Analyze all points where lex) is not continuous. 
3. Sketch 	 all vertical, horizontal and oblique 
asymptotes, if there are any. 
4. Evaluater(x) andj"(x). 
5. Find and plot all critical points, a, where f'(a) 
does not exist or wherer(a) = O. 
6. Find and plot all relative maximum and all 
relative minimum points. 
7. Find and plot all possible inflection points, b, 
wherej"(b) does not exist or wherej"(b) = O. 
8. Find and plot the x-intercepts and the y-intercepts, 
if there are any. 
9. Complete the sketch of the curve. 
D. Rate ofChange 
I. Average rate of change of/over the interval [a,x]: 
/(x)- lea)Ia. s x a . 
b. As x approaches a. the average rate of change 
approachesr(a). 
c. It is the slope of the line containing the 
endpoints of the interval. 
2. Instantaneous rate of change off; 
a.lsr(a)whenx=a. 
b. It is the slope of the unique line tangent to the 
graph of/ at point a. 
c. It measures how fast/ increases or decreases at 
point a. 
d. Instantaneous velocity is 1'(1), where s is the 
position, s = let), and I is time. 
e. Instantaneous acceleration isr(I), where" is 
velocity, " = f(1), and I is time. 
3 
INTEGRATION 
I. Area Under a Curve 
A. If a function, f(x), is a curve graphed in the 
interval [a, b], then the area bounded by 
the curve, the x-axis, and the vertical lines 
containing the endpoints of the interval [a, b] 
may be approximated through the following: 
I. Rectangular methods 
a. Divide the interval 	 into rectangles with 
equal width of b;; a . 
b. This results in n + 1 points on the x-axis. 
. b-a 
c. These pomts are Xo = a, XI = a+ --;;- , 
x 2 =a+2(b~a), .•• ,xo =a+n( b~a). 
d. Find the sum of these n rectangles of equal 
width in the bounded region. 
e. Left-endpolnt method 
i. 	The height of each rectangle is the 
vertical left side. 
ii. The height of each rectangle is f(x,) for 
o:s i:S (n - 1). 
iii .The sum of these rectangle areas is b;;a 
[f(Xt) + f(x.) + f(X2) +,..+ f(x_.)] = 
b n-I 
;a Lf(xJ 
;=0 
f. Right-endpolnt method 
i. 	 The height of each rectangle is the 
vertical right side. 
ii. The height of each rectangle is f(xj) for 
1 :s i:S n. 
g. The sum of these rectangle areas is b-a 
o n 
[f(x.) + f(X2) +,..+ f(x.)] = b~a L f(xj) . 
h. Midpoint method ,~I 
i. The height of each rectangle is the 
'vertical line segment from the midpoint 
of the rectangle base to the midpoint of 
the opposite side. 
ii . The height of each rectangle is 
f( Xj +2Xj+1 ) for O:S i:S (n -I). 
iii.The sum of these rectangle areas is 
b:a [f( Xo ;XI )+f( xI :x2 )+,..+ 
( X0-1 +x)]=-=.!!b 0-1L f )f 	 (x-1...........l:+x
0 
2 n j~O 2 
0- 1 (x. +x. ) b
or!'J.x L f -'__1+_1 where !'J.x = ;a. 
j~O 2 
y I 1 I 
JL , ~ 
, ,1/ , 
, , , \ 
, , ,IL , , , :'\, , ,j , , , , , 
, , , , , ,
,, , , , , x 
a l x l x2 l x, 1 X4 b 
I I I 
i. Riemann sums 
i. 	 The height of each rectangle is f(c,), 
where Cj is any point in each subinterval, 
[Xj, Xj+l], for 0 :s i:S (n - I). 
o 
ii. Riemann sum is Lf(c,)(x, - XH) or 
o j~1 b 
!'J.xLf(c,), where /j.x=(Xj-Xi-\) = ;a. 
