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1 Calculus 4/14/03 4:08 PM Page 1

SPARKCHARTSTM

CALCULUS I

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CHARTS

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BACKGROUND AND FUNCTIONS
Calculus is the study of “nice”—smoothly changing—functions
...

• Integral calculus studies areas enclosed by curves
...


FUNCTIONS
WHAT IS A FUNCTION?
A function is a rule for churning out values: for every value
you plug in, there’s a unique value that comes out
...

• The set of all the values that can be output is the range
...
Its
graph is a horizontal line at height c
...
The
slope of a straight line measures how steep it is; if (x1 , y1 )
and (x2 , y2 ) are two points on the line, then the slope is

y2 − y1
change in y

...
There may
not be a universal equation that describes such a function
...
It is defined everywhere except at the roots of
q(x)
q(x)
...

• If the degree of p(x) is greater than the degree of q(x)
(deg p(x) ≥ deg q(x) ), then at points where |x| is very
large, f (x) will behave like a polynomial of degree
deg p(x) − deg q(x)
...
See Limits and Continuity
...


EXPONENTIAL AND LOGARITHMIC FUNCTIONS—Very fast
or very slow growth
Simple exponential functions can be written in the form

y = ax , where the base a is positive (and a �= 1)
...
The domain is all the reals; the range is the
positive reals
...
The basic shape of the graph is
always the same, no matter the value of a
...
The number loga b is “the power to which you raise a to get b”:
loga x = y if and only if ay = x
...
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...

All rights reserved
...

A Barnes & Noble Publication
10 9 8 7 6 5 4 3 2

REMEMBER: Logarithms are exponents
...

Polynomial of degree 5
1 2
Roots: −2, −1, − 2 , 3 , 2
4 “turns”

SPECIAL CASES OF POLYNOMIALS—Lines and parabolas
A polynomial of degree 0 is the constant function f (x) = a
for a �= 0
...


A polynomial of degree 2 is a quadratic; it can be written in
the form f (x) = ax2 + bx + c
...

x=
2a

√
• If √b2 − 4ac < 0, then the quadratic has no real roots
...

• If b2 − 4ac > 0, then the quadratic has 2 real roots
...


Changing the base of a logarithm is the same thing as
mulplying the logarithm by a constant:

loga x = (loga b) (logb x)
...

b

The number e is a special real
number (approximately
2
...
The
logarithm base e is called the
natural logarithm and is written loge x = ln x
...
Any logarithmic expression can be written in terms of natural logarithms using the
change of base formula:

loga x =

ln x

...


The graph of a quadratic function is a parabola
...

• The vertex is at (h, k)
...
If a < 0, the parabola opens
Parabola
...

down
...
If |a| is small, the
parabola is wide
...
If y = f (x), then
�
plotting many points x, f (x) on the plane will give a picture of the function
...
All polynomials
are defined for all real numbers
...
If n is even, then the polynomial
reaches some maximum if an < 0 or some minimum if
an > 0
...
A polynomial of degree n has at most n roots or zeroes—values of x where the graph crosses the x-axis—and at most
n − 1 “turns” (peaks or valleys) in its graph
...
A graph represents a function as long as it passes the
vertical line test: for every x-value, there is at most one y value
...
If y = x2 + 1, then y is a
“function of x,” and it is the dependent variable
...

π
180

• The unit circle is the circle of radius 1 centered at
the origin O = (0, 0)
...
Since
θ and θ + 2π define the
same angle, all trig functions satisfy f (x) = f (x + 2π); they
are periodic with period 2π (or π)
...

For all θ, −1 ≤ sin θ ≤ 1
...


Cosine: cos θ = x, the x-coordinate of P = (x, y)
...
Cosine is an even function
...


Sinusoidal functions can be written in the form

y = A sin B(x − h) + k
...


• k is the average value: halfway between the maximum
and the minimum value of the function
...
A larger B
B
means more cycles in a given
interval
...


