14Chapter
Contents:
Syllabus reference: 6.1
Introduction to
differential calculus
A Limits
B Limits at infinity
C Rates of change
D The derivative function
E Differentiation from first principles
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HISTORICAL NOTE
OPENING PROBLEM
344 INTRODUCTION TO DIFFERENTIAL CALCULUS (Chapter 14)
In a BASE jumping competition from the Petronas Towers in Kuala Lumpur, the altitude of a
professional jumper in the first 3 seconds is given by f(t) = 452 4:8t2 metres, where
0 6 t 6 3 seconds.
Things to think about:
a What will a graph of the altitude of the jumper in the
first 3 seconds look like?
b Does the jumper travel with constant speed?
c Can you find the speed of the jumper when:
i t = 0 seconds ii t = 1 second
iii t = 2 seconds iv t = 3 seconds?
Calculus is a major branch of mathematics which builds on algebra, trigonometry, and analytic geometry.
It has widespread applications in science, engineering, and financial mathematics.
The study of calculus is divided into two fields, differential calculus and integral calculus. These fields
are linked by the Fundamental Theorem of Calculus which we will study later in the course.
Calculus is a Latin word meaning pebble.
Ancient Romans used stones for counting.
The history of calculus begins with the Egyptian
Moscow papyrus from about 1850 BC.
Archimedes of Syracuse was the first to find the
tangent to a curve other than a circle. His methods
were the foundation of modern calculus developed
almost 2000 years later.
Archimedes
The Greek mathematicians Democritus, Zeno
of Elea, Antiphon, and Eudoxes studied
infinitesimals, dividing objects into an infinite
number of pieces in order to calculate the area
of regions, and volume of solids.
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INTRODUCTION TO DIFFERENTIAL CALCULUS (Chapter 14) 345
The concept of a limit is essential to differential calculus. We will see that calculating limits is necessary
for finding the gradient of a tangent to a curve at any point on the curve.
Consider the following table of values for f(x) = x2 where x is less than 2 but increasing and getting
closer and closer to 2:
x 1 1:9 1:99 1:999 1:9999
f(x) 1 3:61 3:9601 3:996 00 3:999 60
We say that as x approaches 2 from the left, f(x) approaches 4 from below.
We can construct a similar table of values where x is greater than 2 but decreasing and getting closerand closer to 2:
x 3 2:1 2:01 2:001 2:0001
f(x) 9 4:41 4:0401 4:004 00 4:000 40
In this case we say that as x approaches 2 from the right, f(x) approaches 4 from above.
In summary, we can now say that as x approaches 2 from either direction, f(x) approaches a limit of
4, and write
limx!2
x2 = 4.
INFORMAL DEFINITION OF A LIMIT
The following definition of a limit is informal but adequate for the purposes of this course:
If f(x) can be made as close as we like to some real number A by making x sufficiently close to
(but not equal to) a, then we say that f(x) has a limit of A as x approaches a, and we write
limx!a
f(x) = A.
In this case, f(x) is said to converge to A as x approaches a.
It is important to note that in defining the limit of f as x approaches a, x does not reach a. The limit
is defined for x close to but not equal to a. Whether the function f is defined or not at x = a is notimportant to the definition of the limit of f as x approaches a. What is important is the behaviour of the
function as x gets very close to a.
For example, if f(x) =5x+ x2
xand we wish to find the limit as x ! 0, it is tempting for us to
simply substitute x = 0 into f(x).
Not only do we get the meaningless value of 00 , but also we destroy the basic limit method.
Observe that if f(x) =5x+ x2
x=
x(5 + x)
x
then f(x) =
5 + x if x 6= 0
is undefined if x = 0.
LIMITSA
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346 INTRODUCTION TO DIFFERENTIAL CALCULUS (Chapter 14)
The graph of y = f(x) is shown alongside. It is the straight
line y = x+ 5 with the point (0, 5) missing, called a point
of discontinuity of the function.