;=1 
2. Trapezoid method 
a. Divide the total 	 interval into trapezoids 
with parallel sides of equal subinterval 
b-alengths, --,,-- . 
b. This results in n + 1 points on the x-axis. 
. 	 b-ahc. T ese pomts are x. = a, XI =a+--,,-, 
x 2 =a+2(b-a) x =a+n(b-a)n , ... , n n ' 
d. The area of each trapezoid is the average 
of the vertical left and right (parallel) sides 
multiplied by the horizontal subinterval 
length (distance between them). 
e. The average of the two parallel sides is 
f(xj)+ f(xj+l ) forO :S i:S(n-l), 
2 
f. The sum of these trapezoid areas is 
b-;,a [f(xo); f(x\) + f(xl ); f(x2 ) + ... + 
f(xn-I~+ f(xo)] = 
b [0 0-1 ]2-na j~f(xj)+j~f(x,) . 
3. Simpson's Parabolic Rule 
a. The interval [a, b] is divided into 
subintervals of length, b-a, where n is a 
positive even integer. n 
b. This results in n + 1 points on the x-axis. 
. 	 b-aThc. ese pomts are x. = a. XI =a+--,,-. 
x 2 =a+2(b;a), .. . , xo=a+n(b;a) . 
d. Every three consecutive points, (Xj, f(xj)). 
on the curve are also points on a parabola 
when 0 < i<n. 
e. The su~ of these parabola areas is b3-na 
(f(xo) + 4f(x.) + 2f(X2) + 4f(x,) +.. ..+ 
2f(x_2) + 4f(x_.) +f(x.)]. 
f. 	j) [NOTICE: Inside the brackets, the 
coefficients of f(xo) and f(x.) are both 1, 
but the coefficients of the f(xj)'s with odd 
subscripts are all 4, and the coefficients of 
thef(x,),s with even SUbscripts are all 2.] 
II. Definite Integral 
A. If the limit exists, then the function has an 
integral on the interval [a, b]. 
B. J is the integral sign. 
C.f(x) is the integrand. 
D.a is the lower limit of integration. 
E. b is the upper limit of integration. 
1. Using Riemann sums [see bottom left ofpage]. 
f: f(x)dx= Iim!'J.xi:, f(cJ 
n--+.... ;=1 
2. Using the trapezoid method, 
s: f(x)dx =!~'?o t; [j~ f(xj)+:f f(x j )1 
3. Using the Simpson's Parabolic Rule, 
fb f(x)dx= lim !'J.x [((Xt) + 4f(x.) + 2f(X2) 
fI n --+co 3 
+ 4f(x,) +...+ 2f(x_2) + 4f(x_tl+ f(x.)]. 
4. fb f(x)dx is the net signed area between the 
c~rve and the x-axis. 
a. When this area is 	above the x-axis, it is 
considered to be ~. 
b. When this area is below the x-axis, it is 
considered to be ~. 
c. @JForexample, when a is a positive number, 
the curve for y = .Ja2 - x 2 is a semicircle with 
a radius ofaunitsanda centerof(O, 0); therefore, 
in the interval [-iI. a], this semicircle and thex­
axis create a bounded region having an area of 
2 1t~ square units, so the integral orone-halfof 
this region is f: .Ja2 _x2 dx= 1t: . 
Ill. Indefinite Integral 
A.F(x) is called an antiderivatlve off(x) if r(x) 
=f(x). 
B. There is a family 	of antiderivatives F(x) + C, 
where C is a constant, because all such functions 
have the same derivative. 
C. & Jf(x~ = F(x) + C. IF and onlv IF, r(x) 
=f(x). 
IV. Fundamental Theorem of Calculus 
A. If f(x) is an integrable continuous function on 
the interval [a, b], and if F(x) is a continuous 
function in [a, b] such that rex) = fix), then 
f: f(x)dx= F(x)l! =F(b)- F(a). 
