Calculus
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Arrangements for the course
 Teaching hours: 64 hrs (4hrs/wk, 16 weeks)
 Self-Study hours: 2 times above (8hrs/wk)
 TA platform: Textbook, extra materials,
also for attendance, for submit your
homeworks, etc
 No acceptance for late submission!
 Zero tolerance for cheating (homeworks,
exams) No Scholarship!
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Study Tips (Good habits)
 Read before you attend class
 List three questions you don’t understand
 Attend the lecture regularly, and taking
handwritten notes
 Do a lot of exercises (including Optionals)
 “Practice makes perfect”
 Ask Questions
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The magic of Calculus
Sir Isaac Newton: explain the motion of the planets
around the sun.
Today: Calculus is used in
calculating the orbits of satellites and spacecraft,
 predicting population sizes,
 estimating how fast oil prices rise or fall,
 forecasting weather,
 measuring the cardiac output of the heart,
 calculating life insurance premiums,
 and in a great variety of other areas.

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The beauty of Calculus
 Calculus is an exciting subject, justly
considered to be one of the greatest
achievements of the human intellect.
 I hope you will discover that it is not only
useful but also intrinsically beautiful.
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The extreme brief history of


Calculus
In this course we will cover the calculus of real functions,
which was developed during more than two centuries.
The pioneers were Isaac Newton (1642-1737) and
Gottfried Wilelm Leibniz (1646-1716).
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Enter Newton…
Isaac Newton (English) is credited with many of the beginnings
of calculus. He introduced product rule, chain rule and higher
derivatives to solve physics problems.
He used calculus to explain many physics problems in his book
Principia Mathematica
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…and Leibniz



Gottfried Wilhelm Leibniz (German) systemized the ideas of calculus of
infinitesimals. Unlike Newton, Leibniz provided a clear set of rules to
manipulate infinitesimals.
Leibniz spent time determining appropriate symbols and paid more
attention to formality.
His work leads to formulas for product and chain rule as well as rules for
derivatives and integrals.
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Newton vs. Leibniz
There was much controversy over who (and thus which country)
should be credited with calculus since both worked at the same
time.
Newton derived his results first, but Leibniz published first.
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Newton vs. Leibniz
Today it is known that Newton began his work with derivatives
and Leibniz began with integrals. Both arrived at the same
conclusions independently.
The name of the study was given by Leibniz, Newton called it
“the science of fluxions”.
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Calculus

Calculus is a branch of mathematics that has tremendous
application and is phenomenally vast. It is essentially
covered in two segments namely differential caculus and
integral calculus.
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Differential
Fundamental Thm of Calculus
Integral
Limits
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Why study calculus



Calculus is the study of mathematically defined change. You get a
series of mathematical equations that come together to tell you how
things change over a period of time.
It will introduce you to the basic concepts of mathematics used to
study almost any type of changing phenonema with a controlled
setting.
Studying calculus will develop invaluable scientific sense and practical
problem solving skills in you. You will understand how to think
logically to reduce a complex system to a few interacting components.
Your mind will develop into a powerful systematic instrument.
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Chapter 1
Functions
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1.1
Review of Functions
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Figure 1.1
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A function is defined by three elements:

A domain: The set of numbers which may be '' fed into the machine''.

A range: The set of numbers that may be '' emitted by the machine''.

A transformation rule: The crucial point is that to every number in its
domain corresponds one and only one number in its range.
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Example 1 Determine if each of the following are functions.
2
y
=
x
+1
(a)
2
(b) y = x + 1
Solution
(a) The first one is a function. No matter what values of x you put into
the equation, there is only one possible value of y .
(b) It is not a function. Choose a value of x ， say x = 3 and plug this
into the equation. y 2 = 3 + 1 = 4 . Now there are two possible values of y
that we could use here. We could use y = 2 or y = −2 . Since there are two
possible values of y that we get from a single x this equation is not a
function.
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Graphs
We can associate every function a graph. What is a graph? A graph has to
be thought of as a subset of the plane. For a function f , we denfine the
graph of f to be the set
{( x, y ) : x ∈ D, y = f ( x)}
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Careful!, not all squiggles are
functions!, e.g what is f(7) ?
The requirement that a function assigns a unique value of the dependent variable to
each value in the domain is expressed in the vertical line test
The requirement that a function assigns a unique value of
the dependent variable to each value in the domain is
expressed in the vertical line test
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Figure 1.2
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Figure 1.3 (a)
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Figure 1.3 (b)
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Figure 1.3 (c)
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Figure 1.3 (d)
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Recall that, the domain of a function is the set of all values that can be
plugged into a function and have the function exist and have a real number
for a value. So, for the domain we need to avoid division by zero, square
roots of negative numbers, logarithms of zero and logarithms of negative
numbers, etc. The range of a function is simply the set of all possible
values that a function can take.
Example 2 Find the domain and range of each of the following
functions.
(a) f ( x) = x 2 + 1
2
(b) g (t ) = 4 − t
(c) h(u ) =
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1
u −1
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Figure 1.4
domain : (−∞,+∞)
range : [1,+∞)
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Figure 1.5
domain : [−2,2]
range : [0,2]
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Figure 1.6
domain : {u : u ≠ 1}
range : {w : w ≠ 0}
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Composite Functions
Functions may be combined using sums ( f + g ) , differences ( f − g ),
products ( fg ) , or quotients ( f / g ). The process called composition also
produces new functions.

