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Engineering Mathematics first semester CALCULUS printed notes

CHAPTER 1

Numbers and Functions
The subject of this course is “functions of one real variable” so we begin by wondering what a real number
“really” is, and then, in the next section, what a function is
...
What is a number?
1
...
Different kinds of numbers
...

Together these form the integers or “whole numbers
...

These are the so called fractions or rational numbers such as
1 1 2 1 2 3 4
, , , , , ,
or

2 3 3

4 4 4 3

, ···

1
1
2
1
2
3
4
− , − , − , − , − , − , − , ···
2 3 3 4 4 4 3

By definition, any whole number is a rational number (in particular zero is a rational number
...

One day in middle school you were told that there are other numbers besides the rational numbers, and
the first example of such a number is the square root of two
...
e
...
e
...

n
Nevertheless, if you compute x2 for some values of x between 1 and 2, and check if you
get more or less than 2, then it looks like there should be some number x between 1
...
5 whose square is exactly 2
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This raises several questions
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4 and 1
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44
1
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It turns out to be rather difficult to give a precise
description of what a number is, and in this course we won’t try to get anywhere near the bottom of this
issue
...

One can represent certain fractions as decimal fractions, e
...

279
= 1116 = 11
...

100
25

Not all fractions can be represented as decimal fractions
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333 333 333 333 333 · · ·
3
It is impossible to write the complete decimal expansion of 1 because it contains infinitely many digits
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An electronic calculator, which always represents
numbers as finite decimal numbers, can never hold the number3 1 exactly
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If the decimal
expansion doesn’t end, then it must repeat
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142857 142857 142857 142857
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A real number is specified by a possibly unending decimal expansion
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414 213 562 373 095 048 801 688 724 209 698 078 569 671 875 376 9
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To give a precise description of a real number (such as 2) you have to
explain how you could in principle compute as many digits in the expansion as you would like
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√
1
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A reason to believe in 2
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In
middle or high school you learned s√
omething similar to the following geometric construction
of a line segment whose length is 2
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The figure you get
consists of 5 triangles of equal area and by counting triangles you see that the larger
square has exactly twice the are√
a of the smaller square
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Why are real numbers called real? All the numbers we will use in this first semester of calculus are
“real numbers
...
No real number has this property since the square of any real number is positive, so
it was deci√
ded to call this new imagined number “imaginary” and to refer to the numbers we already have
(rationals, 2-like things) as “real
...
3
...
It is customary to visualize the real numbers as points
on a straight line
...
We also
decide which direction we call “left” and hence which we call “right
...
”
To plot any real number x one marks off a distance x from the origin, to the right (up) if x > 0, to the
left (down) if x < 0
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In particular, the
distance is never a negative number
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To draw the half open interval [−1, 2) use a filled dot to mark the endpoint which is included
and an open dot for an excluded endpoint
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To find 2 on the real line you draw a square of sides 1 and drop the diagonal onto the real line
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we write down in this course will be true for some
values of x but not for others
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Below are some examples of sets of real numbers
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The collection of all real numbers between two given real numbers form an interval
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set of all real numbers x which satisfy a ≤ x < b
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set of all real numbers x which satisfy a ≤ x ≤ b
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E
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(−∞, 2] is the interval of all real numbers
(both positive and negative) which are ≤ 2
...
4
...
A common way of describing a set is to say it is the collection of all real numbers
which satisfy a certain condition
...
(A,B,C,D,
...
e
...
This set consists of two parts: the interval (−∞, −1) and the interval (1, ∞)
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Some sets can be very difficult to draw
...
In this course we will try to avoid such sets
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Or the set
E = x | x3 − 4x2 + 1 = 0 }
which consists of the solutions of the equation x3 − 4x2 + 1 = 0
...
)
If A and B are two sets then the union of A and B is the set which contains all numbers that belong
either to A or to B
...


}

Similarly, the intersection of two sets A and B is the set of numbers which belong to both sets
...


}

2
...
What is the 2007th digit after the period in the expansion of 1 ?

4
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Is it always true that
A ∩ B is an interval? How about A ∪ B?

7

2
...


5
...
}
Are these sets the same?