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COMBININ
• Two fun
subtracte
function i
original f
undefined

LIM

LIMIT OF A
If function
x gets clos
approache
or the value
the existen

limx→a f (

type of disc

• One-side
approache

limx→a− f
when x is c
limx→a+ f

to and larg
• If the lim
sided limi
Contrariwi
its of f (x

limx→a f (

sided limit

A NOTE ON
1
...
We can
limit at infi

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CHARTS

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“BUT HOW IS ONE TO MAKE A SCIENTIST UNDERSTAND THAT
THERE IS SOMETHING UNALTERABLY DERANGED ABOUT
ANTONIN ARTAUD
DIFFERENTIAL CALCULUS
...

Ex: If f (x) = x + 2 and g(x) = 4x, then

(f ◦ g)(x) = 4x + 2
...


• The unit circle is the circle of radius 1 centered at
the origin O = (0, 0)
...
Since
θ and θ + 2π define the
atisfy f (x) = f (x + 2π); they
r π)
...

Sine is an odd function
...

Cosine is an even function
...


cos θ
sin θ

=

x
y

written in the form

− h) + k
...


CHANGING A SINGLE FUNCTION
1
...
The new function y = f (x) + c has the same
shape and the same domain as the original function
...
Horizontal translation: The function y = f (x − c) is a
shift of the original function c units horizontally (to the right
if c is positive, left if c is negative)
...

3
...
If c < 1, then y = cf (x)is a
compression of the original function by a factor of c
...

4
...
Vertical
distances remain the same
...
The domain of the new
function includes only points that are in the domain of both
original functions
...
)

Vertical Stretch

Horizontal Stretch

5
...
The
function y = f (−x) is a reflection of the original function
over the x-axis
...

Ex: cos x is an even function
...
A
reflection over the x-axis is the same as a reflection over

LIMITS AND CONTINUITY
LIMIT OF A FUNCTION
If function f (x) comes infinitely close to some value L as
x gets close to a, we say that “L is the limit of f (x) as x
approaches a” and write limx→a f (x) = L
...
Rather, comparing the
limx→a f (x) and f (a) tells about the continuity or the
type of discontinuity of f (x) at x = a
...
The left-hand limit,
limx→a− f (x) exists if the f (x) is close to some value
when x is close to and smaller than a
...

• If the limit of f (x) as x → a exists, then so do both onesided limits, and the three limits have the same value
...

A NOTE ON INFINITY
1
...

Ex: lim

1
2
x→0 x

the y -axis
...

Ex: sin x is an odd function
...
This is
sometimes denoted by (f ◦ g)(x)
...
Here, the line x = 0 is a vertical asymptote:

the function tends towards the line but never quite reaches it
...
We can also look at the limit of f (x) at +∞ or −∞
...


gets very large, positively or negatively
...
If limx→±∞ f (x) = L exists and is finite, the
line y = L is a horizontal asymptote to the graph of f (x)
...
The line y = 0 is a horizontal asymp-

x→−∞

tote to the function y = ex
...
However, the statement “limx→a f (x) exists”
could mean that this limit is infinite
...

CONTINUITY AND DISCONTINUITY
• If f (a) exists and is equal to limx→a f (x), we say that
f (x) is continuous at x = a
...
If f (x) is continuous at every real x, we say that f (x)
is continuous on the whole real line or simply continuous
...
Ex: the function
2
y = x −4 is indistinguishable from y = x + 2 everyx−2
where except when x = 2, where it is undefined
...

Vertical asymptote: If either of the one-sided limits
limx→a− f (x) or limx→a+ f (x) exists and is infinite, then
f (x) has a (possibly one-sided) vertical asymptote at
x = a
...

Often, functions teachers use will have vertical asymptotes

Odd Function

Even Function

6
...
Such a function has a unique inverse f −1 (x) whose
domain is the range of f (x), and vice versa
...

• The inverse � the inverse function is the original
of
�
−1
(x) = f (x)
...
Ex: y = ex and y = ln x are inverse functions
...

