introduction of calculus
TRANSCRIPT
ESSAY ( ANIS NABILA BT ISMAIL )
Introduction of Calculus
The word calculus itself comes from Latin and means a small stone or pebble
used in gaming, voting, or reckoning. Calculus is concerned with comparing quantities
which vary in a non-linear way. It is used extensively in science and engineering since
many of the things we are studying (like velocity, acceleration, and current in a circuit)
do not behave in a simple, linear fashion. If quantities are continually changing, we need
calculus to study what is going on. Calculus was developed independently by the
Englishman, Sir Isaac Newton, and by the German, Gottfried Leibniz. They were both
working on problems of motion towards the end of the 17th century. There was a bitter
dispute between the men over who developed calculus first. Because of this
independent development, we have an unfortunate mix of notation and vocabulary that
is used in calculus. From Leibniz we get the dy/dx and ∫ signs.
There are two main branches of calculus. The first is differentiation derivatives),
which helps us to find a rate of change of one quantity compared to another. The
second is integration, which is the reverse of differentiation. We may be given a rate of
change and we need to work backwards to find the original relationship (or equation) between
the two quantities. In our daily life, there are many applications of calculus that we use for
example in manufacture, business and so on. There are very interesting, important and useful to
mankind.
Differentiation
Example 1
Designing a Soda Can That Uses a Minimum Amount of Material
A soda can is hold 12 fluid ounces. Find the dimensions that will minimize the amount of
material used in its construction, assuming that the thickness of the material is uniform
(i.e., the thickness of the aluminum is the same everywhere in the can)
To solve this problem, first, we draw and label a picture of a typical soda can (see
Figure 1.0). Here we have drawn a right circular cylinder of height h and radius r.
assuming uniform thickness of the aluminum, notice that we can minimize the amount of
material by minimizing the surface are of the can. We have
Area = area of top + bottom + curved surface area
=2πr+2πrh
We can eliminate one of the variables by using the fact that the volume ( using 1 fluid
ounce 1.80469in ) must be
12 fluid ounces = 12 fl oz × 1.80469 ¿ .3
fl oz = 21.65628in.
Further, the volume of a right circular cylinder is
Vol=π r2h
And so,
h=vol
πr2=21.65628
πr2
Thus, from (7.2) and (7.3), the surface area is
A (r )=2πr2+2πr 21.65628πr2
¿2π ¿)
So, our job is to minimize A( r ), but here, there is no closed and bounced interval of
allowable values. In fact, all we can say is that r¿ 0. We can have r as large or small as
you can imagine, simply by taking h to be corresponding small or large, respectively.
That is, we must find the absolute minimum of A ( r ) on the open and unbounded
interval ( 0.∞). To get an idea of what a plausible answer might be, we graph y=A ( r )
( see Figure 1.1 ). There appears to be a local ( and possibly an absolute ) minimum
located between r =1 and r =2. This minimum value seems to be slightly less that 50.
Next, we compute A' (r )= d
dr[2π (r2+ 21.65628πr )]
¿2π (2 r−21.656 .28πr2 )
¿2π ¿
Notice that the only critical numbers are those for which the numerator of the fraction is
zero: 0=2πr3−21.56528
This occurs if and only if
r3=21.65628
2π
And hence, the only critical number is
r=rc= 3√ 21.656282 π≈1.510548
Further, notice that for 0¿ r r, A ( r ) 0 and for r ¿r, A ( r ) 0. That is, A ( r )is
decreasing on the interval (0, r ) and increasing on the interval ( r ,∞). Thus, A ( r ) has
not only a local minimum, but also an absolute minimum at r = r . Notice, too, that this
corresponds with what we expected from the graph of y = A ( r ) in Figure 1.1. This says
that the can that uses a minimum of material has radius r and height
h=21.65628πr2c
3.0211
Differentiation
Example 2
Business
Another example of application of calculus in daily life is in business which is to
find rate of change of revenue, cost, and profit. For a company making stereos, total
revenue from the sale of X stereos is given by
R ( ) = 1000 - ,
And total cost is given by
C ( ׿ = 3000 + 20
Suppose the company is producing and selling stereos at the rate of 10 stereos per day
at the moment when the 400th stereo is produced. At the same moment, what is the rate
of change of ( a ) total revenue? ( b ) total cost? ( c ) total profit?
