Calculus 1 with Dr. Janet
Semester 1, 2021/22
1.1 What is Calculus?
Syllabus
• Arithmetic, algebra, geometry, etc. are useful for
describing quantities that are not changing or moving.
1. Functions, Limits & Continuity
• But we live in a world full of change!
2. Differentiation
• Calculus gives us tools to describe change.
3. Applications of Differentiation
Examples:
4. Integration
5. Applications of Integration
3
Calculus is the best way to describe most of the 'laws
of nature' as well as many relationships in finance,
engineering and other fields.
Chapter 1 Functions, Limits
and Continuity
1.1
1.2
1.3
1.4
1.5
1.6
1.7
What is Calculus?
Straight Lines. Equations of Lines
Functions and Graphs
New Functions from Old Functions. Inverse Functions
Parametric Curves
Definition of a Limit. One-sided Limits
Laws of Limits. Evaluating Limits. The Squeeze
Theorem
1.8 Limits Involving Infinity
1.9 Continuity
1.10 The Intermediate Value Theorem
2
"Today calculus is used in
calculating the orbits of satellites and spacecraft,
in predicting population sizes,
in estimating how fast coffee prices will rise,
in forecasting weather,
in measuring the cardiac output of the heart,
in calculating life insurance premiums,
and in a great variety of other areas."
James Stewart
4
Example 1
1.2 Straight Lines. Equations of
Coordinates and Graphs
Lines
Draw line segments with slope
a) 3, b) 5, c) 1/2, d) -1.
O is the origin
Ox is the x-axis
Oy is the y-axis
(x, y) are the coordinates
of a point
y = x2
The graph of an equation is the
set of all points (x, y) whose
coordinates satisfy the equation.
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The SLOPE of a Straight Line
Example 2
Consider any two points (x1, y1) and (x2, y2) on a straight
line segment. On the interval [x1, x2],
is the change in x
is the change in y.
Find the slope of the graph
a) on the interval (2, 5)
The slope (or gradient) of the
line is
b) on (0, 2)
• E.g. if m = 5 then Dy = 5 Dx,
so for every unit increase in x,
y increases by 5 units.
• m is a constant, characteristic of the line segment.
• m tells us the rate of change of y with respect to x.
http://www.mathwarehouse.com/algebra/linear_equation/interactive-slope.php
+slope gif
c) at x = 6
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8
The Equation of a Straight Line
Example 3
• Suppose a straight line with slope m crosses the y -axis at
y = c. We call c the y-intercept.
• For any two points on the line,
• Setting (x1, y1) = (0, c) and letting (x2, y2) be a
Sketch the following graphs:
(a) y = x + 2
(c) y = 1 – x
(b) y = 2x – 6
(d) 2y = x + 2
general point (x2, y2) = (x, y),
we get
and so
This is the most common
way of writing the equation
of a straight line. It is called
the slope-intercept form.
The slope-intercept form is very convenient for graphsketching.
slope
y
y = 3x
y=x+1
Point-Slope Form
For a line with slope m passing through point (x1, y1):
y=x
y=x
2
1
1
1
-1
OTHER FORMS
The equation of a straight line can also be rearranged
or written in other ways, for example:
intercept
y
3
11
9
x
y=x–1
Two Point Form
For a line passing through points (x1, y1) and (x2, y2):
x
-1
(These results follow directly from
y=-x
10
)
12
Example 4
1.3 Functions and Graphs
Find the equation of the straight line passing through
points (2, 0) and (0, 3).
1.3.1 Functions
A function arises when one quantity depends on
another. E.g.
 the height H of a child varies with age t.
 the cost C of mailing a parcel depends on its mass m.
 the area A of a circle depends on the radius r.
Given the value of x, there is a rule which determines
the value of f. We say f is a function of x.
It is like a machine:
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Practical Application
x is called an independent variable.
f(x) is a dependent variable. (It depends on x.)
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Functions are often expressed by formulae.
