______________________•
FUNCTIONS, LIMITS & DERIVATIVES FOR FIRST YEAR CALCULUS STUDENTS
~
FUNCTIONS
DEFINITIONS
• Function. A correspondence that assigns one
value (output) to each member of a given set.
The given set of inputs is called the domain. The
set of outputs is called the range. Onevariable
calculus deals with rcalvalued functions whose
domain is a set of real numbers. If a domain is
not specified, it is assumed to include aLi inputs
for which there is a real number output.
• Notation. If a function is namedf, thenf(x)
denotes its value at x, or "/ evaluated at x ." If
a function gives a quantity y in terms of a
variable quantity x, then x is called the
independent variable and y the dependent
variable. Given a function by an equation
such as y = x 2 , one may think of y as
shorthand for the function's expression. The
notation xl~.~ ("x maps to xl") is another
way to refer to the function. The expression
f(x) for a ~nction at an arbitrary inputx often
stands in for the function itself.
• Graph. The graph of a function / is the
set of ordered pairs (x, f(x), presented
visually on a Cartesian coordinate system.
The vertical line test states that a curve is the
graph of a function if every vertical line
intersects the curve at most once. An equation
y = f(x) often refers to the set of points (x,y)
satisfying the equation, in this case the graph
of the function f The zeros of a function are
the inputs x for which/(x) = 0, and they are
the Xintercepts of the graph.
• Even & Odd. A function/is even if
f(x) = j(x), e.g., xl and odd if f(x) = f(x).
Most are neither.
NUMBERS
• Rational numbers. A rational number is a
ratio p/q of integers p and q, with q f. O. There
are infinite ways to represent a given rational
number, but there is a unique "lowestterms"
representative. The set of all rational numbers
forms a closed system under the usual
arithmetic operations; except division by
zero.
• Real n umbers. In this guide, R denotes the
set of real numbers. Real numbers may be
thought of as the numbers represented by
infinite decimal expansions. Rational
numbers terminate in all zeros or have a
repeating segment of digits. Irrational
numbers are real numbers that are not
rational.
r~
• Machine numbers. A calculator or computer
approximately represents real numbers using
a fixed number of digits, usually between 8
and 16. Machine calculations are therefore
usually not exact. This can cause anomalies in
plots. The precision of a numerical result is
the number of correct digits. (Count digits
after appropriate rounding: 2.512 for 2.4833
has two correct digits.) The accuracy refers
to the number of correct digits after the
decimal point.
• Intervals. If a < b, the open interval (a,b) is
the set of real numbers x such that a < x < b.
The closed interval [a,bl is the set of x such
that a ~x ~ b. The notation (cI), a) denotes the
"halfline" consisting of all real numbers x
such thatx < a (or cI)
are intervals of the form (oo,a) and (a, (0).
The symbol 00 is not to be thought of as a
number. The entire real number line is an
interval, R = (cI), 00).
NEW FUNCTIONS
• Arithmetic. The scalar multiple of a
function/by a constant c is given by
(cf)(x) = c • f(x). The sum / + g, product /g,
and quotient fig of functions / and g are
defined by:
(f+g)(x) = f(x) + g(x),
(fg)(x) = j(x)g(x),
(f/g)(x) = f(x)/g(x).
In each case, the domain of the new function
is the intersection of the domains of/ and g,
with the zeros ofg excluded for the quotient.
• Composition. If / and g are functions, "/
composed with g" is the function / 0 g given
by (f " g)(x) = f(g(x)) with domain (strictly
speaking) the set of x in the domain of g for
which g(x) is in the domain off
• Translations. The graph of xl~(xa) is the
graph of/ translated by a units to the right;
e.g., (a, f(0)) would be on the graph. The
graph of xl~j(x)+b is the graph of /
translated b units upward.
or, f(x) = y has only one solution; or, any
horizontal line intersects the graph of/at most
once. E.g., xl is onetoone, xl is not. Strictly
increasing or decreasing functions are oneto
one.
There can be only one inverse defined on the
range off, denoted/I. For any y in the range
off, /1 (y) is the x that solves f(x) = y.
If the axes have the same scale, the graph of
/ I is the reflection of the graph of/across
the line y = x.
• Implicit functions. A relation F(x,y) = c often
admits y as a function of x, in one or more
ways. E.g., xl + y2 = 4 admits y = /4  x 2 .
Such functions are said to be implicitly
defined by the relation. Graphically, the
relation gives a curve, and a piece of the curve
satisfying the vertical line test is the graph of
an implicit function. Often, there is no
expression for an implicit function in terms of
elementary functions. E.g., xl 2y + y2 2x = 4
admits y = f(x), but there is no formula for /(x).
ELEMENTARY ALGEBRAIC
FUNCTIONS
• Constant and Identity.
A constant function has only one output:
f(x) =c.
The identity function is:
xl~x, or f(x) = x.
• Absolute value.
Ixl
= {
m
z
7l.S g
The above is an example of a piecewise
definition. For any x, R = Ixl.