The solution of this problem is like this:
( a ) = 1000. Differentiating with respect to time
= 1000 . 10 – 2( 400 ) 10 Substituting 10 for dx / dt and 400 for x
= $2000 per day
( b ) = 20 . Differentiating with respect to time
= 20 ( 10 )
=$200 per day
( c ) Since P = R – C,
= $2000 per day - $200 per day
= $1800 per day
Differentiation
Example 3
Graphing Where Some Features Are Difficult to See
Draw a graph of f ( x) = e
1x¿¿ showing all significant features.
The solution:
The default graph produced by our computer algebra system is not particularly helpful
( see Figure 2.0a ). The default graph produced by most graphing calculators ( see
Figure 2.0b ) is certainly better, but we can’t be sure if this is adequate without further
analysis. First, notice that the domain of f is ( -∞, 0 ) U ( 0, ∞ 0. Thus, we consider
limx→0+¿ e
1x=∞, ¿
¿(6.20)
Since 1x→∞as x→0
+¿ . Also ,since 1x→−∞as x→ 0−¿¿ ¿¿
limx→0−¿ e
1x=0 ,¿
¿ (6.21)
From (6.20) and (6.21), there is a vertical asymptote at x = 0, but an unusual one, in that
f (x) ∞ on one side of 0 and f (x) 0 on the other side. Next,
f ' x=e1x ddx
( 1x)
¿e1x (−1x2 )<0 , for all x≠0 (6.22)
Since e1x 0, for all x. From (6.22), we have that f is decreasing for all x ≠ 0. We
also have f' ' ( x )=e
1x (−1x2 )(−1x2 )+e
1x ( 2x3 )
¿e1x ( 1x4+ 2x3 )=e1 / x( 1+2 xx4 )
{ ¿0 , on(−∞,−12 )concave down¿0 , on(−12 ,0)U (0 ,∞ )concave up
(6.23)
Since x = 0 is not in the domain of f, the only inflection point is at x = . Next, note that
limx→∞
e1x=1 , (6.24)
Since 0 as x ∞ and 1 as t . Likewise,
limx→−∞
e1x=1, (6.25)
From (6.24) and (6.25), y = 1 is a horizontal asymptote, both as x ∞ and as x .
Finally, since
e1x>0 ,
For all x ≠ 0, there are no x-intercepts. It is worthwhile nothing that all of these features
of the graph were discernible from Figure 3.66b, except for the inflection point at x = -
1/2. Note that in almost any graph you draw, it is difficult to see all of the features of the
function. This happens because the inflection point (-1/2, ) or ( -0.5, 0.135335…) is
so close to the x-axis. Since the horizontal asymptote is the line y = , it is difficult to see
both of these features on the same graph ( without drawing the graph on a very large
piece of paper). We settle for the graph seen in Figure 3.67, which shows all of the
features except the inflection point and the concavity on the interval (-1/2, 0). To clearly
see the behavior near the inflection point, we draw a graph that is zoomed-in on the
area of the inflection point ( see Figure 3.68). Here, while we have resolved the problem
of the concavity near x = 0 and the inflection point, we have lost the details of the “big
picture”
The important of differentiation in our daily life
These three examples of calculus in daily life are use differentiation to solve the
problems. Differentiation is all about finding rates of change of one quantity compared to
another. We need differentiation when the rate of change is not constant. There are many
applications of differentiation in science and engineering. Differentiation is also used in
analysis of finance and economics. One important application of differentiation is in the
area of optimization, which means finding the condition for a maximum (or minimum) to
occur. This is important in business such as for cost reduction and make the profit
become increase. Optimization is also important in engineering for maximum strength
and minimum cost.
