Notes on Applied Mathematics
Math 332
By: Sarah A. Al-Sheikh
Mathematics Department
King Abdulaziz University
First Semester 2009
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Introduction
Applied mathematics is a branch of mathematics that concerns itself with the
mathematical techniques typically used in the application of mathematical knowledge
to other domains.
Divisions of applied mathematics
There is no consensus of what the various branches of applied mathematics are.
Such categorizations are made difficult by the way mathematics and science change
over time, and also by the way universities organize departments, courses, and
degrees.
Historically, applied mathematics consisted principally of applied analysis, most
notably differential equations, approximation theory, and applied probability. These
areas of mathematics were intimately tied to the development of Newtonian Physics,
and in fact the distinction between mathematicians and physicists was not sharply
drawn before the mid-19th century. This history left a legacy as well; until the early
20th century subjects such as classical mechanics were often taught in applied
mathematics departments at American universities rather than in physics departments,
and fluid mechanics may still be taught in applied mathematics departments.
Today, the term applied mathematics is used in a broader sense. It includes the
classical areas above, as well as other areas that have become increasingly important
in applications. Even fields such as number theory that are part of pure mathematics
are now important in applications (such as cryptology), though they are not generally
considered to be part of the field of applied mathematics. Sometimes the term
applicable mathematics is used to distinguish between the traditional field of applied
mathematics and the many more areas of mathematics that are applicable to real-world
problems.
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Mathematicians distinguish between applied mathematics, which is concerned with
mathematical methods, and the applications of mathematics within science and
engineering. A biologist using a population model and applying known mathematics
would not be doing applied mathematics, but rather using it. However,
nonmathematicians do not usually draw this distinction. The use of mathematics to
solve industrial problems is called industrial mathematics. Industrial mathematics is
sometimes split in two branches: techno-mathematics (covering problems coming
from technology) and econo-mathematics (for problems in economy and finance).
The success of modern numerical mathematical methods and software has led to the
emergence of computational mathematics, computational science, and computational
engineering, which use high performance computing for the simulation of phenomena
and solution of problems in the sciences and engineering. These are often considered
interdisciplinary programs.
Utility of applied mathematics
Historically, mathematics was most important in the natural sciences and
engineering. However, after World War II, fields outside of the physical sciences have
spawned the creation of new areas of mathematics, such as game theory, which grew
out of economic considerations, or neural networks, which arose out of the study of
the brain in neuroscience, or bioinformatics, from the importance of analyzing large
data sets in biology.
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The RQWQCQ Strategy for Solving Math
Word Problems
RQWQCQ is a good strategy to use when solving math word problems. Each of
the letters in RQWQCQ stands for a step in the strategy.
Read
Read the entire problem to learn what it is about. You may
find it helpful to read the problem out loud, form a picture
of the problem in your mind, or draw a picture of the problem.
Question
Find the question to be answered in the problem. Often the
question is directly stated. When it is not stated, you will have
to identify the question to be answered.
Write
Write the facts you need to answer the question. It is helpful to
cross out any facts presented in the problem that are not needed to answer the
question. Sometimes, all of facts presented in the problem are needed to answer the
question.
Question
Ask yourself "What computations must I do to answer the question?"
Compute
Set up the problem on paper and do the computations. Check your computations for
accuracy and make any needed corrections. Once you have done this, circle your
answer.
Question
Look at your answer and ask yourself: "Is my answer possible?" You may find that
your answer is not possible because it does not fit with the facts presented in the
problem. When this happens, go back through the steps of RQWQCQ until you arrive
at an answer that is possible.
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Chapter I
Applications of Calculus
1. Functions
We write y = f(t) to express the fact that y is a function of t. The independent variable
is t, the dependent variable is y, and f is the name of the function. The graph of a
function has an intercept where it crosses the horizontal or vertical axis. Horizontal
intercepts are also called the zeros of the function.
Example 1.1: The value of a car V, is a function of the age of the car, V = f(a).
(a) Interpret the statement f(5) = 9 if V is in thousands of dollars and a is in years.
(b) In the same units, the value of a Honda is approximated by f(a) = 13.24-0.9a. Find
and interpret the vertical and horizontal intercepts of the graph of this function.
