Calculus

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Contents
1 Functions 2
1.1 The Concept of a Function . . . . . . . . . . . . . . . . . . . . 2
1.2 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . 12
1.3 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . 19
1.4 Logarithmic, Exponential and Hyperbolic Functions . . . . . . 26
2 Limits and Continuity 35
2.1 Intuitive treatment and definitions . . . . . . . . . . . . . . . 35
2.1.1 Introductory Examples . . . . . . . . . . . . . . . . . . 35
2.1.2 Limit: Formal Definitions . . . . . . . . . . . . . . . . 41
2.1.3 Continuity: Formal Definitions . . . . . . . . . . . . . 43
2.1.4 Continuity Examples . . . . . . . . . . . . . . . . . . . 48
2.2 Linear Function Approximations . . . . . . . . . . . . . . . . . 61
2.3 Limits and Sequences . . . . . . . . . . . . . . . . . . . . . . . 72
2.4 Properties of Continuous Functions . . . . . . . . . . . . . . . 84
2.5 Limits and Infinity . . . . . . . . . . . . . . . . . . . . . . . . 94
3 Dierentiation 99
3.1 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.2 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.3 Dierentiation of Inverse Functions . . . . . . . . . . . . . . . 118
3.4 Implicit Dierentiation . . . . . . . . . . . . . . . . . . . . . . 130
3.5 Higher Order Derivatives . . . . . . . . . . . . . . . . . . . . . 137
4 Applications of Dierentiation 146
4.1 Mathematical Applications . . . . . . . . . . . . . . . . . . . . 146
4.2 Antidierentiation . . . . . . . . . . . . . . . . . . . . . . . . 157
4.3 Linear First Order Dierential Equations . . . . . . . . . . . . 164
i
ii
CONTENTS
4.4 Linear Second Order Homogeneous Dierential Equations . . . 169
4.5 Linear Non-Homogeneous Second Order Dierential Equations 179
5 The Definite Integral 183
5.1 Area Approximation . . . . . . . . . . . . . . . . . . . . . . . 183
5.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . 192
5.3 Integration by Substitution . . . . . . . . . . . . . . . . . . . . 210
5.4 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . 216
5.5 Logarithmic, Exponential and Hyperbolic Functions . . . . . . 230
5.6 The Riemann Integral . . . . . . . . . . . . . . . . . . . . . . 242
5.7 Volumes of Revolution . . . . . . . . . . . . . . . . . . . . . . 250
5.8 Arc Length and Surface Area . . . . . . . . . . . . . . . . . . 260
6 Techniques of Integration 267
6.1 Integration by formulae . . . . . . . . . . . . . . . . . . . . . . 267
6.2 Integration by Substitution . . . . . . . . . . . . . . . . . . . . 273
6.3 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . 276
6.4 Trigonometric Integrals . . . . . . . . . . . . . . . . . . . . . . 280
6.5 Trigonometric Substitutions . . . . . . . . . . . . . . . . . . . 282
6.6 Integration by Partial Fractions . . . . . . . . . . . . . . . . . 288
6.7 Fractional Power Substitutions . . . . . . . . . . . . . . . . . . 289
6.8 Tangent
x/
2 Substitution . . . . . . . . . . . . . . . . . . . . 290
6.9 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . 291
7 Improper Integrals and Indeterminate Forms 294
7.1 Integrals over Unbounded Intervals . . . . . . . . . . . . . . . 294
7.2 Discontinuities at End Points . . . . . . . . . . . . . . . . . . 299
7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
7.4 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . 314
8 Infinite Series 315
8.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
8.2 Monotone Sequences . . . . . . . . . . . . . . . . . . . . . . . 320
8.3 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
8.4 Series with Positive Terms . . . . . . . . . . . . . . . . . . . . 327
8.5 Alternating Series . . . . . . . . . . . . . . . . . . . . . . . . . 341
8.6 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
8.7 Taylor Polynomials and Series . . . . . . . . . . . . . . . . . . 354
CONTENTS
1
8.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
9 Analytic Geometry and Polar Coordinates 361
9.1 Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
9.2 Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
9.3 Hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
9.4 Second-Degree Equations . . . . . . . . . . . . . . . . . . . . . 363
9.5 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 364
9.6 Graphs in Polar Coordinates . . . . . . . . . . . . . . . . . . . 365
9.7 Areas in Polar Coordinates . . . . . . . . . . . . . . . . . . . . 366
9.8 Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . 366
Chapter 1
Functions
In this chapter we review the basic concepts of functions, polynomial func-
tions, rational functions, trigonometric functions, logarithmic functions, ex-
ponential functions, hyperbolic functions, algebra of functions, composition
of functions and inverses of functions.
1.1 The Concept of a Function
Basically, a
function
f
relates each element
x
of a set, say
D
f
, with exactly
one element
y
of another set, say
R
f
. We say that
D
f
is the
domain
of
f
and
R
f
is the
range
of
f
and express the relationship by the equation
y
=
f
(
x
).
It is customary to say that the symbol
x
is an
independent variable
and the
symbol
y
is the
dependent variable
.
Example 1.1.1 Let
D
f
=
{a,b,c}, R
f
=
{
1
,
2
,
3
}
and
f
(
a
) = 1
, f
(
b
) = 2
and
f
(
c
) = 3. Sketch the graph of
f
.
graph
Example 1.1.2 Sketch the graph of
f
(
x
) =
|x|
.
Let
D
f
be the set of all real numbers and
R
f
be the set of all non-negative
real numbers. For each
x
in
D
f
, let
y
=
|x|
in
R
f
. In this case,
f
(
x
) =
|x|
,
2
1.1. THE CONCEPT OF A FUNCTION
3
the absolute value of
x
. Recall that
|x|
=
x
if
x
0
−x
if
x <
0
We note that
f
(0) = 0
,f
(1) = 1 and
f
(
−
1) = 1.
If the domain
D
f
and the range
R
f
of a function
f
are both subsets
of the set of all real numbers, then the
graph
of
f
is the set of all ordered
pairs (
x,f
(
x
)) such that
x
is in
D
f
. This graph may be sketched in the
xy
-
coordinate plane, using
y
=
f
(
x
). The graph of the absolute value function
in Example 2 is sketched as follows:
graph
Example 1.1.3 Sketch the graph of
f
(
x
) =
p
x−
4
.
In order that the range of
f
contain real numbers only, we must impose
the restriction that
x
4. Thus, the domain
D
f
contains the set of all real
numbers
x
such that
x
4. The range
R
f
will consist of all real numbers
y
such that
y
0. The graph of
f
is sketched below.
graph
Example 1.1.4 A useful function in engineering is the unit step function,
u
, defined as follows:
u
(
x
) =
0 if
x <
0
1 if
x
0
The graph of
u
(
x
) has an upward
jump
at
x
= 0. Its graph is given below.
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