4 
B. Iff(x) is a continuous function on the interval 
[a. b]. then the function F(x)= f: f(t)dt is 
an antiderivative off(x) on [a. b]. 
V. Mean Value Theorem for Integrals 
A.If f(x) is a continuous function on the 
interval [a, b], then there is a point m in the 
interval such that f(m)=If--fbf(x)dx or 
(b-a)f(m)= f:f(x)dx . -a ' 
VI. Basic Definite Integral Theorems 
A. f; f(x)dx=O 
B. f: f(x)dx=-f: f(x)dx when a < b. 
C. f!cf(x)dx=c f: f(x)dx when c is a constant. 
D. f:[f(xl±g(x)]dx=f: f(x)dx±f:g(x)dx 
E. f!cdx=db-a) 
F. 	 f: f(x)dx <f: g(x)dx whenf(x) < g(x). 
G. f: f(x)dx=f: f(x)dx+ f: f(x)dx 
H.@JForexample: 
e.slnxdx =f~.sinxdx+ f;sinxdx= 
- f~.(-sinx)dx+f;sinxdx= 
-f;sinxdx+f;sinxdx=O because 
f~.sinxdx=- f~.(-sinx)dx; and, from a 
geometric point of view, 
y 
2 
/ "\ \ y = x - slnx 
-It: !} / 71 
2 
y 
2 
y=slnx / "\ 
-If\. / 
-\ 
71 x 
2 
f~. (-sinx)dx can be considered the region 
bounded by the x-axis and the curve of y = 
-sinx. which has the same area as the region 
bounded by the x-axis and the curve y = sin x; 
therefore, f~. (-sinx)dx= f;(sinx)dx, and 
hence, r.sinxdx = f~.sinxdx+ f;sinxdx= 
- f;sinxdx+ f;sinxdx= O. 
VII. Common Integration Formulas 
A,fdu =u + C 
"..+1B. f"..du = -- when,;t - 1. 
,+1 
C. Jeos udu = sin u 
D,fsin udu = -eos u 
E. Jtan udu = Inlsee ul 
F. Jeot udu = Inlsin ul 
G.Jsee udu = Inlsee u + tan ul 
H. Jese udu = Inlese u - eot ul 
I. Jsee2udu = tan u 
J. Jese2 udu = ~ot u 
K,fsec u tan udu =see u 
L. Jese u eot udu = ~se u 
M.Je"du= e" 
N·f!du=f d,: =Inlu! 
O fbudu=L 
. Inb 
P. Jlnluldu =u(lnlul-l) 
-0)' 
f 
has a 
at an 
on 
a is 
1<' · 
a. 
here 
aylor 
a =O, 
or all 
: )xn 
)=1, 
Integration (con tinued) 
Q.f .Ja~~u2 =arCSin(*)=sin-. (*) 
R.f a2~u2 =~arctan(*)=~tan-I(*) 
S f~=sinh-I(.!!.)=ln(U+~)
. 	 .Ja2 +u2 a a 
T f~=COSh-I(.!!.)=ln(U+~) 
. 	 .Ju2-a2 a a 
U f ~=.ltanh-l (.!!.)=-.Lln(a+u)
. 	 a2-u2 a a 2a a-u 
when u2<a2 
V. J~=-.lcoth-I (.!!.)=_...Lln(a+u)u2_a2 a a 2a u-a 
when a2 < u2 
W.f.Ju2±a2du= .!!..Ju2+a2±E..lnlu+.Ju2+a2 12 - 2 ­
X. J) Note that the Fundamental Theorem tells 
how to evaluate the integral of a function J by 
using any antiderivative ofJ(x); and, the value 
of the integral can be obtained no matter which 
antiderivative ofJ(x) is used. 
I.~For example, the integral f: cosxdx can 
be evaluated using the antiderivative sin x or 
sin x + 7, since D..(sinx) = cosx and D..(sinx 
+ 7) = cosx. 