• If f (x) takes the same value more than once, we
restrict the domain before taking the inverse
...

Inverse trigonometric functions: sin x, cos x, and tan x
are periodic and take on the same value many times
...

Arcsine, the principal inverse of sine, is often denoted
1
sin−1 x (not to be confused with sin x = csc x)
...

2 2
Arccosine, or cos−1 x, is defined on [−1, 1] and takes
values in the range [0, π]
...

2 2

where the two one-sided limits tend to infinities of opposite
sign
...


CONTINUITY OF BASIC FUNCTIONS
• All linear, polynomial, exponential, logarithmic, and
sinusoidal functions are continuous
...
At those roots of q(x) which are
also roots of p(x), they have removable discontinuities if
the multiplicity of the root in p(x) is at least as great as the
multiplicity of the root in q(x)
...
See graph on page 4 in Sketching
Graphs: Summary
...
The
2

1
x
trigonometric functions cot x = cos x and csc x = sin x
sin
are continuous everwhere except at the zeroes of sin x—

integer multiples of π
...


TIP: In practice, functions encountered in the classroom are
discontinuous only at isolated points
...

Continuity: Function f (x) is said to be continuous at x = a if and only if for every ε > 0,
there exists some δ > 0 such that whenever x0 is within δ of a, f (x0 ) is within ε of f (a)
(that is, |x0 − a| < δ implies that |f (x0 ) − f (a)| < ε )
...


x→a

�

x→a

x→a

If lim g(x) �= 0 , then lim

Quotient:

Trigo
know
the tr

��
�
lim f (x)
lim g(x)
x→a

x→a

limx→a f (x)
f (x)

...


x→a

x→a

x→a

LIMIT LAWS

The classic application of this theorem establishes that

Suppose f (x) and g(x) are two functions, a is a point (possibly ±∞) near which both
f (x) and g(x) are defined
...


Frequently-encountered limits:

Sum:

Scalar multiple:

x→a

lim

sin x
x

lim

�
�
lim f (x) ± g(x) = lim f (x) ± lim g(x)
x→a
x→a
�
�
lim cf (x) = c lim f (x)
Here, c is any real
...
This is also the slope of the line through the points a, f (a) and
�
b, f (b) on the graph of f (x)
...
Equivalently, the derivative is the slope
of the�tangent�line to the graph of f (x) at the point x = a—the unique line through the
point a, f (a) that touches the graph at only that point near x = a
...
Equivalently, f � (a) is the limit as
�
�
�
h → 0 of the slope of the line through a, f (a) and a + h, f (a + h)
...

f (x+h)−f (x)

is the derivative function of f (x)
...


If f (x) is differentiable at x = a, then f (x) is continuous at a
...
There are two cases where this occurs:

1
...
The
function is continuous at x = 0 since

=0

f � (0) undefined

2
...
” If f (x)
has a vertical tangent at x = a, then
the derivative f � (a) is undefined and
the graph of f � (x) will have a vertical asymptote at x = a
...
The derivative func1
tion, f � (x) = √ 2 , goes to infinity
3
3 x
at 0
...
”

d
dx

Quotient:

Vertical tangent at x

=0

f (x)
g(x)

�

f � (x)g(x) − f (x)g � (x)
=
g 2 (x)

MNEMONIC: “Ho d hi minus hi d ho over ho ho
...
Here are two ways of writing it:
�
1
...


2
...

dx
du dx

Implicit differentiation uses the product and chain rules to find slopes of curves when it is
difficult or impossible to express y as a function of x
...
Take the derivative of each term in the equation with
dy
dx
respect to x
...