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Example 1.2: A biologist is studying the effect of a new drug. Suppose that y is the
concentration of the drug in the blood after x hours. He tested several people for
several values of x. The biologist seeks to find a mathematical model in the form of a
function y = f(x) where
i.f(0) = 0
ii. f shoud be continuous for x ≥ 0 (the change in concentration is gradual)
f ( x)  0 .
iii. After a period of time the effect of the drug should be negligible, i.e. lim
x 
He made the following chart:
Show that the function
(between 3/4 and 4/5 %)
has the properties described above.
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2 Linear Functions
The equation of a line of slope m through the point (xo , yo) is
y - yo= m(x – xo)
Example 1.3: Suppose that a graph of an unknown cost function is known to be a line.
Find an equation of the line if the fixed cost is 10 S.R. and if each item costs an
additional 2S.R. above the fixed cost.
Example 1.4: Studies of some biological environments require knowledge of the two
customary scales of measurements of temperatures. The graph of the equation that
relates the temperature y in degrees Celsius to the temperature x in degrees Fahrenheit
is known to be a line. The lines passes through (32 , 0) and (212 , 100). Find the slope
of the line, an equation of the line and then find the Celsius temperature that
corresponds to 50 degrees Fahrenheit.
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Example 1.5: An art object is purchased for 50,000S.R. and is expected to appreciate
in value at a constant rate of 5,000 S.R. per year for the next five years. Find the
equation for the value of the object. After three years what will its value be?
Example 1.6: The solid waste generated each year in the cities of US is increasing.
The solid waste generated in millions of tons, was 205.2 in 1990 and 229.2 in 2001.
(a) Assuming that the amount of waste is a linear function of time, find a formula
for this function by finding the equation of the line through these points.
(b) Use this formula to predict the amount of solid waste generated in the year
2020.
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3. Applications of sets
Example 1.7:
Cartoons
Fill in the Venn Diagram that would represent this data.
A study was made of 200 students to determine what TV shows they watch.







1.
2.
3.
22 students don't watch these cartoons.
73 students watch only Tiny Toons.
136 students watch Tiny Toons.
14 students watch only Animaniacs and Pinky & the Brain.
31 students watch only Tiny Toons and Pinky & the Brain.
63 students watch Animaniacs.
135 students do not watch Pinky & the Brain (for some completely
incomprehensible reason).
How many students watch all three shows?
How many students t only Pinky & the Brain?
How many students don't watch Animaniacs?
4. Applications of logic to switching circuits
An interesting application of truth tables is to switching circuits.The electric system in
your home is an example of a switching circuit. When you turn a light switch on, the
electrical current will flow, causing the light to burn. When you turn the switch off,
the electrical current is interrupted and the light goes off.
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The application of logic to electronic devices such as computers, lies in the restriction
of the variable to two possible condition 'On and Off' or 'True or False' .
In practice, electronic engineers use the language of logic as follows.
We associate two logical operations 'AND' and 'OR' operations with switching circuits
in 'series' and 'parallel' respectively.
Logical 'AND' Operation
Let us refer to a circuit consisting of two switches p and q connected in series with a
lamp and battery as shown in figure.
The lamp will glow, only if switch p and switch q are closed. If we replace the word
'closed' by T and 'open' by F, the switch will glow only if p = T and q = T.
Table 1and Table 2 describes all possible states of the switches for the series
connection.
Switches in Series
Table 1
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Table 2 - Truth table for p and q
It is clear that 'AND' operation stipulated that with two input variables, the output is
true only when both the inputs are true.
Logical 'OR' operation
Let us refer to a circuit consisting of two switches p and q connected in parallel with a
lamp and battery as shown in figure.
In this case, the lamp will glow if and only if at least one of the switches is closed.
Table 3 and Table 4 describe all possibles states of the switches for the 'OR' operation.
Switches in Parallel - Table 3
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Truth Table for p OR q -Table 4
The OR operation stipulates that with two binary input variables, the output is true if
either or both the inputs are true.
More complicated circuits can be constructed by using different combinations of
parallel and series circuits.
Example 1.8: Two switches p and q are connected in series. This combination of two
switches is itself connected in parallel with another switch r. Write down the truth
table for this circuit.
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