In the composition y = f ( g ( x)), f is called the outer function and g is the
inner function.
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Figure 1.8 (a) & (b)
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1
Example Let f ( x) = 3x 2 − x and g ( x) = , find f(g(x)) and g(f(x)).
x
2
3
Example Given f ( x) = x and g ( x) = x − x − 6 , find g  f and g  g .
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Figure 1.13 (a)
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Figure 1.13 (b)
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Figure 1.13 (c)
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Identifying symmetry in functions
Example Identify the symmetry, if any, in the following functions.
(a) f ( x) = x 4 − 2 x 2 − 20
3
(b) g ( x) = x − 3x + 1
(c) h( x) =
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x3 − x
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Figure 1.14
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Figure 1.15
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Figure 1.16
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1.2
Representing Functions
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We consider four different approaches to definging and representing
functions:




formulas
graphs
tables
words
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Using formulas



n
n −1
Polynomials: f ( x) = an x + an −1 x +  + a1 x + a0 , where the coefficients
a0 , a1 , an are real numbers.
Rational functions: f ( x) = p( x) / q( x) where p ( x), q ( x)are polynomials
Algebraic functions: constructed using the operations of algebras:
addition, substraction, multiplication, division and roots. e.g.
f ( x) = 2 x 3 + 4
Exponential functions: f ( x) = b x where b ≠ 1 is a positive real number
 Logarithmic functions: f ( x ) = log b x where b > 0 and b ≠ 1
 Trigonometic functions: sin x, cos x, tan x, cot x, sec x, csc x

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Using formulas
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Using graphs
Figure 1.18 Technology or analytical methods?
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Linear functions
The equation of a line is
y = mx + b
where the slope m and the y-intercept b are constants. This function has a
straight-line graph and is called linear function.
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Piecewise functions
Functions have different definitions on different part of the domain are
called piecewise functions.
If all of the pieces are linear, the function is piecewise linear.
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Power functions
Power functions are a special case of polynomials
where n is a positive integer.


f ( x) = x n
when n is even, the function values are non-negative and the graph
passes through the origin, opening upward.
where n is an odd integer, the function has values that are positive
when x is positive and negative when x is negative.
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when n is an even integer
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when n is odd
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Root functions
Root functions are a special case of algebraic functions.
f ( x) = x1/ n
where n>1 is a positive integer.


when n is even, the domain and range consists of nonnegative
numbers. Their graphs begin steeply at at the origin and then flatten
out as x increases.
when n is odd, the domain and range consists of all real numbers.
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when n is even
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when n is odd
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Rational functions
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Transformations of functions and graphs
There are several ways to transform the graph of a function to produce
graphs of new functions. Four transformations are common:

shifts in the x- and y-directions

scalings in the x- and y-directions
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Figure 1.37
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Figure 1.38
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Figure 1.39
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Figure 1.40
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Figure 1.41
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Figure 1.42
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f ( x) = x 2
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Figure 1.44
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Graph 𝑦𝑦 = 𝑥𝑥 2 + 4𝑥𝑥 − 3
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1.3
Trigonometric Functions
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Figure 1.45 (a) & (b)
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Figure 1.46
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Figure 1.47
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Figure 1.48
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Figure 1.49
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Figure 1.50
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Figure 1.51 (a) & (b)
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Figure 1.52 (a) & (b)
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Figure 1.53
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Figure 1.54
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