Calculus equations answers

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.
~ 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!

Lh

FUNCTIONS
I. Definitions
A. A relation is a set of order 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 of y = 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.±a are the slopes of the two sides of the graph.

C. Square root: f(xl=a.Jx-h +k
I. (h, k) is the endpoint.
If a> 0, the graph goes to the right.

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

2.
3.

f2
p

f(x) = o.
3. There are, at most, n - 1 points of change or turns

in the graph.
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

4.

x-intercepts:

f. f(x) = log, x = In x; this is the natural log.

r-----------------~
g.log.

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.

2a


x-axis wherey = O.
3. If degree p(x) = degree q(x), the asymptote is y =
(lead coefficient ofp(x»/(lead coefficient of q(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

ii . I degree

= (]

degrees

;0) radians

tanS=~

~

0
~
~

III

Z
~

(d) cscS=.!.
y
(e) secS=.!.

x

(f) cotS=,,!.

y

vi . j) use ful values

r =x",n

8 = degrees; t = radians', {) = unde fined

d - ", __1_
-

= (] !O)

(c)

xn

.X

i. I radian

iii.proportion conversion of angle measurements:
angle i n 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
P(I,O).
(f) j ) Angles rotating one or more full times
require adding ±21t for each rotation.

v. function definitions
(a)sinS=y
(b) cos S =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

c. (x'"

log. x logx Inx
10gb a = loga =III/l;

this is the change-of-base rule.
h. log.xy = log.x + log.y

X= -b+~ .


()
F. Rational: f(x) = p x

X=

x'"

8

0

30

4S

60

90

180

t

0

.n.

.n.

.n.

.n.

6

4

3

2

It

g'(;T=~:

sin

0

l

J2

.J3

2

2

2

1

0

h.x~ =~ =(!!/Xr

cos

I

.J3

J2

l

2

2

2

0

-]

j. ~=~

tan

0

.J3

1

.J3

{)

0

k.~ ="'!!/X

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

e. _1_= x"'

x - ",

f. (xy)"' =x"'y"'

i. ~=!!/X.~

I. Ifx' =x', then a = b.
m. Ifa x = b x , 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. a ftx )= 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

1

=y .

3

full cycle of the curve, or

2:.

c. Horizontal shift or phase shift is c.
d. Vertical shift or average value is d.

e. Sine:f(x)

= aslnb(x- c) + d

f. Cosine:f(x) = acosb(x - c) + d
g. Tangent:f(x) = atanb(x - c) + d
[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

Lh

~

0

"
W

m

Z
~


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=

iii. tan'

U=

H. 0/'
~ Forexample, when finding lim
x 3 -28 such thaI
x-+l x­

K.Cosine:f(x) = cosx
y

I
I
I

r

=+

-

-

I
I

"
-I

If

j

I

I

j x


x"# 2 x 3 - 8 becomes (x - 2)(x 2 + 2x + 4) and then
, x-2
x-2'

2"

(x' + 2x + 4); consequently, when x is close to 2,

"'~
-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 vertical stretch if a> 1; vertical shrink if

0< a < 1; x-axis reflection if a is negative.
j(ax) has horizontal shrink if a> 1; horizontal stretch
ifO < a < 1; y-axis reflection if a is negative.

LIMITS & CONTINUITY

2. n > m, the lim j(x) = lim P «x»
X--,)ooo
X----)
3.n

cr eX_I=1

I. Definitions
A. A is the limit of f(x) as x approaches a, written
~!T. j(x) = A; means that for each neighborhood

such that If(x) - AI < E when Ix-aIis defined for all x in an open interval containing a.
C. Geometrically, ~i,"! j (x) is the y-value of the point in
D.

·x~ x

D.lim
X----)<><>

x:
e

=0

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

which the graph ofj should intersect the line x = a.
( 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, if j(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
xJ,Q

X-)Q+

C. Absolute value:
j(x) = Ix!

l.n=m, the limj(xl=![
x----)oo
b

X--> ~

°

B. Linear identity:j(x) =x G.Rational: j(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

j ( ) p(x)
x = Q(x) and Q(x) "# 0, then when:

x-..

(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 >

l-cos2u

1 +cos2u

Ill. Basic Common Function Graphs

A.Cons*F. Cubic:f(x) =x'


as x gets close to a from the right; that is, x gets
close to a but remains greater than a.