• Linear functions. For a linear function, the
difference of two outputs is proportional to the
difference of inputs. The proportionality constant,
i.e., the ratio of output difference to input
difference
m = Y2 )'1
x 2 xI
is called the slope. The slope is also the change in
the function per unit increase in the independent
variable. The linear function
f(x) =mx+b
has slope m andy;nterceptf(O) = b, the graph is a
straight line. The linear function with valueyo atxo
and slope m is
f(x) = YII + m(."I:xtl·
• Quadratics. These have the form
,
f(x) = /L\..2+bx+c where (a f. 0).
The normal form is f(x) = a(x_II)2 + k.
Also, h = b/(2a) and k = f(1I). The graph is a
parabola with vertex (h,k). opening up or down
accordingly as a > 0 or a < 0, and symmetric
about the vertical line through vertex. A
C
a
• Inverses. An inverse ofa function/is a function
g or/I such that g(f(x))= l (f(x)) = x for all x
in the domain off A function/has an inverse if
and only if it is onetoone: for each of its
values y, there is only one corresponding input;
r
m
Z
Functions (continued)
quadratic has two, one, or no zeros accordingly
as the discriminant b2  4ac is positive, zero, or
negative. Zeros are given by the quadratic
formula
b±~
x=
2a
and are graphically located symmetrically on
either side of the vertex .
• Polynomials. These have the form
p(x) = ax" + bx"i + ... + dx + e.
Assuming a "I 0, this has d egree n, leading
coefficient a, and constant term e = prO). A
polynomial of degree n has at most n zeros. If
Xo is a zero of p(x) , then x  Xo is a factor ofp(x):
p(x) = (x  xI) q(.r:)
for some degree nI polynomial q(x) . A
po lynomial graph is smooth and goes to ± 00
when Ixl is large.
• Rational functio ns. These (~Jve the form
f(x) = :(x)
where p(x) and q(x) are polynomials. The
P = Plltl l = Poe'l.
• Pure exponentials. The pure exponential
function with base a (a > 0, a "I I) is
f(x) = aX.
The domain is R and the range is (0,00 ). The y
intercept is a O = I. If a < 1, the funct ion is
decreasing; if a > I, it is increasing. It changes
by the factor a!1x over any interval of length ~x.
Exponentials turn addition into multiplication :
aO= I
a"+Y = aXaY
am, = (u')m
aXJ' = aXlaY
ax = lla x
• Loga rithms. The logarithm with base a is the
inverse of the base a exponential :
log" x = "the power of a that yields x " ; thus.
logax = y if a)' =x
Equivalently, x = a'0ga' or log"a)' = y.
domain excludes the zeros of q. The zeros off
The dom a in of/og" is (0, 00) and the range is R.
are the zeros of p that are not zeros of q. The
If a > I , then log"x is negative for 0 < x < I ,
graph of a rational function may have vertical
positive for x > I, and always increasing. The
common logarithm is log lo' Examples:
asymptotes and removable discontinuities, and
is
similar to
some
polynomi a l (perhaps
constant) when ~r:1 is large.
nh tor
• nth Roots. These have the form y = x~ ==
some integer n > I. If n is even, the domain is
10,
such thatyn =x.lfn is odd, the domain is Randy
• Algebra ic vs, transcendental. An algebraic
= f(x)
satisfies a twovariable
polynomial equation P(x,y) = O. The functions
above are algebraic. E.g. , y =
x 2  y2
Ixl satisfies
= O. Sums, products, quotients, powers,
and roots of algebraic funct ions are algebraic.
Functions
that
are
not
algebraic
(e.g.,
exponentials, logarithms, and trig functions) are
called tran scendental.
• Rational p owers. These have the form
f(x) = x7. == (x m )~ == (x~ )m where it is assumed
m and n are integers, n > 0, and Iml/n is in
lowest terms. If m < 0 then x'" = Ilxlml . The
domain of x min is the same as that of the nth root
function, excluding 0 if m < O. For x > 0, as p
decreases in absolute value, graphs of y = x P
move toward the line y = xiI = I; as p increases
in absolute value, graphs ofy
= x P move
from the line y = 1 and toward the line
x
log JI) (1110) =  J
Logarithms turn multiplication into addition:
log" 1 = 0
loga xy = loga x + loga Y
log"xn = mlog"x
is the unique real number such thatyn =x.
function y
log" a= 1
log2 32 = 5,
away
= 1.
log" (xly)
= log" x  log"y
log" (I/x) = Iog"x
The third identity holds for any real number Ill.
For a change of base, one has
log x
loghx=~
• Natural exponential an7f logarithm. The
natural exponential function is the pure
exponential whose tangent line at the point (0,1)
on its graph has slope I. ~ts base is an irrationa l
*1'"
number e = lim (I +
2.718.
The natural'iogarithm is /11 ; loge' the inverse
toxlHr':
In x = y means x = eY .
There are identities
In eX =x,
e1nx =x,
In e = 1,
and In has the properties of a logarithm.