2

Ex 2: x cos y − y 2 = 3x

d(cos y)
dy
dx
dx
Differentiate to obtain first dx cos y + x dx − 2y dx = 3 dx ,
cos y − x sin yy � − 2yy � = 3
...

sin y+2y

and

then

DERIVATIVES OF BASIC FUNCTIONS

DISPLA
MOTIO

Suppos
seconds
• The fi
instanta
particle
for the
• The fi
the inst
tion: a(
“meters

MAXIM

A local
�
point c
least (o
tion in s
minimu
a relativ

The glo

f (x) a

conside
the abs
If th
inclu
alwa
poss
Valu

Constants:

d(c)
=0
dx

A constant function is always flat
...


The wo
or maxi

Powers:

d(xn )
= nxn−1
dx

True for all real n �= 0
...
The most
dy
d
common notations in calculus are f � (x), y � , dx f (x), and dx
...
The derivative can now be found for any point on
y
the curve, even though it is not actually a function
...


limh→0−
No tangent at x

�
d �
f (x) ± g(x) = f � (x) ± g � (x)
dx
�
d �
cf (x) = cf � (x)
dx
�
d �
f (x)g(x) = f � (x)g(x) + f (x)g � (x)
dx

Scalar Multiple:

2

limx→0+ |x| = limx→0− |x| = 0,
but the derivative f � (0) is undefined

since the left-hand slope limit,
|h|
h = −1 , does not equal
the
right-hand
slope
limit,
|h|
limh→0+ h = 1
...


Sum and Difference:

�y

change in y

notation; dx evolved from �x = change in x , or slope
...

• Higher-order derivatives can be written in “prime” notation: f � (x), f �� (x), f ��� (x),

This is why e is called the “natural”
logarithm base: Aex are the only functions that are their own derivatives
...


• The derivative at a particular point a is most often expressed as f (a) or
�

This downloadable PDF copyright © 2004 by SparkNotes LLC
...

dx �
x=a

d(a )
= ax ln a
dx

When in doubt, convert ax to ex ln a
...


d(loga x)
1
=
dx
x ln a

f (4) (x), or in Leibniz notation:

loga x to

x

Logarithmic:

When in doubt, convert
ln x
ln a
...
Chec
defined

f (x) =

it may b
2
...
Check endpoints: If the domain is a closed interval [a, b],
always check f (a) and f (b) when looking for extrema
...

Approximating f (x) near a: For small h,

�

(x)

riting it:

hen it is
easiest
on with

, which
point on
rst solve

then

flat
...


f (a + h) ≈ f � (a)h + f (a)
DISPLACEMENT, VELOCITY, ACCELERATION:
MOTION IN ONE DIMENSION
Suppose a particle’s position on a line in meters at time t
seconds is determined by the function s(t)
...
The units
for the first derivative are “meters per second,” m/s
...
The units for acceleration are
“meters per second per second,” m/s2
...
A local
minimum or maximum is also called
a relative minimum or maximum
...
The global minimum or maximum is also called
the absolute minimum or maximum
...
(This is the Extreme
Value Theorem
...
The plural of extremum is extrema
...

All extrema—that is, all minima and maxima—happen
either at endpoints or at critical points
...


ex ln a
...


How to find extremum points:
1
...
Such a point may be a local extremum, as in
f (x) = |x| at x = 0
...
Or
it may be neither
...
Check critical points where f � (x) = 0
...

• If the sign of f � (x) switches from + to − at x = a,
then f (a) is a local maximum
...

• If the sign of f � (x) does not switch around x = a,
then f (a) is neither a maximum nor a minimum
...
Ex: f (x) = (x2 + 1)e−x ,
graphed under “Maxima and Minima”, has a local minimum at
x = 0 but no global minimum, though f (x) > 0 for all real x
...
Equivalently, a function has only one family of
antiderivatives
...
See
the Calculus II SparkChart for more on antiderivatives
...