Example 5 On a certain day, the temperature of air at
ground level was 20 ºC and the temperature at a height of 1 km
was 10 ºC. Assume temperature varies linearly with height.
a) Sketch a graph of the temperature T (in ºC) as a function of
height h in kilometres. b) Find the equation of the line.
c) What is the slope? What are its units? What does it mean?
d) Find the temperature at a height of 2.5 km.
Example 6
Given
, find:
a)
b)
c)
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Definition
A function f is a rule that assigns to each element in
some set D(f) exactly one element f(x) in a set R(f).
The element f(x) is called the value of f at x.
Many functions can be represented by their graph.
The graph of a function f is the graph y = f(x).
It can also be visualized as an arrow diagram:
But not every graph or equation represents a function!
The domain D is the set of values x can take,
the range R is the set of values f(x) can take.
If not explicitly given, D(f) is the set of numbers for which
f(x) makes sense.
To be a function, each x must correspond to a single
value of y = f(x).
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Example 7
19
Vertical Line Test. A curve in the xy-plane is the
graph of a function of x if and only if any vertical line
intersects the curve not more than once.
State the domain and range of the given functions.
a) f(x) = x2 + 3
Yes!
No!
b)
Example 8
Sketch the graphs (a) y = x2, (b) y2 = x. State whether or not
each curve represents a function of x.
c) h(x) = 2 + 3 sin(πx)
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Example 10
Representing Functions
A box with an open top is made from a rectangular piece of
card, 15 cm  20 cm, by cutting out squares of side length x at
each corner, then folding up the sides, as shown in the figures.
Find a formula for the volume of the box as a function of x.
A function can generally be represented in one or
more of the following four ways:
(1) a verbal description
(2) a table of values
(3) a graph
(4) a formula
You need to be able to move between these forms.
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Example 9
Functions and Mathematical Modelling
a) Sketch an approximate graph of your height H as a function
of your age t.
In many practical situations, data does not fit a formula
exactly, but we can use an approximate formula to ‘model’
the data.
When we plot this
data, we find it lies
approximately on a
straight line
CO2 level (ppm)
b) Find a formula for the area A of a circle as a function of the
circumference l.
For example, the table
shows the CO2 level
measured at a certain
place 1980 – 2002.
22
year
24
So we could assume a linear model for this data.
- We could find the equation of
the straight line through the
end points.
- Then use our equation to
predict the 2021 CO2 level, etc..
C = 1.545t - 2721
This is an example of
mathematical modelling.
real
problem
formulate
maths
model
A polynomial of degree 1 has form f(x) = mx + c
so is a linear function.
A polynomial of degree 2, f(x) = ax2 + bx + c,
is called a quadratic function.
A polynomial of degree 3 is called a cubic function.
Example Sketches of four polynomials are shown below.
What degree do you think each has?
solve
maths
solution
interpret
real
prediction
test
Here, data was modelled with a linear function.
Sometimes other functional forms will be appropriate.
Models are never absolutely accurate but a good model
yields predictions close to reality.
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1.3.2 Some Common Functions
We will revise some common classes of functions.
You should be able to sketch these types of functions
quickly and know their basic properties.
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POWER FUNCTIONS have the form
a is a constant.
where
You should know the graphs of common functions such as:
y = x3
POLYNOMIALS
A polynomial is a function of the form
y = x2
where n is a non-negative integer
and the numbers an are constants.
The numbers an are called coefficients.
The value of the highest power, n, is the degree of
the polynomial
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TRIGONOMETRIC FUNCTIONS
• You should know the sine (sin), cosine (cos) and
tangent (tan) functions
EXPONENTIAL FUNCTIONS have the form
x is the exponent (or power or index)
a is the base
• Also
The most common exponential function (often called the
exponential function) is f(x) = ex.
e is an irrational number called the exponential constant,
e = 2.7182818…. (Its importance will become clearer later!)
• In calculus, USE RADIANS unless told otherwise.
• Complete the table:
Graphs
sin q
q
0
.
cos q
tan q
y = ex
You should know the graphs of
a) y = ex (exponential growth)
p/6
b) y = e-x (exponential decay)
p/4
p/3
p/2
y = e-x
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Trigonometric functions are periodic.