H. Exponential:
j(x) =e
X

3.

l'

limj(x) exists and limj(x)=A,

~ x-).

X-)(I

lE..AWl

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
,
h-.O
h
provided the limit exists.
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 of j at the point (a, /{a»

has s Iope j '()
a WI'th j'()
II =I'1m j(a + h)- j(a) .
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. r P For example, when y==.J4-x 2 , the
slope of the tangent line to the curve at point
(x,y) is - ; ; therefore, the derivative of y is

H. The functionjis continuous at the point a ifj(a)
is defined and

t!T. j(xl= j(a).

y'=f'Cx)=-~=~ .
y .J4-x'

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

°

D.Square root:
j(xb-.JX

I. Logarithmic:
j(x)=lnx

= 12.

B. lim c = c, when c is a constant.

~A

l+cos2u

2


xl-:

(xl + 2x + 4) is close to 12; therefore, lim
III. Rules
x----)2 x-

y

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. X_CHI
lim[j(x). g(x)]=(limj(x»)(limg(x»)
X----)(I
X-HI

E. Quadratic function:
f(x) =x'

j(x) lim j(x)
D lim--=~ provided g(x "# 0) and
. X-->. g(x) limg(x) '
X-->.

limg(x) "" 0.


J. Sine:
j(x) = sinx
y

f--

2+--+-+-f-IH

1'\--­

I t-:

r ......

"+-t---+:v.:,,,x

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

-2

-\

2

C. The first derivative function notations include
dy, DxY, and ..!L j(x); second
dx
.
fix.
• d1 y
derivative notations mclude j (x), y, dX'

f'(x), y',

and D;y.
D. The derivative atx = a is usually written as:
/'(11), D(f)(II), or

1;L.


II. RuleslFormulas
A. Assume fix) and g(x) are differentiable functions,

:x

D,f(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)
x--+«

X--+d'

t-

0, then

lim lex) = lim f'(x) .
x~.

x~.

g(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
.

open mterval (a, b) such that

/(b)- lea)
()
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)=_...L
dx 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
dx

x..- 1 is defined.


14. L

dx

£ =

I,

2vx

15. L JI (x)
I
; derivative
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=l
dx

t- 0,

x

19.A... log x=_I_

dx
•
xlna

20. L(sinx) = cosx
dx
21. L(cosx)=-sinx
dx

22. L (tanx)=sec l x
dx

23. L (cotx)=-csc l x

dx

24. L(secx) = secx' tanx
dx
25. L(cscx)= -cscX'cotx
dx

,....!-.,
27.L(arccosx)=- ,....!-.,

dx
vI-xl
26. L(arcsinx) =
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)
dx

1
x.Jxl-l

of

an

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=
g(x)

Jx~\

with

.Jx+l'

f(;)[;(X] dx g(x)

1) and

becomes

L[ (x-I)]= .Jx+IDx (x-l)-(x-I) Dx .Jx+I ;
dx .Jx+I
[.Jx+lf
then, using the powerformula to find D x .Jx + I ,

the statement becomes

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.~Y For example, on the graph of y = 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.

s

y

.Jx+l (1)- (x-I)
2.Jx+l

L

x+1
finally, using algebra to simplify the expression, the
derivative becomes
B. Graphs

p

(

x+3
or (x+3),Jx+t
2(x+ If.
2(x+ 1)1 .

I. Increasing/decreasing
a. A function/ is increasing in an interval (a, b)
iff(a) 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 that are 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. If j" (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.

3

I

-

/

"

R

\.

./
Q

f"

./
-

I-

-­

\

\ T

N

x

g.

I

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.
[CAUTION! This test does not apply if
f"(a) = 0.]
Helpful Hints for Sketching a Curve
C.
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 of Change
I. Average rate of change of/ over the interval [a,x]:

6

P

a. I s

/(x)- lea)

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, " =
and I is time.

f(1),


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
a.
equal width of

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(x l ) ; f(x 2 ) + ... +

b;;

f(xn-I~+ f(xo)] =

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_.)] =
n-I
b
;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

n
L f(xj ) .
o

[f(x.) + f(X2) +,..+ f(x.)] = b~a
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