In example 1, the type of differentiation is shown when designing a soda can that
uses a minimum of material. The important of this differentiation is it can save and
minimize the cost to design a soda can. As we all know, nowadays, they use a
maximum of material in order to design a soda can but in this new strategy it can give a
lot of benefits to us for example to our environment because it can reduce our pollution.
Besides, designing a soda can that uses a minimum of material can improve the
productivity of soda can so that it can be more quality to produce. Another important is it
can save the energy and time for the workers. It shows study of calculus is very
interesting.
The second example shows how important study of calculus in business
especially in manufacture. This is because the uses of differentiation can improve the
quality production of stereos. It is very important and interesting of studying calculus for
businessman. It may help them calculate how to get the minimum cost of the stereos.
As a result, it can help them to save their money and get higher profit. It can also
estimate the budget and save the cost of the company. This is important in making the
profit become increase from day to day in order to minimize the cost without use a lot of
money in business.
In example 3, graphing where some features are difficult to see uses a type of
differentiation. It can make the objects become clear to see. Besides, the important is to
automate tedious and sometimes difficult algebraic manipulation tasks. It can also
graphing equations and provide a programming language for the user to define his or
her procedures. In addition, it can simplify rational function, factor polynomials; find the
solution to a system of equation and various manipulations.
Based on these three examples, clearly show that study calculus is very
important and interesting. Without we noticed, actually in our daily life we are using
calculus for solve our daily problems. It makes our life easier and become useful to us.
Why the study of calculus is important and interesting
The study of calculus is important and interesting because with historical
hindsight, it can be said that the development of calculus is certainly one of the greatest
intellectual achievements of the past two millennia. Without calculus, most of the
incredible advances in science and engineering which occurred in the twentieth century
and have become part of everyday life, such as air and space travel, television,
computers, weather prediction, medical imaging, nuclear bombs, wireless phones, the
internet, microwave ovens, etc., could not have happened.
Calculus provides the language and basic concepts used to formulate most of
the fundamental laws and principles of the various disciplines throughout the physical,
mathematical, biological, economic and social sciences, as well as electrical,
mechanical, computer, bio-, civil, and materials engineering. Of course, within
mathematics, calculus serves as the inescapable gateway to all higher level courses.
Calculus is also an essential tool used widely throughout business and industry, such as
in the financial, insurance, transportation, manufacturing, and pharmaceutical industries,
and in the development of computer, communications, and medical technologies.
Calculus provides us with a framework to analyze the most basic and essential
properties of functions. The reason calculus has such a monopoly in describing our
world is that almost any quantitative model of a physical, chemical, biological,
engineering, industrial, or financial system involves the use of functions. This is not
surprising, however, given the basic and fundamental nature of the tools developed in
calculus. For example, in physics and engineering one regularly encounters functions
describing the position of a particle or object as a function of time t, such as x(t) for
motion in one dimension.
Calculus provides the language and concepts used to formulate most of the
fundamental laws and principles of science and engineering. This is because these laws
and principles are usually, and most naturally, expressed locally in terms of rates of
change or derivatives, or globally in terms of integrals (or in some cases in terms of
derivatives and integrals). One of the interesting ways in which this happens is through
the transference of ideas and techniques that are developed in one field of study, and
are then subsequently applied to another.
The operating system for analyzing functions provided by calculus is what allows
a particular set of ideas such as Fourier analysis and the heat equation to achieve such
wide influence. From experience with personal computers you know that any application
which works on one computer running Windows (or MAC OS or Linux) as its operating
system, will (in principle) work on any other computer running the same operating
system, even if that computer is on the other side of the world, down the hall, or on a
trip to Jupiter. It is this type of standardization, as well as standardization of protocols for
how computers communicate with one another, which led to the explosion of personal
computers and the internet in the late 1900’s. Similarly, calculus standardized how
functions in any field are quantitatively analyzed, in terms of how to calculate rates of
change and integrals.