• sin x, cos x have period 2p, e.g. sin x = sin(x + 2p)
• sin wx has period T = 2p/w,
LOGARITHMIC FUNCTIONS
If x = ay then y = loga x. This is a logarithmic
function. a is again called the base.
Graphs:
If no base is given, log x should be understood to
mean log10 x (log to the base 10).
-p
y = sin x
1
p
0
But in calculus we almost always natural logs,
notated ln, which are logs to the base e.
That is ln x = loge x.
2p
-1 y = cos x
y = tan x
-p
0
p
Graphs
You should know the
graph y = ln x
2p
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1.3.3 Piecewise Functions & Symmetry
Extra note: CIRCLES
PIECEWISE FUNCTIONS
A circle of radius r centred at (a,b) has
equation
A piecewise function is defined by different formulae in
different parts of its domain. Two common examples are:
Note that a circle cannot be described
by writing a single function of x or y. (Why not?)
1) The Modulus Function
|x| is called the modulus or absolute value of x.
However we can write functions for
the upper half of the circle
and the lower half
We have
2) A Step Function
You should be able to sketch circles from their equations.
You may need to first rearrange an equation into the
standard circle form using the technique of 'completing
the square'.
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On graphs,
indicates that the end point is included,
indicates that the end point is not included.
Example 11
Example 12
Sketch the graph of the equation
and describe it in words.
The table below gives the cost C of mailing a parcel as a
function of its mass m. Write a formula for C(m) and sketch
the graph of the function.
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Mass of Parcel
Cost (USD)
Up to 100 g
1.25
100 to 250 g
2.30
250 to 500 g
4.10
500 to 1000 g
6.90
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Example 13
Example 15
a) Sketch the graph of the function
Give examples of even and odd functions. Draw their graphs.
b) Write a formula for the function g.
c) State the value of i) f(3)
ii) g(5)
Symmetry An even function satisfies
An odd function satisfies
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37
fe(-x) = fe(x)
fo(-x) = – fo(x)
Note: The graph of an even function is symmetric with respect
to reflection in the y-axis. The graph of an odd function is
symmetric with respect to rotation by 180° about the origin.
Example 14
Show that f(x) = x3 – 1/x is an odd function.
It is easily proved that:
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•
Any sum of two or more even functions is even
•
Any sum of two or more odd functions is odd
•
For products, even × even = even
odd × odd = even
odd × even = odd
Example 16
1.4 New Functions from Old
Sketch a)
Functions
1.4.1 New Graphs from Old Graphs
Suppose we know the graph of a certain function.
We can quickly obtain the graphs of some related
functions by some simple transformations.
b)
Investigation Exercise
Plot the following graphs. What patterns do you notice?
1. a) y = x2,
b) y = x2 + 3,
c) y = (x – 3)2.
2. a)
b)
c)
http://www.meta-calculator.com/online/
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TRANSLATIONS For a function f(x) and positive constant c,
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Example 17
to obtain the graph of
Figure A is the graph of f(x) = x2.
What is the equation of graph B?
y = f(x) + c, shift the graph of y = f(x) UP by c units
y = f(x) – c, shift the graph of y = f(x) DOWN c units
y = f(x + c), shift the graph of y = f(x) LEFT c units
y = f(x – c), shift the graph of y = f(x) RIGHT c units
A
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B
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Investigation Exercise
Example 20 The graph of f(x) is shown. Match the
Plot the following graphs. What patterns do you notice?
other graphs with their equations:
1. a) y = sin x,
b) y = 3 sin x,
c) y = sin 2x.
2. a)
b)
c)
STRETCHES
To obtain the graph of
y = cf(x), stretch y = f(x)
vertically by a factor c
y = 2f(x)
y = f(2x)
y = f(x)
y = f(cx), compress y = f(x)
horizontally by a factor c
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Example 21 Sketch: (a) y = 1 – sin x ,
REFLECTIONS
(b) y = |sin x|
To obtain y = – f(x),
reflect y = f(x) in the x-axis
To obtain y = f(–x),
reflect y = f(x) in the y-axis
Example 19
Sketch: a) y = – x2 ,
b)
Note that y = |f(x)| means
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So where f(x) is positive, the graph is unchanged. Where f(x) is
negative the graph is reflected in the x-axis (to become positive).