E.g., In(Ilx) = In x . Specia l values are:
In 1= 0, In 2::::: 0.6931, In 10::::: 2.303.
Any exponential can be written aX = e(tn lI)X.
Any logarithm can be written log"x=:~ ~.
• General exponential fu nctions. These have the
Rational Powers
formf(x) = Poa" and have the property that the
ratio of two outputs depends only on the
X I/2
difference of inputs. The ratio of outputs for a
.~~~~
   _ _ _ X · I /2
x 2
unit change in input is the base a .
The y intercept is frO) = Po'
• Exponential growth. A quantity P (e.g.,
invested money) that increases by a factor
a = e' > lover each Lillit of time is described by
Ove r an interva l M the fi.lctor is a"". E. g., if P
increases 4% each half year, then a '1;= 1.04, and
P = PI/(I.fJ4)21::::: PoeO.078t (t in yrs) .
The doubling time D is the tim e interva l ove r
whi.ch the quantity doubl es:
aD = e,D = 2 D = l!!.1. = .l!!.1.
,
In (/
r
If the doubling time is D, the n P = Po2i1D.
• Continuous compoundin g at
the annua l
percentage rate r x 100'Y.. yields the ann ual
growth factor
/1
= lim (I + ii)
II
e' ; al so, a= Pe'
II  . 0..
• Exponential d ecay. A quantity Q (e .g., of
radioacti ve materi a l) that decreases to a
proportion b = ek < lover eac h unit of ti me is
desc ribed by Q = Qob t = QlJekt.
Ove r an interva l M the proportion is
E.g., if
b /)l
Q decreases 10% every 12 ho urs, then b l2 = 0.90.
and Q = Q,/O. 90)1112 ::::: Qoe·0088t (t in hrs).
The halflife H is the tim e interva l over which
the quantity decreases by the fac to r onehalf:
b ll=ekll = .11.'2 . H. = .l!!.1.
In b
If the halflife is H . th en Q
=
= .l!!.1.
k'
Qi1/2)IIH
• Irrational p owers These may be de fined by
f(x) = x" = eptll x where (x> (J).
• HY I)e rbolic fu nctio ns. The hyperbolic cosi ne
is cosh
x = eX +2/''"
It ha domain R, range 1/,the restricted domain 10,(0 ), it has inverse
arccosh
x == COShIX =
In
(x + R"=I).
The hype r b olic sine is sinh.\' = e'"  / .,
It has domain R, range R, and is odd. Always
strictly increasing, it has inverse
arcsinh
x
== sinhIx = In
The bas ic identity is
(x + j x 2 + I).
cosh 2.\ 
sinh 2x =
I.
TRIGONOMETRIC FUNCTIONS
• R adian s. The radi an meas ure o f an ang le () is
the ratio of le ngth s to radius r of a
con'e ponding circular arc:
e = f.
Radian M.casurc
211: radian s = 360"
,./ '\s
IBO
." = ...lL..
In calculus, it is normally
ass umed (and necessary
for standard deriva tiv e
formul as) that argu ments ....     .....
to trig functions are in radi ans.
~
• Cosine, sine, ta ngent. Consider a real number t as
the radian meas ure of an angle: the distance
meas ured
counte rcloc kwi se
along
the
circumference of the unit circle from the po int
(1,0) to a terminal point (x,y). Then cos t = x ;
•
sin t
Y C·
'
SID t = y ; tan t = cos t = ~
os IDe an d sIDe
have domain R and range
the fangent excludes ±
II,ll. The domain
A
of
tn, ± i n ... , and its
range is R. The cosine is even, the sine and
tangent are odd.
LIMITS
• Inequalities. Iff(x) ~ M for x near a, then
lim /(x) ~ M if the limit exists. Likew ise if
DEFINITIONS
f(x) ~ m.
• Sandwich Theorem. If g(x) ~J(x) ~ h(x) for
x near a, and lim g(x) = lim h(x) = L, th en
Cosine & Sine
x  u
• Limit. Intuitively, the limit of j(x) as x
approaches a is the number thatj(x) gets close
to when x gets close to a. Precisely, a number L
is the limit, written:
lim /(x) =L or/(.:.;) I.tL asx .t a
x  a
• Secant, cosecant, cotangent:
sec I = cols / ; csc I = si ~1 t
=
cot I = _1_
tan I
·
Special0 va.lues:
H
cos t
sin t
tan
/3
Ii
I
0
"2
SlIl t
H
H
"3
"2
I
T
0
"2
Ii
I
0
T
/3
T
1t
@
0
sin ] t + cos2 I = 1
Ian} t + 1 = sec21
sin( a + h) = sill a c'os h + cos a sill h
('os(a + h) = cos a cos h  sill a sill h
at x = a, the graph of /
eventually lies
• Onesided limits. The lefthand limit is equal to
L, written
Other identities are obtained from the above.
£
lim /(x) = L or /(a) = L, if every
> 0 admits~
ui > 0 such that l/{x)LI< £ when
D < x < a.