[f(

b:a

f ( X

0-1

Xo ;XI )+

f(

xI

:x

2

)+,..+

0-1

+x)]
b

+x
)
0
=-=.!!
L f (x
-1...........l:
2
n j~O
2

+x.1+_1) where !'J.x = b ;a.
or!'J.x0L- 1f (x.
- '__
j~O

2

I 1
JL ,
,
1/ ,

y

IL

j,
,
,

,
,,,
,
,
,

I
~

,
,
,,
,,
,
,

,
,,
,,
,
,

a l x l x2 l x, 1

I

I

\

:'\
,
,

,,,
X4

[0

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 < ie. The su~ of these parabola areas is b3-na
(f(xo) + 4f(x.) + 2f(X2) + 4f(x,) +....+

n--+....

s: f(x)dx =!~'?o t; [j~ f(x j )+ :f f(x

;=1

2. Trapezoid method
a. Divide the total interval into trapezoids
with parallel sides of equal subinterval
b-a
lengths, --,,--.
b. This results in n + 1 points on the x-axis.
.
b-a

c. T hese pomts are x. = a, XI =a+--,,-,

x 2 =a+2(b-a)
n , ... , x n =a+n(b-a)
n '

j )

F. f: f(x)dx
1< ' ·

G. f: f(x)dx=f: f(x)dx+ f: f(x)dx
H. @J Forexample:

e.slnxdx = f~.sinxdx+ f;sinxdx=


- f~.(-sinx)dx+ f;sinxdx=
-f;sinxdx+f;sinxdx=O because

f~.sinxdx=- f~.(-sinx)dx;

2

1t~ square units, so the integral orone-halfof

1t: .

this region is f: .Ja 2 _x 2 dx=

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.
Jf(x~ = F(x) + C. IF and onlv IF, r(x)
C.
=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

a.

and,

from

a

geometric point of view,

here
aylor

y

2

a =O,

\

/ "\

y = - slnx

!}

-It:

x

/ 71

2

y

2

y=slnx

1

the curve for y = .Ja 2 - x 2 is a semicircle with
a radius ofa units and a centerof(O, 0); therefore,
in the interval [-iI. a], this semicircle and thex­
axis create a bounded region having an area of

on

E. f!cdx=db-a)

;=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,

at an

a is

f: f(x)dx= Iim!'J.xi:, f(cJ
2. Using the trapezoid method,

f

D. f:[f(xl±g(x)]dx=f: f(x)dx±f:g(x)dx

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 of page].

I

ii. Riemann sum is Lf(c,)(x, - XH) or
o

j~1
b
!'J.xLf(c,), where /j.x=(Xj-Xi-\) = ;a.

C. f!cf(x)dx=c f: f(x)dx when c is a constant.

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-a
.
c. Th ese pomts are x. = a. XI =a+--,,-.

x

o

0-1 ]

-0)'

has a

B. f: f(x)dx=-f: f(x)dx when a < b.

b a
2-n j~f(xj)+j~f(x,) .

b

i. Riemann sums
i. The height of each rectangle is f(c,),
where Cj is any po int in each subinterval,
[Xj, Xj+l], for 0 :s i:S (n - I).

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

-If\. /

/

"\71

x
or all

-\

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
B. f".. du = - - when,;t - 1.
"..+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. Jsee 2 udu = 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 . f budu=L
Inb

P. Jlnluldu

= u(lnlul-l)

:)xn
)=1,


Integration (con tinued)
Q.f

.Ja~~u2

R.f

a2~u2 =~arctan(*)=~tan-I(*)

S

=arCSin(*)=sin-. (*)

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.

II. General solutions represent a family of curves

T f~=COSh-I(.!!.)=ln(U+~)
. .Ju2-a2
a
a

x

U . f ~=.ltanh-l
(.!!.)=-.Lln(a+u)
a2-u2 a
a
2a
a-u
V. J ~=-.lcoth-I
(.!!.)=_...Lln(a+u)
u2_a2
a
a
2a
u-a
when a2 < u2

f:

f:

f:

3. Also, 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), so rex) = 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
b .... ­
exists.
5. If .f(x) is continuous on (-integrable on (a, b for all a < b, then
f~_J(x)dx= lim fJ(x)dx, if the limit
exists.
•....-­
6. IfJ(x) is continuous on (-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

1

f:[J(x)- g(x)]dx .

if it has unspecified constants.
III. A basic differential equation involving the
dependent variable isr(x) = kf(x) or y ' = ky or

lYl = kt + e.
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
ddY = ky, giving dy = kdt where
I

when u 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 cosxdx can
be evaluated using the antiderivative sin x or
sin x + 7, since D..(sinx) = cosx and D..(sinx
+ 7) = cosx.
2. So, cosxdx=(sinx)l: = sinb-sina.