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1.4.2 Combinations and Compositions of Functions
Compositions of Functions
Let f and g be functions with domains A and B respectively.
These functions can be combined or composed to make
new functions.
Suppose
and
By substitution,
This procedure is called composition.
The new function is called the composition or
composite of f and g, denoted f ० g.
Combinations of Functions
Algebraic operations on f and g are defined as follows:
(f+g)(x) = f(x)+ g(x) with domain A  B
(f ० g)(x) = f(g(x))
(f – g)(x) = f(x) – g(x) with domain A  B
(fg)(x) = f(x)g(x)
with domain A  B
(f /g)(x) = f(x)/g(x)
with domain A  B  {x: g(x)  0}.
Addition and subtraction of functions can also be done
graphically.
f ० g is defined whenever both f and g are
defined. I.e. Its domain is the set of all x in the
domain of g such that g(x) is in the domain of f.
Note: In general f ० g  g ० f
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Example 22
Example 23
Let
Let
a) State the domains of f and g.
Find
a) f ० g ,
b) g ० f ,
c) (f ० g ० f )(0) .
b) Find f + g and its domain.
c) Find f / g and its domain.
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One-to-One Functions
1.4.3 Inverse Functions
• We know that if y is a function of x then for every x
there is exactly one value of y = f(x) (see slide 19-20).
Remember a function can be thought of as a machine:
• If it is also true that for every y there is exactly one
value of x, then f(x) is called a one-to-one function.
Examples
y=x
y = x2
y is NOT a
function of x
y is a function of x
y is a function of x
but is NOT one-to-one and is one-to-one
Q: Can we have another machine which does the
reverse process?
f(x) ?
x
?
A: Yes if the original function is one-to-one.
The ‘reverse’ function is called the inverse function.
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Definition
A function f is called one-to-one if it never takes the
same value twice. That is, f(x1) ≠ f(x2) whenever x1≠ x2.
Horizontal Line Test
A function is one-to-one if and only if no horizontal
line intersects its graph more than once.
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Definition
Let f be a one-to-one function with domain A and
range B. Then its inverse function, f –1, is defined by
for any y in B, and has domain B and range A.
x
f(x)
Example 24
Are the following functions one-to-one?
a) y = sin x
b) y = x3 + 1
f -1
x
Notes
1. f –1 is a special symbol for the inverse.
The -1 is NOT an exponent.
f –1(x)  [f(x)] –1 = 1/ f(x).
2.
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Finding an Inverse Function
Graphs of Inverse Functions
To find the inverse of a given function f(x):
• If f maps a onto b, then f –1 maps b onto a.
• So if the graph of f includes (a, b)
then the graph of f –1 includes (b, a).
1. Write y = f(x).
2. Solve the equation to find x in terms of y.
3. To express f –1 as a function of x, interchange x and y.
This gives y = f –1(x).
• Point (b, a) is obtained from (a, b) by
reflecting in the line y = x.
Example 25
a) Find the inverse of the function f(x) = x2 + 3, x ≥ 0.
• So the graph f –1 is obtained
by reflecting the graph f in the
line y = x.
Your answers to Ex25 should illustrate this! 59
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Example 25, cont.
Non-one-to-one functions and Inverses
b) Find the inverse of the function g(x) = e x
Many important functions are not one-to-one!
But if we restrict the domain (as in Example 25a) we can
obtain a one-to-one then find the inverse of this function.
For example … Inverse Trigonometric Functions
Inverse Sine Function
The function
is one-to-one.
The inverse of this restricted sine function is denoted by
sin-1 or arcsin:
c) Sketch graphs of the functions f and g and their inverses.
NOTE: do not confuse
58
with
60
Inverse Cosine Function
is one-to-one on [0, p], so we define
Inverse Tangent Function
For tangent we take the interval (-p/2, p/2), and define
Example 26
a) Sketch the curve
x = t2 – 2t ,
y = t + 1.