SKETCHING GRAPHS: SUMMARY
L’HÔPITAL’S RULE
1
...

function at the endpoints
...
Horizontal asympand g � (x) �= 0 on an interval near a (except perhaps at a)
...
Evaluate f (0)
lim g(x) = 0
and
If lim f (x) = 0
to find the y -intercept
...
Gaps: Find all isolated points x = a where f (a) is not
lim g(x) = ±∞ ,
If lim f (x) = ±∞ and
defined
...
A vertix→a
x→a
f � (x)
f (x)
cal asymptote will appear if limx→a± f (x) = ±∞
...

removable discontinuity (hole in the graph) will appear if
• L’Hôpital’s Rule can also be applied if the limit is onelimx→a f (x)exists and is finite
...

3
...
If not, evaluating the function at the critical points
x approaches infinity (x → ±∞)
...

limit is well-defined
...
Rise and fall: Determine
limit is infinite and the bottom limit is zero, or vice versa
...
Since lim ln x = 0 and lim x − 1 = 0 , use
x→1
x→1
x→1
decreasing by looking at
�
L’Hôpital’s Rule:
the sign of f (x)
...

f � (x) > 0 , then f (x) is
x−1 1
x→1
x−1 x−1
dx
increasing
...

1
Horizontal asymptote: y = 2
nate forms, such as ±∞ · 0
...
Local extrema: Find all Vertical asymptote: x = −1 and x = 2 expression to 0 or ±∞
...
Convert to the expression limx→−∞ e−x ,
the critical points where
�
�
which is an indeterminate form −∞
...

∞
1
convert to limx→−∞ −e−x = 0
...
Concavity: Determine when the function cups up or
��
��
down by looking at the sign of f (x)
...
If f �� (a) = 0, then the function is temporarily not
��
curving at x = a; if f (x) is changing sign near x = a,
A: Vertical tangent
B: Local minimum
then this is a point of inflection (change in concavity)
...
This is a completely intuitive statement!
Rolle’s Theorem: If f (x) is continuous on the closed interval
[a, b], differentiable on the open interval (a, b), and satisfies
f (a) = f (b), then for some c in the interval (a, b), we have
f � (c) = 0
...


SPARKCHARTS

TIP: Often, but not always, f �� (a) = 0 means that f (a) is
neither a minimum nor a maximum
...
The second derivative
test tells you nothing, but the changing sign of the first derivative indicates that f (0) = 0 is a local minimum
...
If f (x) is continuous on the closed interval
[a, b] and differentiable on
the open interval (a, b),
then there exists a point
c ∈ (a, b) such that the
slope of the tangent to f (x)
at x = c is the same as the
slope of the secant line
f (b)−f (a)
MVT: f � (c) =
b−a
through � the �two points
�
�
f (b)−f (a)
�
a, f (a) and b, f (b) : that is, f (c) = b−a
...
95 CAN

The tangent line gives a (very) crude approximation to

f (x): If h is small, then f (a + h) ≈ f � (a)h + f (a)
...


Alternatively, you can use the second derivative test:
• If f �� (a) < 0, then f (a) is a local maximum
...

• If f �� (a) = 0, then you must check whether f � (x)
switches sign around x = a
...
95

The tangent line to a curve y = f (x) at the point
�
�
a, f (a) is given by the equation
y = f (a) + f � (a)(x − a)
...
Williams
Illustration: Matt Daniels
Series Editors: Sarah Friedberg, Justin Kestler

1
d(tan−1 x)
=
1 + x2
dx

USING DERIVATIVES
THE TANGENT LINE: APPROXIMATING f (x) NEAR x = a

)g (x)

�

d(cos x)
= − sin x
dx
d(cot x)
= − csc2 x
dx
d(csc x)
= − csc x cot x
dx

d(sin x)
= cos x
dx
d(tan x)
= sec2 x
dx
d(sec x)
= sec x tan x
dx

Inverse Trigonometric: A pain
...


Report errors at
www
...
com/errors

Trigonometric: Found using the definition of derivative and the Squeeze Theorem
...


C: Point of Inflection
D: Point of Inflection
E: Point of Inflection
F: Local maximum
• The function is decreasing
from the y -axis to B,
increasing from B to F, and
then decreasing from F on
...


SPARKCHARTS™ C++ page 4 of 4



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