I. Differential equations are equations that
involve derivatives of unknown functions.

f~=sinh-I(.!!.)=ln(U+~)
a
a
. .Ja2 +u2

2 +a 2 ±E..lnlu+.Ju2 +a 2 1
W.f.Ju 2 ±a2 du= .!!..Ju
2
2
­

DIFFERENTIAL

EQUATIONS


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.]

;: = J(x)g(y) orJ(x)dx= g(ylv·
VI. Chain Rule Equation: DJ(u) = D'/(u)DxU.
VII. If F'(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

y

satisfy the differential equationy' = lr.
But, in most applications of differential

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
C.

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 and x = b,
the graph has length L= f: ~1 + [j'(x)]2 dx.

5

&

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) where y = e when x = 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 .
x

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
2rccos(a - a) + c' = al, has
radius of a and polar coordinates center of(e, a).
3. Polar equation r = 2ecos(a - a); if the circle
contains the origin, then c' = a2•

r-


Polar Coordinates & Graphs (continued)

z
w
~

O
.oiIIII
'I11III

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.
C. Cardioids & Limacons
I. Polar equations r = a ± bcos8 or r = a ± bsin8.

2. When ~ > I, the Iimacon graph has an inner loop.

3.j)

~=I, the limacon graph has

When

NO

inner loop, is heart-shaped, and is specifically
called a cardioid.
a.
For example, r = 2(1 - cos8).
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)2 dS .
2 (l
2 (l
B. The area bounded by two polar graphs is

fiP

A=.!f~(r.2
-r.I2)d8.
2
2
(l

V. Arc Length
A. The length of arc r = f(8) where 8 is in the interval

~j is L = f~

[a,

+(~~r dS.

r2

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

(x(S),y(8»

dy

sinS dr +rcos8
of dy =AJi.=
dS
.
dx dx cosS dr -rsinS
dS
d8

SEQUENCES & SERIES
t.iIIII
'I11III

Z
W
ft
~

O
.oiIIII
'I11III

I. Sequences are functions that have domains that are
integers and ranges that are all real numbers.
A. The integer in the nth position is called a term and
denoted by the symbol a. rather than a(n}.
B. Consecutive arithmetic sequence terms have a
common difference, d, with each term the result of

a.=a....l+d=al+d(n-I}.
C. Consecutive geometric sequence terms have a
common ratio, r, with each term the result of a. =
a...ir} = al(rr l .
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:

~r-(a. +b.)=~it,!!a. +~t,!!b.
2. Iim(a.b.) = lima. limb.

B. Ifthe partial sums ot;.a sequence {a.} converge to the

number, then

. f-

. f­

a
lima

• .t_ .
hen I'1m b
·
3 . IImb=~'w

"1_ ,,

1I11'~

4. limca =clima

,.roo "

"roo

it.

~

Z
W

ft

~

o

°

~.

,wherecmaybeanynumber.

II

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.
Monotone sequences converge. IF and only
IE. they are bounded.
1. The limit of an increasing sequence is its least
upper bound.
2. The limit of a decreasing sequence is its greatest
lower bound.
II.A series is a sequence obtained by adding the terms of

another sequence.
A. A sequence, {S.}. whose terms are defined by

it.

S
....

'I11III

II

= fa =0..
k -=I '

I

+a +a3 +... +a ,
1

k=O

6.

C

7. lf f(x)=k'~{k(x-a)

(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.

t-

9.!'(x)=I,kck(x- a 1 is integrable on
k=1
(a - r, a + r); its integral vanishing at a is
f:f(t)dt=k~O kC:I (x-atl+C when ~-al
f

f (cak + db. )=c fa k+d bk and it converges.
k=1
k=1
k=1
G.lf
ak is a convergent series, then Iim(aJ=O; if
k=1
. tII. does not approach zero, then it diverges.
H.
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_ and the series hI k

f

10. Taylor Series

a.

it.

then

!b

d. If r >
~--al

~

K. If the series L~kl converges, then the series
k=1
converges absolutely.

f

it.

centered at 0;

La.

is a sequence of

"

partial sums of each sequence {a.}.
l.If the sequence, {S.}. converges, then the series
converges; otherwise, the series diverges.