2. So, f: cosxdx=(sinx)l: = sinb-sina. 
3. Also, f: cosxdx=(sinx+7)1: = 
(sinb+ 7)-(sin a+ 7)= sinb-sina. 
4. Therefore, since the solutions are the same, it 
is wise to choose the simplest antiderivative 
when stating an integration formula and 
solution. 
VIII. Integration Techniques 
A. Substitution 
I. This is a method of using the Chain Rule to 
calculate integrals and find antiderivatives. 
2. Use f.f(g(x»)g'(x)ttt = J.f(u)du, where u = g(x). 
3. IfJi(x)ttt= F(x) + C, then useJ.f(g(x»g·(x)dx 
= F(g(x» + C. 
B. Integration by Parts 
I. Use 	 the abbreviated notation, Ji(x)g'(x)dx 
= .f(x)g(x) - Jg(xV'(x)ttt, from the 
Fundamental Theorem of Calculus; set u = 
J(x) and v = g(x), sorex) = DxU and g'(x) = 
Dxv; then, use the resulting formula fudv = uv 
-Jvdu. 
2. For definite integrals, use f: J(x)g'(x)dx= 
J(x)g(x)l: - f: g(x) j'(x)dx. 
C. Partial Fractions 
I. Decompose rational functions whenever the 
denominator is factorable. 
2. Integrate each partial fraction that results. 
D. Improper Integrals 
I. Integrals over an infinite interval or having an 
infinite range. 
2. Some converge, having a finite limit that 
exists. 
3. Some diverge, having a finite limit that does 
not exist. 
4. If J(x) is continuous on [a, 00) and 
integrable on [a, b) for all b > a, then 
J;" J(x)dx=limf!J(x)dx,if the limit 
exists. b .... ­
5. If .f(x) is continuous on (-<Xl, b] and 
integrable on (a, b1 for all a < b, then 
f~_J(x)dx= lim fJ(x)dx, if the limit 
exists. •....- ­
6. IfJ(x) is continuous on (-<Xl, 00) and integrable, 
then [j(x)dx=f~_ J(x)dx+f: J(x)dx, 
for any a, if the limit exists. 
IX. Applications of Integrals 
A.Areas 
I . The area, viewed as horizontal rectangles, in 
the interval [a, b] between two curves with 
J(x) > g(x), making .f(x) the top curve, is 
f:[J(x)- g(x)]dx. 
2. The area, viewed as vertical rectangles, in 
the interval [V" Y2] where y, < Y2 between 
two curves.f(v) > g(y), makingJ(y) the right 
curve, is f:[J(y)- g(y)]dy. 
x 
B.Volumes 
1. Solids oriented to an axis with cross-sections 
on planes that are perpendicular to that axis; 
the area of a cross-section is given by the 
function A(x). 
2. The volume of a cross-section slice of 
thickness I:!.p is A(x)l:!.p. 
3. The volume of the solid bounded by the 
planes at the ends of the interval [a, b] is 
V = f: A(p)dp. 
4. Disk Method: The volume of a solid of 
revolution created by the curve .f(x) as it 
revolves around an axis in the interval [a, b] 
is V= 7t f:[J(xW dx. 
J) [NOTICE: [f(x)P is the radius squared.] 
y 
5. Washer Method: The volume ofthe solid of 
revolution created between two curves with 
J(x) >g(x) as they revolve around an axis in the 
interval [a, b] is V= 7tf:[J(X)2 -g(x)2]dx. 
J) [NOTICE: [f(X)2 - g(X)2] is outer radius 
squared minus inner radius squared.] 
6. Shell Method: This method is used when it is 
difficult to compute the inside or the outside 
radius of a cross-section. 
a. A radial coordinate r, with a < r < b, 
along an axis perpendicular to the axis of 
revolution, produces the heights her) of 
cylindrical sections or shells of the solid 
parallel to the axis of revolution. 
b. The area of this shell at r is A(r) = 
27trh(r). 
c. The volume of this solid shell is 
V =f: A(r)dr=f:27trh(r)dr. 
