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Calculus
Jianzhao (Cliff) Gao
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School of Mathematical Sciences,
Nankai University
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Jianzhao (Cliff) Gao
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：EBJ6TC
Text Book
Weir, Hass and Giordano，
THOMAS’ Calculus,
PEARSON,11th Edition.
 40%(classroom
practices)+60%(final exams)

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Chapter 1
Preliminaries
1.1
Real Numbers and the Real Line
Real Number
Real numbers can be represented
geometrically as points on a number line
called the real line.
 R denotes real number system.

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Property of R

Algebraic properties:
 Real
numbers can be added, subtracted,
multiplied, and divided(except by 0) to produce
more real number.
Order property: (see next slide).
 Completeness property:

 There
are enough real numbers to “complete”
the real number line, in the sense that there are
no “holes” or “gaps” in it.
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Subsets of real numbers
Natural numbers (N), namely 1, 2, 3, 4,…
 Integers (Z), namely …,-3,-2,-1,0,1,2,3,…
 Rational numbers(Q), namely the numbers
that can be expressed in the form of a
fraction m/n, where m and n are integers,
and n≠ 0.
 Irrational numbers, namely the real numbers
that are not rational .

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set
A set is a collection of objects, and these
objects are the elements of the set.
 If S and T are sets, then

 S∪T
is their union and consists of all elements
belonging either to S or T.
 S∩T is their intersection and consists of all
elements belonging to both S and T.

Empty set Ø
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Intervals
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Ex.1

Solve following inequality and show their
solution sets on the real line.
(1) 2x-1<x+3; (2)

𝑥
−
3
<2x+1; (3)
6
≥5
𝑥−1
Solution:
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𝑥<4
2
𝑥 > −3 7
2
1 < 𝑥 < −11 5
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Absolute value
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Absolute Value Properties
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The inequality |x|<a says that the
distance from x to 0 is less than the
positive number a.
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Ex. 6 solve the inequality and show the solution set on the real
line.
(a) |2x-3|≤1; (b) |2x-3|≥1.
Solution:
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1.2
Lines, Circles and Parabolas
Rectangular coordinate system or Cartesian coordinate system
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If x changes from x1 to x2, the
increment in x is Δx=x2-x1.
Ex1. In going from the point
A(4,-3), to the point B(2,5) the
increments in the x- and ycoordinates are :Δx=x2-x1=24=-2.
Δy=y2-y1=5-(-3)=8.
Increments from C(5,6) to
D(5,1):
Δx=x2-x1=5-5=0.
Δy=y2-y1=1-(6)=-5.
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The direction and steepness of a line can also be
measured with an angle.
If φ is the inclination of a line , then0 ≤ 𝜑 < 180°
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The relationship
between the slope
m of a nonvertical
line and the lines’
angle of
inclination φ.
m=tan φ
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Vertical line x=a
Horizontal line y=b
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Ex1. A line through two
points:
Write an equation for the line
through (-2,-1) and (3,4).
The line’s slope is
With(x1,y1)=(-2,1-)
y=-1+1*(x-(-2))
y=-1+x+2
y=x+1
You can also try to use (3,4).
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The y-coordinate of the point where a
non vertical line intersects the y-axis
is called y-intercept of the line.
X-intercept of a non horizontal line is
the x-coordinate of the point where it
crosses the x-axis.
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Lines that are parallel have equal angles of inclination, so
they have the same slope. Conversely, lines with equal
slopes have equal angles of inclination and so are parallel.
If two non vertical lines L1 and L2 are perpendicular, their
slopes m1 and m2 satisfy m1*m2=-1. So each slope is the
negative reciprocal of the other.
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𝑎
𝑚1 = ;
ℎ
ℎ
𝑚2 = − ;
𝑎
𝑚1 ∗ 𝑚2 = −1；
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Copyright © 2008
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Parabola
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Ex.1 compute the
axis, vertex,
intercept of
y=-0.5x2-x+4.
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1.3
Functions and Their Graphs
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Identifying domain and range
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Another way to
visualize a function is its
graph.
Graph of f(x) is
x, f x x ∈ 𝐷
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Another way to represent a function is numerically,
through a table of values. The graph of only the tabled
points is called a scatterplot.
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Vertical Line Test
Not every curve a graph is the graph of a function. A
function have only one value f(x) for each x in its
domain, so no vertical line can intersect the graph of
function more than once.
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Piecewise-Defined Functions
Absolute value function
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Ex.1. Graphing
Piecewise-defined
function.
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Greatest integer function
[2.3]=2, [1.9]=1,[-0.3]=-1
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Least integer function
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Ex.1 Write a formula
for the function y=f(x)
, whose graph consists
of the two line segment
in Figure 1.33
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1.4
Identifying Functions;
Mathematical Models
Linear Functions: a function of the form
f(x)=mx+b, for constants m and b, is called a linear
function.
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Power functions: A function f(x)=xa, where a is a
constant, is called a power function.
(a) a=n, a positive integer.
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(b) f(x)=xa, a=-1, or a=-2.
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(b) f(x)=xa, a=1/2 (square root), 1/3 (cube root),
3/2 and 2/3.
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Polynomials. A function p is a polynomial, if
𝑝 𝑥 = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 1 + 𝑎0
Where n is a nonnegative integer and the numbers
𝑎0 , 𝑎1 , … , 𝑎𝑛 are called coefficients of polynomial. N is
called degree of polynomial.
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Rational functions: is a quotient or ration of two
𝑝(𝑥)
polynomials: 𝑓 𝑥 =
𝑞(𝑥)
Where p and q are polynomials. The domain of a rational
function is the set of all real x for which q(x)≠0.
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Algebraic functions: is a function constructed from
polynomials using algebraic operations(addition,
subtraction, multiplication, division and taking roots).
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Trigonometric function
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Exponential functions: functions of the form f(x)=ax, where
the base a>0 is positive constant and a ≠ 1, are called
exponential functions. Domain(-∞,∞), Range(0, ∞)
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Logarithmic function: f x = log 𝑎 𝑥 , where
a≠1 is a positive constant.
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Transcendental function: These are functions that
not algebraic. They include the trigonometric,
inverse trigonometric, exponential, and
logarithmic functions, and many other functions as
well.
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Increasing versus decreasing functions
If the graph of a function climbs or rises as you move
from left to right, the function is increasing. If the
graph descends or falls as you move from left to right,
the function is decreasing.
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The graph of an even
function is symmetric about
y-axis.
The graph of an odd
function is symmetric about
the origin.
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Mathematical Models
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Most models simplify reality and can only
approximate real-world behavior. One simplifying
relationship is proportionality.
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Ex.3. Kepler’s Third Law. It is postulated by the German
astronomer Johannes Kepler. If T is the period in days for a
planet to complete one full orbit around the sun, and R is the
mean distance of the planet to the sun. Then Kepler postulated
that T is proportional to R raised to the 3/2 power. That is, for
some constant k, 𝑇 = 𝑘𝑅3/2
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1.5
Combining Functions;
Shifting and Scaling Graphs
Sums, Differences, Products, and
Quotients
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1.6
Trigonometric Functions
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Law of Cosines
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