We can construct a table of values and thus plot the curve:
t
x
y
-2
-1
0
1
2
3
4
8
3
0
-1
0
3
8
-1
0
1
2
3
4
5
b) Eliminate the parameter to find a Cartesian equation for
the curve in the form x = f(y).
The graphs are the reflections of the original graphs in the line y = x. 61
1.5 Parametric Curves
Introduction
Suppose a particle moves
along the curve C.
C cannot be described by
an equation of the form
y = f(x). (Why not?)
But the x- and y- coordinates of the particle are both
functions of time: x= f(t) and y= g(t).
t is called a parameter. C is called a parametric curve.
C has parametric equations x= f(t) and y= g(t).
We can also write c(t) = (f(t), g(t)).
Generally, a parameter may be any quantity on which two other
quantities depend. Time and angle are common parameters. 62
63
Notes
• The parameter can sometimes be eliminated (as in
Example 26). But this is not always possible.
• The direct equation and parametric equations describe
the same curve.
• But the parametric equations also tell us when the
particle was where, i.e. how the curve is traced.
• The parameter domain can be
restricted.
E.g.
x = t2 – 2t, y = t + 1,
0 ≤ t ≤ 4.
• Parametric forms are especially useful for complicated
curves which are not functions (or not one-to-one).
64
Parametric curves are easily drawn by computers and
are widely used in computer-aided design (CAD).
Some Common Parametrizations
1) A circle of radius R centred at the origin
has Cartesian equation x2 + y2 = R2.
Letting t be the angle a point makes
with Ox, parametric equations to
traverse the circle once anti-clockwise
are:
2) The straight line segment that joins (x1, y1) and (x2, y2)
can be described by the parametric equations
For example, for the line segment from (1, 2) to (4, 9),
we can write
[Graphs drawn at https://www.desmos.com/ ]
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Example 27
1.6 Definition of a Limit
Sketch the curve with parametric equations
x = sin t
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Introduction
y = sin2 t
Suppose a scientist wants to know the value of a
certain physical quantity at zero air pressure. In his
laboratory he can produce low air pressures but he
cannot achieve a perfect vacuum. What might he do?
We are often interested in the value of a function f(x) when
x is very close to a value x0 but not necessarily equal to x0.
This requires the concept of the limit of a function.
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Limits: A Working Definition
One-Sided and Two-sided Limits
We ask:
As x gets closer and closer to x0 (but x  x0), does
f(x) get closer and closer to some finite number L?
For the function above, we get the same answer whether
we approach from above or below. This is not always the
case. So we need the concept of one-sided limits.
If ‘yes’, we say the limit of f(x) as x approaches x0 equals L.
Written
A limit from the right
(x approaching x0 from above):
or
Equivalently: we can make the value of f(x) as close as
we like to L by taking x sufficiently close to x0.
Note:
A limit from the left
(x approaching x0 from below):
depends only on the values of f(x) near x0.
The two-sided limit
exists if and only if
both one-sided limits exist and are the same, i.e.
if and only if
The value of f(x0) is not relevant! f(x0) may have a different
value or be undefined.
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Three Examples
Example 31
We will consider the following functions:
(II) Sketch a graph of
What is the value of
,
and
?
These functions are not defined at x = 0.
But we can look at their behaviour close to x = 0.
(I) Consider f(x). Using a calculator or computer we can
draw a table of values or plot the graph.
It seems that
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(III) The graph of
is shown below.
What can we say about
,
and
Limits: Formal Definition [Optional]
?
The definition given above is rather informal. More
formally, the concept of a limit may be defined as follows.
Definition
Let f be a function that is defined on an open interval
containing x0, except possibly at x0. We say
if for every small quantity e > 0 there exists a d > 0 such
that | f(x) – L |< e for all x satisfying 0 < | x – x0|< d.
• As x  0+, 1/x gets bigger and bigger …
... and sin(1/x) continues to oscillate in the range [-1,1].
I.e. the function does not tend towards any fixed value.