C. Surface Area 
1. Solids of revolution created by revolving 
the function y =J(x) around the x-axis in the 
interval (a, b). 
2. Then, between x = a and x = b, the surface 
area is S = f:27tJ(x)~1 + [j'(X)]2 dx . 
3. If the generated curve C is parametrized 
by (x(t), yet»~, with a :s t :s b, and revolves 
around the x-axis, then the surface area is 
S= f:27ty(t)~[X'(t)]2 +[y'(t)]2 dt. 
D.Arc Length 
I. Ifa graphy = J(x) has a continuous derivative in 
the interval (a, b), then, between x = a andx = b, 
the graph has length L= f: ~1 + [j'(x)]2 dx. 
5 
DIFFERENTIAL 
EQUATIONS 
I. Differential equations are equations that 
involve derivatives of unknown functions. 
II. General solutions represent a family of curves 
if it has unspecified constants. 
III. A 	 basic differential equation involving the 
dependent variable isr(x) = kf(x) ory ' = ky or 
ddY = ky, giving dy= kdt where lYl = kt + e . 
I y 
IV.A differential equation that is 	 linear in the 
dependent variable and involves only the first­
order derivative isr(x) + p(I)J(x) = q(t). 
V. Separation of variables can be written as 
;: =J(x)g(y) orJ(x)dx= g(ylv· 
VI. Chain Rule Equation: DJ(u) = D'/(u)DxU. 
VII. IfF'(x) =J(x), then F(x) is an antiderivative of 
J(x). 
VIII.F'(x) =J(x) is solved when the Fundamental 
Theorem of Calculus is used to evaluate 
f:J(x)dx . 
IX. Always consider if the differential equation has 
a solution and if there is only one solution. 
A.~For example, to solve the differential 
equation y' = J(x), use the Fundamental 
Theorem of Calculus to evaluate the 
integral f: J(x)dx as F(b) - F(a) (for this 
purpose, any solution will do). 
S.Such as, for f.23x2dx, F(x) can be x' or 
x1 +t or x'- 5, and so on, since all of them 
satisfy the differential equationy' = lr. 
C. & But, in most applications ofdifferential 
equations, it is not true that any solution will 
do; only one particular solution also satisfies 
some given initial conditions. 
D. Thus, in addition to the differential equation, 
specific numbers, a and e, might be given, 
such that the differential problemy' =J(x), 
with y = e when x = a is obtained. 
X.lf J is continuous in an interval containing 
point a, then y= f: J(t)dt+e satisfies some 
differential equation problems. 
XI. IfJand g are continuous andg(y) f. 0 fory in some 
interval containing e, then the differential problem 
, J(x) .. 
y = g (x) wherey = ewhenx = a III some Illterval 
containing a can be solved by solving the equation 
f: g(t)dt=f: J(t)dt fory. 
POLAR COORDINATES 
& GRAPHS 
I. Point P = (r, a) where: 
A. The pole, 0, is the center point (like the origin) 
of the polar coordinate system. 
B. r is the length of segment Op, 
C. 9 is the angle formed by segment OP (the terminal 
side) and the initial side (usually the positive x-axis). 
D. 9 is positive when segment OP moves counterclockwise; 
negative when it moves clockwise. 
II. Conversions 
A, J) From polar to Cartesian, use x = rcosa and 
y = rsina. 
S. J) From Cartesian to polar, use a =tan- ' 1'. 
and r =.Jx2+y2 . 
III. Graphs & Equations 
A.Circles 
1. Cartesian equation (x - ecosa)2 + (y- esina)' 
= a2, has radius of a and polar coordinates 
center of (e, a). 
2. Polar equation r - 2rccos(a - a) + c' = al, has 
radius ofa and polar coordinates center of(e, a). 