2

.=o2n+1 xl
~
(d) arctanx=x-T+ ~ - ...
X .+
f (- I)·
2n+1
2

=

1

. =0

r

•

Xl

I.

-

(-I)" x 2n "
( )
2n+ 1 !
(p)x-

p(p-I) 2
(l+x) =1+px+--,-x
+ ... =!
2
forp 1= and Ix! < 1. .
. 0 n

°

ii. The binomial coem:ients are (:) = I ,

, (p)= p(p-)
( p)=
I
P 2
2'
is

and"p choose

k"

(p)= P(P-I)(p-2) ... (p-k+J) .

~~ ( : ) = °for k > p.

iii.lfp i! a positive integer

converges absolutely.

k=1
2. When L > 1 or when L is the symbol co, the series
diverges.

3. WhenL = I, more information is needed to detennine

!a

6

x~

f . .Binomi~ series

II

k=!

Ix! < 1.

(g) slDx=x-3T+ST-"'= L
for all real x. .
.
.=0

P. Root test:lftheupperlimitlim~kl/ k:lim~=L,
then:
kt_
k .... ~

whether or not the series
k converges.
~
Q. Power Series
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
number x outside the closed interval [a - r, a + rj;
r is the radius of convergence.

when

­
_
x 2 xl

- x"
-I +x+2f+3f+'''= .~o -;;r for all
real x.
( ). _.1
x2 X4
-1 x-.
(f) cosx=l-x+Tr. +4'. - ... =~ ( ),
. - 0 2n.
for all real x.
(e)

determine whether or not the series fa. converges.
k=1

k=1

x· when Ixi < 1.

11=0

(C)ln!~~=2(X+~ +x; + ... )
211+1
= 2 L _x__ when Ix! < 1.

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, the series fak diverges.
k=1
3. When L = I, more information is needed to

fak

3

f -

f

fak

.rk~O~(x) .

(b)ln(l+x)=x-X; +~ - ... =
( 1)·+1 •
x when-I < x:5l.
11=1
n

k=1

1 is the harmonic series, from the p-series,
-k
k=1

where p = 1; always diverges.

I. When L < I, the series

where

=~.

-x

- 1
converges when p > I and diverges whenp :5 I.
LkP

-

f c (x-4
k=Ok

i. Basic MacLaurin series
1 =1+x+x2+ ... = f
(a}-1

r

I.

f(x)=

e. MacLaurin series is the Taylor series with a = 0,

f

a.

and

.r
fak converges but ~kl diverges,
k=1
. =1
is conditionally convergent.
then the series
k=1
M.
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 f(xflx converges.
N .p-Series

L. If the series

°

< r, then the coefficients are the Taylor

coefficients ck

hI

Ok

converges.

2

b. It is centered at a.
c. It converges at a or some interval around a.

~

_ .t_
-

k.

--z!(x-a) +...

f

I. ak

- .rk
L -,-(x-a) =f(a)+/'(a)(x-a)+

k=O

f"(a)

converges,
then the series
Lak also converges;
_
_
k=1
_ if
Lak diverges, then Lbk diverges; the series Lbk
k=1
.=1
k=1
dominates the series IIk •
k=1
J.
Limit comparison
test: Given the
_
a series of
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,

it.

has a radius of ~

k

convergence, r, then it is differentiable on

series,
k=1
and d are any real numbers, then

b.

)(x- 4

-

• I

°

k=O

ck (x-4l~0 dk(x-4)=
k _,

a(l-r·)

S = I. a (r) - = - - - , when r1= I.
• k=1 k
l-r
F. If !ak and fb k are convergent
k=1
and if

Ct

k=O

k~OC~. c,d

f

I"

•

2.

G. Increasing sequences have every term a", wi1h a. :5 a...I'
H. Decreasing sequences have every term a... wi1h "" ~ 11,,;-1'

'I11III

5. fC k(x-4 + fdk(x-a)'= f(c k +dk)(x-at

+a2+a3 + ... =limS.;

k=1

" j- "

II

=al

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.
The geometric series
ak (r)k-I is convergent,
E.
IF and only IF, < 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.

.t-

S= L ak

6

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