However, even though this point is missing, the limit of f(x)
as x approaches 0 does exist. In particular, as x ! 0 from
either direction, y = f(x) ! 5.
We write limx!0
5x+ x2
x= 5 which reads:
the limit as x approaches 0, of f(x) =5x+ x2
x, is 5.
In practice we do not need to graph functions each time to determine limits, and most can be found
algebraically.
Evaluate: a limx!2
x2 b limx!0
x2 + 3x
xc lim
x!3
x2 9
x 3
a x2 can be made as close as we like to 4 by making x sufficiently close to 2.
) limx!2
x2 = 4.
b limx!0
x2 + 3x
x
= limx!0
x(x+ 3)
x
= limx!0
(x + 3) since x 6= 0
= 3
c limx!3
x2 9
x 3
= limx!3
(x+ 3)(x 3)
x 3
= limx!3
(x + 3) since x 6= 3
= 6
EXERCISE 14A
1 Evaluate:
a limx!3
(x + 4) b limx!1
(5 2x) c limx!4
(3x 1)
d limx!2
(5x2 3x + 2) e limh!0
h2(1 h) f limx!0
(x2 + 5)
2 Evaluate:
a limx!0
5 b limh!2
7 c limx!0
c, c a constant
3 Evaluate:
a limx!1
x2 3x
xb lim
h!2
h2 + 5h
hc lim
x!0
x 1
x+ 1d lim
x!0
x
x
4 Evaluate the following limits:
a limx!0
x2 3x
xb lim
x!0
x2 + 5x
xc lim
x!0
2x2 x
x
d limh!0
2h2 + 6h
he lim
h!0
3h2 4h
hf lim
h!0
h3 8h
h
g limx!1
x2 x
x 1h lim
x!2
x2 2x
x 2i lim
x!3
x2 x 6
x 3
Example 1 Self Tutor
-5
5
missing
point
x
y
x
xxx
25)(
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THEORY OF KNOWLEDGE
INTRODUCTION TO DIFFERENTIAL CALCULUS (Chapter 14) 347
The Greek philosopher Zeno of Elea lived in what is now southern Italy, in the 5th century BC. Heis most famous for his paradoxes, which were recorded in Aristotles work Physics.
The arrow paradox
If everything when it occupies an equal space is at rest, and if that which is in locomotion is
always occupying such a space at any moment, the flying arrow is therefore motionless.
This argument says that if we fix an instant in time, an arrow appears motionless. Consequently, how
is it that the arrow actually moves?
The dichotomy paradox
That which is in locomotion must arrive at the half-way stage before it arrives at the goal.
Achilles and the tortoise
In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach
the point whence the pursued started, so that the slower must always hold a lead.
According to this principle, the athlete Achilles will never be able to catch the slow tortoise!
1 A paradox is a logical argument that leads to a contradiction or a situation which defies
logic or reason. Can a paradox be the truth?
2 Are Zenos paradoxes really paradoxes?
3 Are the three paradoxes essentially the same?
4 We know from experience that things do move, and that Achilles would catch the tortoise.
Does that mean that logic has failed?
5 What do Zenos paradoxes have to do with limits?
We can use the idea of limits to discuss the behaviour of functions for extreme values of x.
We write x ! 1 to mean when x gets as large as we like and positive,
and x ! 1 to mean when x gets as large as we like and negative.
We read x ! 1 as x tends to plus infinity and x ! 1 as x tends to minus infinity.
Notice that as x ! 1, 1 < x < x2 < x3 < :::: and as x gets very large, the value of1
xgets very
small. In fact, we can make1
xas close to 0 as we like by making x large enough. This means that
limx!1
1
x= 0 even though
1
xnever actually reaches 0.
LIMITS AT INFINITYB
If an object is to move a fixed distance then it must travel half that distance. Before it can travel a
half the distance, it must travel a half that distance.