3. Polar equation r = 2ecos(a - a); if the circle 
contains the origin, then c' = a2• 
x 
- -
Polar Coordinates & Graphs (continued) 
4. Polar equation r = 2ccos8 or r = 2csin8 if a = O. 
B. Roses 
I. Polar equations r = ccosn8 or r = csinn8. 
2. n determines the number of petals on the rose. 
z C. Cardioids & Limacons I. Polar equations r = a ± bcos8 or r = a ± bsin8. 
w 2. When ~ > I, the Iimacon graph has an inner loop. 
3.j) When ~=I, the limacon graph has NO~ 
inner loop, is heart-shaped, and is specifically O 
called a cardioid. 
.oiIIII a. fiP For example, r = 2(1 - cos8). 
'I11III IV. Area 
A. The area bounded by the curve r = f(8) and enclosed 
by the rays 8 = a and 8 = ~ may be found using 
A=.!f~r2d8=.!f~ f(S)2dS . 2 (l 2 (l 
B. The area bounded by two polar graphs is 
A=.!f~(r.2 -r.2)d8.2 (l 2 I 
V. Arc Length 
A. The length of arc r = f(8) where 8 is in the interval 
[a, ~j is L = f~ r2 +(~~r dS. 
VI. Slope of Tangent 
A. The curve r 	= f(8), with coordinates x = f(S)cos8 
and y = f(8)sin8 , has the slope of the tangent at 
dy sinS dr +rcos8 (x(S),y(8» of dy =AJi.= dS . 
dx dx cosS dr -rsinS 
dS d8 
SEQUENCES & SERIES 
t.iIIII I. Sequences are functions that have domains that are 
'I11III integers and ranges that are all real numbers. Z A. The integer in the nth position is called a term and 
denoted by the symbol a. rather than a(n}. 
W B. Consecutive arithmetic sequence terms have a 
ft common difference, d, with each term the result of 
~ a.=a....l+d=al+d(n-I}. 
O C. Consecutive geometric sequence terms have a 
common ratio, r, with each term the result of a. = 
.oiIIII a...ir} = al(rrl . 
'I11III D. lf a sequence has a limit, then it converges; 
otherwise, it diverges. 
E. Ifa sequence converges, then it is bounded. 
F. If {a.} and {b.} are convergent sequences, then: 
I. ~r-(a. +b.)=~it,!!a. +~t,!!b. 
2. Iim(a.b.) = lima. limb. 
.t- . f- . f ­
a lima 
• .t_ . h I' b °3
. 
I· en 1mImb=~'w ~. 
"1_ ,, 1I11'~ II " j- " 
4. limca =clima ,wherecmaybeanynumber. 
,.roo " "roo II 
G. Increasing sequences have every term a", wi1h a. :5 a ... I' 
H. Decreasing sequences have every term a... wi1h "" ~ 11,,;-1' 
I. 	 Together, decreasing sequences and increasing 
sequences form the group of monotone sequences. 
J. 	 A sequence has an upper bound if every term of 
the sequence is less than some fixed number. 
K. A sequence has a lower bound if every term of the 
sequence is greater than some fixed number. 
L. A sequence that has both an upper bound and a 
lower bound is said to be bounded. 
~ M. it. Monotone sequences converge. IF and only 
'I11III IE. they are bounded. 
Z 1. The limit of an increasing sequence is its least 
upper bound. 
2. The limit of a decreasing sequence is its greatestW lower bound. 
ft II.A series is a sequence obtained by adding the terms of 
~ another sequence. o A. A sequence, {S.}. whose terms are defined by 
S = fa =0.. +a +a3 + ... +a , is a sequence of 
.... II k -=I' I 1 " 
'I11III partial sums of each sequence {a.}. 
l.If the sequence, {S.}. converges, then the series 
converges; otherwise, the series diverges. 