• This means
I.e. graphically, if f(x) lies inside the
horizontal strip of the width 2e around
L then x lies inside the vertical strip of
the width 2d around x0 (irrespective or
whether or not point (x0, L) belongs to
the graph of f).
does not exist.
• Similarly
does not exist.
• So also
does not exist.
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Example 32
Similar definitions can be written for one-sided limits.
Use the given graph of the function f to state the value of the
following limits. If a limit does not exist, explain why.
These definitions can be used to find limits.
Example 33 (Optional)
Use the definition above to prove that
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76
Example 34
1.7 Evaluating Limits. Laws of
Using the theorem and laws above, find
Limits.
In section 1.5 we used tables and graphs to ‘guess’
limits. Then we met a formal proof but this is hard work
to use! Now we will develop tools for finding limits
precisely and relatively easily.
1.7.3 Limits of Elementary Functions
1.7.1 An Initial Theorem
Most of the functions we meet are elementary functions:
polynomials, power functions, rational functions (ratios of
two polynomials), exponentials, logarithms, trigonometric
and inverse trigonometric functions, and all the functions
which can be obtained from these by addition,
subtraction, multiplication, division and composition.
E.g.
From the definition of a limit, the following simple but
important result can be proved:
For any constants x0 and c,
and
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1.7.2 Laws of Limits
Direct Substitution Property
If f is an elementary function and x0 is in the
domain of f , then
So if f is elementary and x0 is in its domain, the limit can
be found simply by substituting x0 into the formula for f.
If x0 is not in the domain then this property cannot be used!
In some cases the limit can still be found by algebraic
manipulation. Other techniques will be studied in Chapter 3.
Example 35
(For proofs, see textbooks.)
Find the following limits:
From these basic laws, further results can be derived.
E.g.
can be proved by repeated application of (iii) with f(x)=g(x).
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1.7.4 The Squeeze Theorem (or sandwich theorem)
If f(x) ≤ g(x) ≤ h(x) for all x in an open interval
containing x0, except possibly at x0, and if
then
.
If g is trapped between f
and h, and if f and h have
the same limit L at x0, (i.e.
f and h meet at x0), then g
must also have the same
limit L at x0.
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Example 36
Given
83
Example 37
, find
and
Use the squeeze theorem to show that
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An asymptote is a straight line which a graph approaches
arbitrarily close to at long distances from the origin. It may
be approached in many different ways:
1.8 Limits involving Infinity
A cup of hot tea is placed in a room which is air
conditioned at 25 ºC. After a long time, what will
the temperature of the tea be?
1.8.1 Limits at Infinity
Example
Consider
E.g. For
What happens to the value of f(x) as x becomes arbitrarily
large (approaches infinity)?
so as x  , f(x) approaches the straight line y = 2.
We say y = 2 is a horizontal asymptote of the graph of f.
- Both numerator and denominator become large
- But the quotient does not become large …
Dividing throughout by x2,
f(x)
, we have
(x  0).
As x  , 1/x2 0 so f(x)  2. So we say
.
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Limits at Infinity (Informal Definition)
1.8.2 Infinite Limits
Let f be a function defined on some interval (a, ∞).
Then
means the value of f(x) gets closer
and closer to L1 as x gets bigger and bigger.
Example Consider the function
Let g be a function defined on some interval (−∞, a).
Then
means the value of g(x) gets closer
and closer to L2 as x gets more and more negative.
• As x → 0+ , the value of h(x) gets bigger and bigger,
without bound.
• So h(x) do not approach any fixed value L.
• So the limit does not exist.
[x   may be read as “x approaches infinity”, “x becomes
infinite” or “x increases without bound”.]
Graphically, such a limit corresponds to a horizontal
asymptote, y = L1 or y = L2.
Sketch the graph. What is
?
• However it is convenient to say that
[as x → 0+, h(x) “approaches infinity” or “tends to infinity”]
• Similarly, it is convenient to say that
• The graph of h(x)=1/x has a vertical asymptote x = 0.