The righthand limit is defined similarly, the last
condition being a < x < a + D. E.g.,
(I 
E.g.,
sin(t7tl2) = cos I
cos(2t) = e().~2 I  sill] I = 12 sill] I
f) f
xl~II~+arctan( =
A limit exists if and only if the left and right
tan a + tan b
1 tall a tan b
• Amplitudes, periods, & phases. If/(t) =
A sin (ox +¢) + k with A > 0, OJ> 0, and
,tall(a+b)
< ¢:S!t, then the a mplitude is A, the average
value is k, the period is ~, the frequency
( pe:iod ) is
~
, the angular frequency is (0, and
the phase shift (relative to Asin OJ I) is 4>.
• Inverse trig functions.
The arccosine is inverse to cosine on IO,1t]:
arccos x = "angle in IO,1t] whose cosine is x." It
*'
hand limits exist and are equal.
• Infinite limits. One writes lim f(.lC) = 00 if
every Y > 0 admits aD> 0 ;u~h that f(x) > Y
when 0 < Ixal < D.
Likewi se, there are onesided limits to co, and
= co .
limits to co. E.g., lim _
x  0
Limits at infinity. One writes :
lim /(x) = L
if every £ > 0 aln~it~ an X > 0 such that
If(.lC)LI < £ when x > X.
(±)
has domain [I,IJ and range IO,1tJ.
arccos [
1]
=
arccos [
f ]= 3:
The arcsine is inverse to sine on: [211 ,
LIMIT THEOREMS
~1
.1f [whose sine is x." It
has domain [I,IJ, range [2ft , ~ 1, and is odd.
The arctangen t is inverse to tangent on (  J: ' ~ )
: arctan x = "angle in ( 211 , 1f) with tangent x." It
has domain R, range ( 2 .1f), and is odd.
arctan /3 = Zf, arctan (1) = 411
arcsin x = "angle in [ 211
11
The notation COSlx for arccos x is not to be confused
with Il eos x; likewise sinIx and tanIx.
xa
X 
 
d
X  l1
(1
(!)
= 0 (using h(x) = ~..:I).
••
E.g., xlim
x sin
 ()
• L'Hopital's Rule. (N eeds deri vatives.) If
lim f(x) = 0 = lim g(...:) ,
.\" 
XI./
(1
and ifF(x) and g '(...:) are defined and g '(x)
for x near a, then
r
r
lex) 
x 1_~llI g (x) 
f. 0
f'(x)
l~ll1 g'(x)
X
provided the latter limit exists (or is infin ite).
The rule also holds when the limits o fJ and g
are ± w .
if
X(I
f. I
for x near a (or if F is continuous at I), then
lim F(g(x)) = Jim F(y)
y  I
xa
y  2
I
lim sin 5x = lim sin y = 1
x  0
5x
y
0
x 
y
3
il
If p(x) is a polynomial, lim p(...:) = p(a) .
x 
1I
Let p(x) and q(...:) be polynomials.
•
.
p(x) _ pro)
Illn :(
)  (
).
f. 0, then
If q(a)
x  lIqx
qa
If q(a) = 0 and p(a) f. 0, onesided limits are
± 00 . E .g. , for integer n > 0,
limo. .L
xn
x _
+n =
.lim
= W ',
w (n odd);
x _ 0 X
I.im ~
x = w (n even).
If q(a) = 0 and pea) = 0, first cance l a ll
common factors ofx  a from p(x) and q(x).
E.g.,
x _ 0
= lim
lim x 2 + 2\'  3
x
1
+2
x 3  3x
x_
x+3
1
(x  I) (x
+ 2)
• Rational functions at infinity.
For integer n > 0,
lim xn = w ;
.\" ....... 
00
x" =
 00
xn =
00
lim
lim
+.
lim
x  ± oo x
(n even);
=
O.
ox"+bx" I + ...
lim
x _ ±
(n odd);
00
x   oo
1!.
ex"' + dx ln
00
x" 
lim
x  ± cc
In
1+...
c
(a , (' non zero)
• Arbitrary powers.
Hm x P = aP (when aP is defined)
xa
For p > 0, lim x P =
x · oo
00
and lim x P = O.
x  oo
• Limits for b asic derivatives.
lim xn~  a"' = m a m I (when
x __
1I
a
X
a mI
defined)
lim ell ; I = I (a definition of e)
,,  0
provided the limit on the right exists. E.g .,
lim (xl+1)n = lim y" = 2n
x 
LIMIT FORMULAS
• Polynomials and rational functio ns.
If c is a constant, lim c = c.
x 
• Note. The following theorems have
counterparts involving limits to infinity. Also,
"for x near a" will mean "for all x in some
open interval containing a, except perhaps
x = a."