B. Ifthe partial sums ot;.a sequence {a.} converge to the 
number, then S= L ak =al +a2+a3 + ... =limS.; k=1 nt ­
this sum is an infinite series. 
C. There is the sequence {a.} of terms of the series. 
D. There is also the sequence {S.} of partial sums. 
E. it.The geometric series f ak (r)k-I is convergent, 
IF and only IF, I" < 1. k=1 
I. In 	 this case, the geometric series sum is 
S= fa (rt-I =---1L- . 
k=1 k I-r 
2. The geometric series partial sum is 
• • I a(l-r·)S 	 = I. a (r) - =---, when r1= I. 
• 	 k=1 k l-r 
F. If !ak and fbk are convergent series,k=1 k=1 
and if C and d are any real numbers, then 
f 	 (cak+ db. )=c fa k+d f bk and it converges. k=1 k=1 k=1 
G.lf f ak is a convergent series, then Iim(aJ=O; if k=1 	 . t -
II. does not approach zero, then it diverges. 
H. it. An infinite series of non-negative terms 
converges, IF and only IF, its sequence of partial 
sums is bounded. 
l. Comparison test: If°:5 a.:5 b. and the series !bk 
_ hI 
converges, then the series Lak also converges; if 
_ _ k=1 _ 
Lak diverges, then Lbk diverges; the series Lbkk=1 .=1 k=1 
dominates the series f IIk • 
k=1 
J. it. Limit comparison test: Given the series of 
_ a~ 
positive terms, La. and Lbk, if lim bk =c when 
_ k=1 k=1 kt- k ~ 
c > 0, then ~ak converges, IF and only IF, Lbk 
k - I a - k=1 
converges. If lim..,!- = 0 and if Lbk converges, 
_ .t_ Ok hI 
then I.ak converges. 
~ K. If the series L~kl converges, then the series La. 
k=1 k=1 
converges absolutely. 
L. If the series fak converges but f ~kl diverges,
k=1 . =1 
then the series f a. is conditionally convergent.
k=1 
M. it. Integral test: An infinite series, with terms 
that are the values at the positive integers of a 
decreasing function, f that does not take negative 
values in the interval [I, co) converges, IF and only 
lE, the improper integral r f(xflx converges. 
N.p-Series 
- 1
I. 	LkP converges whenp > I and diverges whenp :5 I. 
k=1 
2. f -k1 is the harmonic series, from the p-series, 
k=1 
where p = 1; always diverges. 
O. Ratio test: If lim ~~+III = L, then: 
kt- I"k ~ 
I. When L < I, the series La. converges absolutely. 
k=1 
2. When L > I, theseries fak diverges.
k=1 
3. When L = I, more information is needed to 
determine whether or not the series 	fa. converges. 
k=1 
- II 
P. Root test:lftheupperlimitlim~kl/k:lim~=L, 
then: kt_ k .... ~ 
I. When L < I, the series fak converges absolutely. 
k=1 
2. When L > 1 or when L is the symbol co, the series 
fak diverges.
k=1 
3. WhenL = I, more information is needed to detennine 
whether or not the series !ak converges. 
Q. Power Series ~ k=! 
I. Form in (x-a) is Lck(x-a) where the terms ofthe 
k=O 
sequence {e.} are called the coefticients ofthe series. 
2. Converges only for the choice x = a. 
3. Converges absolutely for every number x. 
4. There is a positive number r such that the series 
converges absolutely for every number x in the 
open interval (a - r, a + r) and diverges for every 
numberx outside the closed interval [a - r, a + rj; 
r is the radius of convergence. 
5. fCk(x-4+ fdk(x-a)'= f(ck+dk)(x-atk=O k=O k=O 
6. Ctck(x-4l~0dk(x-4)= 
k~OC~.c,dk _ , )(x-4 
- k7. lf f(x)=k'~{k(x-a) has a radius of ~ 
convergence, r, then it is differentiable on 
(a - r, a + r) and /'(x) = fkck(x-at l. 