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Example 39
Infinite Limits (Informal Definition)
Find the following limits:
The notation
means f(x) becomes larger
and larger as x gets closer and closer to x0;
And
means f(x) becomes more and
more negative as x gets closer and closer to x0.
Note: Whenever a limit has the value ∞ or −∞, this
means the limit does not exist.
(∞ is a useful concept but is not a real number!)
Where a function has an infinite limit, the graph has a
vertical asymptote.
I.e. if
and/or
then the
graph y = f(x) has a vertical asymptote at x = x0.
Example 38
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Sketch the graphs of f(x) = 1/x2 and g(x) = ln x.
Asymptotes – summary
i) State the values of the following limits:
• If limx→ f(x) = L1 and/or limx→- f(x) = L2
then y = L1 and/or y = L2 is a horizontal asymptote to the graph
• If limx→a+ f(x) = ± and/or limx→a- f(x) = ± 
then x = a is a vertical asymptote to the graph.
• Horizontal asymptotes can be identified by looking at the
behaviour of the function as x → ± .
• Values where a function is undefined may indicate vertical
asymptotes.
Example 40
Identify the horizontal and vertical asymptotes of
ii) What asymptote(s) does each graph have?
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Example 40
1.9 Continuity
a) Consider again the graph shown.
At what values of x is f discontinuous?
What type of discontinuities are these?
Definition
A function f is continuous at x0 if
.
I.e. To be continuous, f(x) must satisfy three conditions:
1) f is defined on an open interval containing x0
2)
exists
3)
b) Consider
. Is f continuous at x = 1?
Graphically, f is continuous at x0 if its graph extends
some distance to the right and left of the point (x0, f(x0))
and has no break at that point.
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Further DEFINITIONS
A function f is continuous from the right at x0 if
Conversely, f is discontinuous at x0 if there is a break,
or the left and right limits are not equal or do not exist.
Discontinuities are classified into three types:
A function f is continuous from the left at x0 if
(a) Removable Discontinuities
could be ‘removed’ by redefining
the function at a single number.
E.g. in Example 40, at x = 1 is f continuous from the left.
(b) Infinite Discontinuities
A function f is continuous on the open interval (a, b) if it
is continuous at every interior point of the interval.
(c) Jump Discontinuities
A function f is continuous on a closed interval [a, b] if it
is continuous on the open interval (a, b), continuous from
the right at x = a and continuous from the left at x = b.
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Graphically, a function is continuous on (a, b) if you can draw
that part of the graph without lifting your pen off the paper!
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A common use of this theorem is in locating solutions or
roots* of equations: if f(x) is continuous on [a, b] and if
f(a) and f(b) have opposite signs so f(a)f(b) < 0, then
there must exist a number c in (a, b) such that f(x) = 0.
Further Theorems
If functions f and g are continuous at x0, then so are
f + g, f − g, fg, and f /g (provided g(x0)  0).
If g is continuous at x0 and f is continuous at g(x0), then
the composite function f ◦ g is continuous at x0.
Example 41
Show that the equation x4 + x2 − x − 3 = 0 has a root in
the interval (1, 2).
Every elementary function is continuous on its domain.
The inverse of any continuous function is also continuous.
(For proofs, see textbooks.)
The last theorem can be established graphically:
the graph of f −1 is the reflection of f in the line y = x,
so if f has no break then f −1 will also have no break.
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Preparation for Chapter 2: Questions to think about
1.9 The Intermediate Value
Theorem
*A root of a function f(x) is a solution to the equation f(x) = 0.
Theorem
If f is continuous on a finite closed interval [a, b] and
if M is a real number lying between f(a) and f(b), then
there exists a number c in (a, b) such that f(c) = M.
I.e., in the interval [a, b], a continuous function takes on
every value between f(a) and f(b) at least once.
The idea is obvious graphically:
if a graph starts at height f(a) and
finishes at height f(b) and is
continuous, it must cross the a line
of height M at least once.
(For formal proof, see textbooks.)
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• What is meant by the ‘speed’ or ‘velocity’ of a
moving object?
• How is it calculated?
• If we have a graph of distance as a function of
time, how does speed relate to the graph?