3

25

15625

6
...


3
...
Each of these
sets is the union of one or more intervals
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Which of thee sets are finite?
= x}
3x + 2 0
A |
−}
B = x | x2 − 3x + 2 ≥ 0
≤
}

Write the numbers
x = 0
...
,

x2

and z = 0
...

as fractions (i
...
write them as

m,

specifying m and n
...
A similar trick
works for y, but z is a little harder
...


}

Group Problem
...
e
...
273273273273
...
99999999999999999
...
Functions
Wherein we meet the main characters of this semester
3
...
Definition
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The set of numbers for which a function is defined is called its domain
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The rule must be unambiguous: the same
xmust always lead to the same f (x)
...
Here the rule defining f is
“take the square root of whatever number you’re given”, and the function f will accept all nonnegative real
numbers
...
Most often it is a formula, as in
the square root example of the previous paragraph
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Functions which are defined by different formulas on different intervals are sometimes called
piecewisedefined functions
...
2
...
You get the graph of a function f by drawing all points whose coordinates are (x, y) where x must be in the domain of f and y = f (x)
...
The graph of a function f
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y0

P0
−

y

y

Figure 4
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The line is the graph of f (x) = mx + n
...

x1−x0

3
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Linear functions
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Its graph is a straight line
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Conversely, any straight line which is not vertical (i
...
not
parallel to the y-axis) is the graph of a linear function
...

x1 − x 0
This formula actually contains a theorem from Euclidean geometry, namely it says that the ratio (y1 − y0) :
(x1 − x0) is the same for every pair of points (x0, y0) and (x1, y1) that you could pick on the line
...
4
...
” In this course we will usually not be careful about
specifying the domain of the function
...
For instance, if we say that h is the
function
√
h(x) = x

Figure 5
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”
The circle fails both tests
...


A systematic way of finding the domain and range of a function for which you are only given a formula is
as follows:
• The domain of f consists of all x for which f (x) is well-defined (“makes sense”)
• The range of f consists of all y for which you can solve the equation f (x) = y
...
5
...
The expression 1/x2 can be computed
for all real numbers x except x = 0 since this leads to division by zero
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To find the range we ask “for which y can we solve the equation y = f (x) for x,” i
...
we for which y can you
solve y = 1/x2 for x?
If y = 1/x2 then we must have x2 = 1/y, so first of all, since we have to divide by y, y can’t be zero
...
On the other hand, if y > 0 then y = 1/x2 has a solution
√
(in fact two solutions), namely x = ±1/ y
...

3
...
Functions in “real life
...
If some
object is moving along a straight line, then you can define the following function: Let x(t) be the distance
from the object to a fixed marker on the line, at the time t
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There are many examples of this kind
...
Here the domain is
the interval [0, T ], where T is the life time of the cell, and the rule that describes the function is
Given t, weigh the cell at time t
...
7
...
Generally speaking graphs of functions are curves in the plane but
they distinguish themselves from arbitrary curves by the way they intersect vertical lines: The graph of

a function cannot intersect a vertical line “x = constant” in more than one point
...


3
...
Examples
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The collection of points determined by the equation x2 + y2 = 1 is a circle
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See Figure 6
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Inverse functions and Implicit functions
For many functions the rule which tells you how to compute it is not an explicit formula, but instead an
equation which you still must solve
...
”
4
...
Example
...

In this example you can solve the equation for y,
3 − x2
y=


...

Here we have two definitions of the same function, namely
(i) “y = f (x) is defined by x2 + 2y − 3 = 0,” and
(ii) “f is defined by f (x) = (3 − x2)/2
...
You see that with an “implicit function”
it isn’t the function itself, but rather the way it was defined that’s implicit
...
2
...
Define g by saying that for
any x the value y = g(x) is the solution of
x2 + xy − 3 = 0
...

x
Unlike the previous example this formula does not make sense when x = 0, and indeed, for x = 0 our rule for
g says that g(0) = y is the solution of
02 + 0 · y − 3 = 0, i
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y is the solution of 3 = 0
...

√
y = + 1 − x2

√
y = − 1 − x2

Figure 6
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4
...
Example: the equation alone does not determine the function
...