_ k-I k=1 
8. /'(x) = L kCk(x-a) is a 	series that has a k=1 
radius of convergence, r, but may ~ at an 
endpoint where (a - r, a + r) converged. 
9.!'(x)=I,kck(x- at-1 is integrable on 
(a - r, k=1 a + r); its integral vanishing at a is 
f:f(t)dt=k~O kC:I (x-atl+C when ~-al <r. 
10. Taylor Series 
-	 .r<k}(a) k 
a. L -,-(x-a) =f(a)+/'(a)(x-a)+
k=O k. 
f"(a) 2 
--z!(x-a) +... 
b. It is centered at a. 
c. It converges at a or some interval around a. 
d. If r > ° and f(x)= kf =Ock(x-4 where 
~--al < r, then the coefficients are the Taylor 
.r<kl(a) 
coefficients ck =~. 
e. MacLaurin series is the Taylor series with a = 0, 
.r<k}(O) k 
centered at 0; k~O~(x) . 
i. 	 Basic MacLaurin series 
(a}-1=1+x+x2+ ... = f x· when Ixi < 1.1 
-x 11=0 
2 3 
(b)ln(l+x)=x-X; +~ - ... = 
( 1)·+1 •f 	- x when-I < x:5l. 
11=1 n 
(C)ln!~~=2(X+~ +x; + ... ) 
- 211+1 
= 2 L _x__ when Ix! < 1. 
.=o2n+1 xl ~ 
(d) arctanx=x-T+ ~ - ... 
( I)· X 2.+1 
= f - when Ix! < 1. 
. =0 2n+1 ­
_ x 2 xl - x" (e) r -I +x+2f+3f+'''= .~o -;;r for all 
real x. ( ). _.1
x2 X4 - -1 x-. 
(f) 	cosx=l-x+Tr. +4'. - ... =~ ( ),
for all real x. . - 0 2n. 
• Xl x~ - (-I)" x 2n" 
(g) slDx=x-3T+ST-"'= L ( )
for all real x. . . .=0 2n+ 1 ! 
f . . Binomi~ series p(p-I) 2 (p)x-
I. 	(l+x) =1+px+-2-,-x + ... =! 
forp 1= °and Ix! < 1. . . 0 n 
ii. The binomial coem:ients 	 are (:) = I , 
( p)= , (p)= p(p-) and"p choose k" I P 2 2' 
is (p)= P(P-I)(p-2) ... (p-k+J) . 
iii.lfp i!a positive integer ~~ ( : ) = °for k >p. 
.... 11 ri lthl.ll rtuned. No pan ofthiJ pubt;u. 
lion may be reprodueed or [ r :uaJIII 'I~ In 
Iny form, <If b} an) rn-. elrctrQllic or 
mech"niClI, incloom& p/lvl.JoOOfI>, m:onh.. 
or In)' infor:nltion .storqt and Ktricnl 
'YJlcm . ..... ithoul ·.orI( \CO pcnni~ion (rom the: 
puhliJher. C2eM fbtClluts.l .e. 0.409 
NOTE 10 STttOEl\.: Thi,pnOc IS inlmdtd 
for,~-'1JIW11O'G1MIy OuclOilI6 condcmt;.Jfurm;l1. this autikroNlOtL'O'W1,.1,. ltU6,'
fr~~~d'r~2ko~Pnt~s at s::!':r;:,:;~::,,"m-' 
qUICKStUay.com ~~.c;;::':;,,"':;~,:;,~,:; ~ 
U.S.$S.95 Author: Dr. S. B. KLzlik ~~ :/:Ie r:ro:..: 
Customer Hotline # 1.800.230.9522 cr~lttlincd'nthi'illidc. 
ISBN-13: 978-142320856-3 
ISBN-10: 142320856-0 
9 ~11ll~~IIII~~lllJlllJ IlfIIIlllIlil~I 
6