Arithmetic. A limit ofa sum is the sum of the
individual limits, provided each individual
limit exists . Likewise for a limit of a
difference or a product. The limit of a
quotient is the quotient of the individual
limits, provided each individual limit exists
and the limit of the denominator is nonzero. If
c is a scalar, then lim cf(x) = e lim J(x),
• Compositions. If lim g(x) = I and g(x)
Arctangent
X 
completely within the fixed plot range, except
perhaps at x = a. (Compare with Zooming
Formulation under Continuity, page 4.)
• Identities:
1t
held fixed with L in the middle, and the plot
x ...... "
Special case: if If(x)1 ~ h(x) for x near a, then
lim h(x) = 0 implies lim f(...:) = 0
.\" 
if every £ > 0 admits a 0 > 0 such that
If(x)LI < £ when 0 < Ixal < D
It is assumed that f(x) is defined for all x in
some open interval containing G, except perhaps
x = a. If a limit exists, there is only one. The
limit statement says nothing whatever about the
value of/at x = a.
• Zooming formulation. If the plot range for/is
domain is narrowed through intervals centered
/3
T
/3
0
t
4"
T
c?s/
H
6
;
x  " .
lim f(x) = L.
I
a" I
lim  h = I n a
lim sin x = 1
x _ 0 x
,, _ 0
r
x~10
cosxl=O
x
is
......
~
CONTINUITY
DEFINITIONS
• Continuity at a point. A function I is
continuous at a if a is in the domain ofland
lim f(x) = I(a).
x Explicitly,fis defined on some open interval
containing a, and every £ > 0 admits a 8 > 0
such that If(x)  f(a)1 < £ when ~  al < 8
• Zooming formulation. If the plot range fori
is held fixed withf(a) in the middle, and the
plot domain is narrowed through intervals
centered at x = a, the graph of I eventually
lies completely within the fixed plot range.
This must hold for any such plot range.
(l
r.,. /0)+£
~~f(a)
a
a8
a+8
I(a)£
• Onesided continuity. A function f is
continuous from the left at a if a is in the
domain ofI and
lim f(x) = f(a).
A functionIis" c~~tinuous from the right at a
if a is in the domain ofI and
lim f(x) = f(a).

.....
....
• Intermediate value theorem. If f is
continuous on the closed interval la,b], then
lachieves every value betweenl(a) and/(b):
for every y between I(a) and I(b) there is at
least one x in la,b] such thatl(x) = y.
The zero theorem states that if f is
continuous on [a,b) and f(a) and f(b) have
opposite signs, then there is an x in (a,b)
such thatl(x) = O.
• Bisection Method. This a method of finding
zeros based on the zero theorem.
1. With f. a, b as in the zero theorem, the
midpoint XI = 112 (a+b) is an initial
estimate of a zero.
2. Assuming I(x /) is nonzero, there is a new
interval [a, XI] or [x" b] on which opposite
signs are taken at the endpoints. It contains
a zero, and its midpointx2 is a new estimate
of a zero.
3. Repeat step (2) with the new interval andx2.
4. The nth estimate xn differs from a zero by
no more than (ba)/2n.
• Extreme value theorem. Iff is continuous
on the closed interval [a,b], thenfachieves a
minimum and a maximum on la,bl: there are
c and d in [a,bl such thatf(c) s f(x) s f(d) for
all x in [a,b]. The proofs of this and the
intermediate value theorem use properties of
the set of real numbers not covered in
introductory calculus.
\:  0+
• Global conti·nuity. We say a function is
continuous if it is continuous on its domain,
meaning continuous at every point in its
domain, using onesided continuity at
endpoints of intervals. Caution: textbooks
sometimes refer to some points not in the
domain as points of discontinuity. Intuitively,
a function is continuous on an interval if
there are no breaks in its graph.
• Uniform continuity. A function I IS
unifonnly continuous on its domain D if for
every £ > 0 there is a 8 > 0 such that x, y in
D and ~  yl < 8 imply If(x)  fO')1 < £.
Uniform continuity implies continuity. A
continuous function on a closed interval
[a,b] is uniformly continuous.
THEORY
• Arithmetic. Scalar multiples of a
continuous Junction are continuous. Sums,
differences, products, and quotients of
continuous functions are continuous (on
their domains).
• Compositions. A composition of continuous
functions is continuous.
• Elementary functions. Polynomials, rational
functions, root functions, cxponentials and
logarithms, and trigonometric and inverse
trigonometric functions are continuous.
DERIVATIVES
DEFINITIONS
• Derivative. The derivative of f at a is the
number
r I(a + h)  I(a)
1 '/ ,j
,a/
=
h
h I.,?IO
provided the limit exists, in which case I is
said to be differentiable at a. The derivative
ofI is the function
The derivative is also
'/
,j =
r
I(x)
 I(a)
f
r.
,a/
x l.,?Ia
X
xl~ F(x _ a), with F(h) = I(a + h)  I(a)
• Zooming formulation. If the plot !amain for
I is narrowed through intervals centered at
x = a, while the ratio of the plot range to the
plot domain is held fixed, the graph of I
eventually appears linear (identical to the
tangent line at x = a). If r(a) .. 0, the
zoomed graph appears linear with no
constraint on the plot ranges (autoscaling).