If x > 1 or x < 1 then x2 > 1 and there is no solution, so h(x) is at most defined when 1 x 1
...

The rule which defines a function must be unambiguous, and since we have not specified which of these two
solutions is h(x) the function is not defined for −1 < x < 1
...
Here are three possibilities:
h1(x) = the nonnegative solution y of x2 + y2 = 1
h2(x) = the nonpositive solution y of x2 + y2 = 1
(
h1(x) when x < 0
h3 (x) =
h2(x) when x ≥ 0

4
...
Why use implicit functions? In all the examples we have done so far we could replace the
implicit description of the function with an explicit formula
...
For instance, you can define a function f by
saying that y = f (x) if and only if
(1)

y3 + 3y + 2x = 0
...
”
E
...
to compute f (0) you set x = 0 and solve y3 + 3y = 0
...
To
compute f (1) you have to solve y3 + 3y + 2 · 1 = 0, and if you’re lucky you see that y = −1 is the solution,
and f (1) = −1
...
Solving (1) is not easy
...
Here it is:
q
q
√
√
3
3
2
y = f (x) = −x +
1+x − x+
1 + x2
...

4
...
Inverse functions
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So to find y = f −1(x) you solve the equation x = f (y)
...


The graph of f

f (c)
The graph of f −1
c
b
f (b)
a
f (a)
a

b

c

f (b)

f (a)

f (c)

Figure 7
...


4
...
Examples
...
Then the equation f (y) = x works out to
be
2y + 3 = x
and this has the solution
y=

x−3


...

Next we consider the function g(x) = x2 with domain all positive real numbers
...
e
...
If x < 0 then√this
equation has no solutions since y≥0 for all y
...

√
So we see that g−1(x) is defined for all nonnegative real numbers x, and that it is given by g−1(x) = x
...
7
...
The familiar trigonometric functions Sine, Cosine and Tangent
have inverses which are called arcsine, arccosine and arctangent
...
We will avoid the sin −1 y notation because it is ambiguous
...


=

1

,

sin y
5
...
The functions f and g are defined by
f (x) = x2 and g(s) = s2
...
Find a formula for the function f which is defined by
y = f (x) ⇐⇒
What is the domain of f ?

x2 y

+ y = 7
...
Find a formula for the function f which is defined by
y = f (x) ⇐⇒ x2y − y = 6
...
Let f be the function defined by y = f (x) ⇐⇒ y is
the largest solution of
y2 = 3x2 − 2xy
...
What are the domain and range of
f?

for all real numbers x
...
Find a formula for the function f which is defined by
y = f (x) ⇐⇒ 2x + 2xy + y2 = 5 and y > −x
...

where x and t are arbitrary real numbers
...
Use a calculator to compute f (1
...
4
...
2))
14
...


What are the range and domain of f ?
21
...
On a graphing calculator plot the graphs of the following functions, and explain the results
...
)
f (x) = arcsin(sin x),

−2π ≤ x ≤ 2π g(x)
0≤x≤1

= arcsin(x) + arccos(x),
h(x) = arctan

sin x
,
cos x

| x| < π/2

k(x) = arctan

cos x
,
sin x

| x| < π/2

The following exercises review precalculus material involving quadratic expressions ax2 + bx + c in one way or
another
...
Explain how you “complete the square” in a quadratic
expression like ax2 + bx
...
Find the range of the following functions:
f (x) = 2x2 + 3
g(x) = −2x2 + 4x

l(x) = arcsin(cos x),

−π ≤ x ≤ π

m(x) = cos(arcsin x),

−1 ≤ x ≤ 1

16
...
Sketch
≤ ≤
the graphs of both f and f −1
...
Find a number a such that the function f (x) =
sin(x + π/4) with domain a≤x ≤
a + π has an inverse
...

18
...
3
...


24
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For each real number a we define a line la with
equation y = ax + a2
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What are the range

19
...


20
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−2, −1, − 2 , 0, 2 , 1, 2
...

25
...


For which values of m and n
does the graph of
f (x) = mx + n not intersect the graph of
g(x) = 1/x?



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