• Notation. The derivative function itself is
denoted I' or D(f). Ify = f(x), the following
usually represent expressions for the
derivative function:
y', .;t. ,DxY,f'(x), 1; j(x)
The second is the Liebniz notation. Notations
for the derivative evaluated at x = a are
r(a), D(j)(a), .;t. Ix = a' .;t.lx = a f(x).
f
by
h
n be
approximates r(a) if h is small. It is the
slope of the secant line through the points
(a,f(a)) and (a+h,f(a+h)). The average of it
and the 'backward quotient',
I(a)  I(a  h)
e (by
and
'31.
rate
the
I
I
h
E.g.,
is the symmetric quotient
I(a + h)  I(a  h)
l~t
is
g in
the
ents
over
ma ll
2h
usually a better approximation ofr(a).
INTERPRETATIONS
• Rate of change. The derivative 1'(a) is the
instantaneous rate of change of I with
respect to x at x = a. It tells how fast I is
increasing or decreasing as x increases
through values near x = a. The average
rate of change oflover an interval [a,xl
. I(x)  I(a)
IS
x
a
. As x nears a, these average
rates approach F(a). The units of the
derivative are the units off(x) divided by
the units of x.
• Tangent line. The derivative F(a) is the
slope of the tangent line to the graph of I
at the point (a,l(a)). It is a limit of slopes
of secant lines passing through that point.
One
can
• Linear Approximation.
approximate values of I near a according
to f(x) == f(a) + F(a)(x  a)
E.g., since,
..!!....h =
dx
j62 ""
a
by the limit theorem for compositions applied
to
4
• Linearization. The linearization, or linear
approximation, of/at a is the linear function
xl~ I(a) + r(a)(xa)
Its graph is the tangent line to the graph offat
the point (a,l(a)). The derivative thus provides
a 'linear model' of the function near x = a.
• Differentials. The differential of/at a is the
expression df(a) = r(a)dx.
Applied to an increment .t:.x, it becomes
r(a).t:.x. If y =I(x), one writes dy =r(x)dx.
• Difference quotients. The difference quotient
I(a + h)  I(a)
__
1_
tant.
and
,) An
m ite
r to
i two
2h'
1M + 2 ...k64 ( 2) = 7.875
The approximation is better the closer x is
to a and the flatter the graph is near a.
• Differential changes. At a given input, the
derivative is the factor by which small input
changes are scaled to become approximate
output changes. The differential change at a
over an input increment .t:.x approximates
the output change:
F(a).t:.x == I(a+.t:.x)  f(a)
The differential change is the exact change
in the linear approximation.
0 11
I that
(
is
f on
ives
na
a
Differential Changes
f(a+tu)
Ii.~(a)~;;.
'  ;"::'~Ar'.t
(a) tu
~
tu
a
a+tu
derivatives continued next page

derivatives continued
• Velocity. Suppose set) is the position at time
t of an object moving along a straight line.
Its average velocity over a time interval
to
to t\
. s(t l
) 
t
IS
seta)
t
I
~:\. eX = eX,
0
Its instantaneous velocity at time t is
v(t) = s '(t).Its speed is lv(t)l. Its acceleration
is v'(t).
• Interpreting a derivative value. Suppose
T is temperature (in 0c) as a function of
location x (in cm) along a line. The
meaning of, for example, r(8) = 0.31
(oC/cm) is, at the location x = 8, small
shifts in the positive x direction yield small
increases in temperature in a ratio of about
0.31 °C per cm shift. Small shifts in the
negative direction yield like decreases in T.
•
•
•
•
and g are differentiable. Each rule should be
viewed as saying that the function to be
differentiated is differentiable on its domain
and that the derivative is as given. For each,
there is also a functional form ,
e.g., (ej) , = eF, and a Liebniz form,
e.g., ,1x (eu) = e ~;.
Sum.
,1< (f(x) +g(x)) =F(x) +g'(x)
Scalar multiple.
,1x [ef(x)) = ef'(x)
Product.
~~ [((x) g(x)) = F(x)g(x) + f(x)g'(x)
Quotient.
.!L[ f(x) 1=
f'{x) g (x)  f(x) g'(x)
g (x)
g (x)2
dx
APPLICATIONS
• Linear approximations at O.
The following are commonly used linear
approximati?ns valid near x = O.
sin x = x
tan x
=x
tr = 1+x
In (1+x) '" x
(1+x)1I2 '" 1+x/2
l/(1+x) '" Ix
The error in each approximation is no more
than M .IxI2/2, where M is any bound on
If'MI for
lYl
:5
lxi, I
being the relevant
function. E.g., Isin x  xl
• Newton's
method.
:5
.005 for
To
Ixl :5 0.1.
find
an
approximate root of f(x) = 0, select an
appropriate starting point x o' and evaluate
x,,+! = x"  I(xj / F(xj
successively for n
=
0, I, ..., until the
values do not change at the desired
• The Chain Rule (for compositions).
,1< l(fog)(x)] == 1x f(g(x)) = F(g(x)) g'(x)
This says that a small change in input to the
composition is scaled by g '(x) , then by F(g(x)).
In Liebniz notation, if z = fry) andy = g(x) , and
we thereby view z a fimction of x, then
,/z = dz dy dz being evaluated aty =g(x).
,Ix
dy dx' dy
In D notation, D(f 0 g) = fD(f)o gJ D(g).
• Inverse functions. If f is the inverse of a
function g (and g' is continuous and
nonzero), thenF(x) '(/(x))'
To get a specific for~ula directly, start with
y =f(x); rewrite it g(y) = x; differentiate with
respect to x to get g '(y)y' = 1; write this
y '=l/g '(y) and put g '(y) in terms of x, using
the relations y = f(x) and g(y) = x.
E.g., Y = In x; eY = x; eYy' = 1; Y '=1/eY = 1/x.
• Implicit functions. The derivative of a
function defined implicitly by a relation
F(x,y) = e may be found by differentiating
the relation with respect to x while treatingy
as a function of x wherever it appears in the
relation; and then solving for y' in terms of x
and y. The result is the same as obtained
from the formal expression
,
precision. The value on the right hand side
in the above is where the tangent line at
(x",f(xj) meets the xaxis.
Example of Newton's Method
j(x)
I
I
=
x 3x1 )
/ ,,1
fx F (x, y)
y=
:yF(x,y)
where y is treated as a constant in the
numerator, x as a constant in the denominator.
DERIVATIVE FORMULAS
• Constants. For any constant C, ,;~. C =
o.
• Reciprocal function.
d
dx
_~_ _
O._5 :::~_
I
1'5
_.L
1
x
x2
1x [f;X) 1=  f(~)2 r(x)
.!Lrx = _1_
dx
2 rx
The chain rule gives
• Related rates. Suppose two variables, each a
function of 'time; are related by an equation.
Differentiate both sides of the equation with
respect to time to get a relation involving the
time derivatives  the rates  and the original
variables. With sufficient data for the
variables and one of the rates, the derivative
relation can be solved for the other rate.
• Square root.
• Powers. For any real value of n, ixxn = nx,,I,
valid where X"l is defined. The chain rule gives
~~ If(x)]" = n(f(X)],,IF(x)).
• Exponentials. An exponential function has
derivative proportional to itself, the
proportionality factor being the natural
logarithm of the base:
5
,1x aX = (In a) aX
The chain rule gives
f,.
• Logarithms.
,/ In Ixl
dx
ej(xj = elM F(x).
 x'
I
d
dx
log
Ivl 
ul'
I
( Ina).\"
Same rules hold without absolute value, but
the domain is restricted to (0,00). The chain
rule gives
jg; .
f< In It(X)1 =
• Hyperbolic functions.
sinh 'x = cosh x
cosh 'x = sinh x
arcsinh 'x
=
,
~
"I
r x
2
arccoshx /,,2 I
• Trig functions.
sin' x = cos x
cos' x = sin x
tan' x = sec 2 x
cot' x = csc 2 x
sec' x = sec x tan x
csc' x = csc
. , I
x cot x
'
arCSIn x = /1 __ x 2 = arccos x
arctan 'x _1_, = arccot'x
I +x
ANALYSIS
LOCAL FEATURES OF
FUNCTIONS
• Neighborhoods. In the following, "near"
a point means in an open interval
containing the point. Such an open
interval is often called a neighborhood of
the point.
• Continuity. If a function is differentiable
at a point, then it is continuous there.
• Critical points. A point e is a critical
point ofI ifI is defined near e and either
F(e) = 0 orF(e) does not exist.
• Local extrema. A local minimum point
of I is a point c with f(x) ~ f(e) for x near
e. A local maximum point oflis a point ('
withl(x) :5 f(e) for x near c. If e is a local
extremum point, then it is a critical point
(this follows from definitions). Relative
extrema are the same as local extrema.
• First Derivative Test. Suppose e is a
critical point off, and I is continuous at c.
If F(x) changes sign from negative to
positive as x increases through e, then (' is
a local minimum point. If F(x) changes
sign from positive 10 negative as x
increases through e, then e is a local
maximum point. If/'(x) keeps the same
sign, then e is not an extremum point.
• Second Derivative Test. Suppose I is
differentiable near a critical point e. [f
I(e) > 0, then c is a local minimum point. [f
I(e) < 0, then c is a local maximum point.
• Inflection points. If the graph of / has a
tangent line (possibly vertical) at c and
/"(x) changes sign as x increases through c,
then c, or the graph point (cJ(c)) , is called
an inflection point. E.g., X I IJ has a vertical
tangent and inflection point at (0,0). An
inflection point for/is an extremum for/';
the tangent line is locally steepest at such a
point. The only possible inflection points
are where /"(x) = 0 or /"(x:) does not exist.
Inflection Point
APPLICATIONS
TRENDS & GLOBAL FEATURES
• Mean Value Theorem (MVT). If / is
continuous on la,h) and differentiable on
the open interval (a ,b), then there is a point
c in (a, h) with j·'(e) = feb)  f(a)
ba
Graphically, some tangent line between a
and h is parallel to the secant line through
(aJ(a)) ~nd (hJ(h)) . The case with
Ira) = /(h) = 0, whence/'(e) = 0, is Rolle's
Theorem. The proof of the MVT relies on
the Extreme Value Theorem.
Mean Value Theorem
a
i?0
c
b
• Increasing and decreasing. If/' = 0 on an
interval, then / is constant on that interval.
If/ ' > 0 on an interval, then / is strictly
increasing on that interval.
If/ ' < 0 on an interval, then / is strictly
decreasing on that interval. (These follow
from MVT.)
• Concavity. A graph is said to be concave up
[down[ at a point e if the graph lies above
[below] the tangent line near c, except at e. If
/" > 0 oli an interval, then the graph of/is
concave up on the interval (UPPOSITIVE);
also/, is increasing, and the tangent lines are
turning upward asx increases. Itf" < 0 on an
IS8N13: 9781572227965
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interval, then the graph of/is concave down
on the interval (DOWNNEGATIVE); also
/ ' is decreasing, and the tangent lines are
turning downward as x increases.
• Extrema on a closed interval. The global ,
or absolute, maximum and minimum
values of a continuous function on a closed
interval [a,h) (guaranteed to be achieved by
the Extreme Value Theorem) can only
occur at critical points or endpoints.
• Optimization with constraint. Here is an
" outline to approach optimization problems
involving two variables that are somehow
related.
1. Visualize the problem and name the
variables.
2. Write down the objective functionthe
one to be optimized as a function of two
variables.
3. Write down a constraint equation
relating the variables.
4. Use the constraint to rewrite the objective
function in terms of one variable.
S. Analyse the new function of one variable to
find its optimal point(s), and the optimal
value.
E.g., to maximize the area of a rectangle
with perimeter being p , we pose the
problem as maximizing A = Iw subject to
the constraint 21 + 2w = p. The constraint
gives w = p/2  I, when A = 1(P/21). The
maximmn occurs at 1 = p/4, with A = (p/4)2
A verbal result is clearest: it's a square.
For geometric problems, volume formulas
may be needed:
cylinder: 7t r2h,
cone: n r2h/3,
sphere: 47t ,3/3.
• Cubics. A cubicp(x) =~ + hx2 + cx+"dhas
exactly one inflection point: (h,k) where
h = h/(3a) and k = p(h). A normal form is
p(x)
= a(xh)3+m(xh)+k
where m = hh + c is the slope at the inflection
point. If m and a have opposite signs, the
horizontal line through the inflection point
meets the graph at two points, each a
distance jm/" from the inflection point, and
local extrema occur at points I /3 '" 0.6 times
that distance.
INTEGRATION
• Area under a curve. The integral of a
nonnegative function over an interval gives
the area under the graph of the function .
• Average value. The average va lue of /
over an interval la,hl may be defined by
average value = b
quac s u y.com
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.b
v(t) dt= s(h)  s(a).
which is the total displacem ent.
FUNDAMENTAL THEOREM OF
CALCULUS
• Antiderivatives. An antiderivati ve of a
function f is a function P whose
derivative is f r(x) = fix) for all x in
some domain. Any two anti derivatives of
a function on an interval di ffer by a constant.
(This follows from MVT.) E.g., arctan x and
arctan( I/x) are both anti derivati ves of
1/(1+ x 2 ) for x> O. (They di ffe r by 1t/2 .) An
antiderivative is also called an indefini te
integral, though the latter term often refers to
the entire family of anti derivative .
• The Fundamental Theorem. There are two
parts:
I. Evaluating integrals. IIf is continuous on
la,h], and F is any antideri vative off on that
interval, then
1 f(x)dx = P(x) I~ == F(b)  F(a).
.b
2. Constructing
antiderivatives. If /
continuous on [a,b], then the func tion
is
G(x) = .eNw) dw is an antideri vative off on
(a,b): G'(x) = fix). (The onesided derivatives
of G agree with/at the endpoints.)
• Differentiation ofintcgrals. To di fferentiate a
function such
as xl~ [. f (w)dw, view it as a
composition G(xl), with G as above. The chain
rule gives 1; G(.~) = G'(.,.2) • h = 2xf( x 2) .
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Often a rough estimate of an integral can be
made by estimating the average value (by
inspection of the graph. for example) and
multiplying it by the length of the interval.
• Accumulated change. The integral of a rate
of change gives the total change in the
original quantity over the time interval. E.g.,
if v(t)=s '(t) represents ve locity, then v(t)LV is
the approx imate displacem ent occurring in
the time increment t to t + LV. Adding the
displacements for all the tim e increments
gives the approximate change in position over
the entire time interva l. In the limit of sm all
time increments. one gets the integral
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