Calculus
About this book : http://matrix.skku.ac.kr/Cal-Book/
Chapter 1. Functions
1.1 Functions and Graph
1.2 Symmetry
1.3 Common Functions
1.4 Translation, Stretching and Rotation of Functions
1.1 Functions and Graph
http://matrix.skku.ac.kr/Cal-Book/part1/CS-Sec-1-1.htm
The growth of mathematics parallels the development of human civilization. Numbers were used for counting, while fractions were used for measuring and trading. The relationship or dependence between one object or quantity on another is represented by functions.
Calculus is one of the branches of the mathematics focused on functions, series, limits, derivatives, integrals, and their relations. Intrinsically, the mathematical study of change, motion, growth or decay is the essence of calculus. Calculus has broad applications in sciences, economics, and engineering. Functions are fundamental objects in calculus. The problem of computing the area enclosed by curve was the main thema in the integral calculus, whose origin goes back to 2,500 years ago to the ancient Greeks. The differential calculus, which had not been invented until more than 2,000 years after integral calculus, deals with problems with derivatives. The main ideas behind differential calculus are attributed to Pierre Fermat (1601-1665, French mathematician) and were developed by John Wallis (1616-1703, English mathematician), Isaac Barrow (1630-1677, English mathematician), Isaac Newton (1642-1727, English mathematician) and Gottfried Leibniz (1646-1716, German mathematician).
Before going into details of calculus, we give a brief introduction on sets, functions and numbers which are essential for this book.
Sets
In this section, we denote some common terminology about sets that will be occasionally used throughout the textbook. Note that we only restrict ourselves with the main terminology, we refer students who are interested in a deeper study of set theory, to the books in the reference.
A set 
is a collection of elements. If an element 
belongs to 
, then we write 
. If 
is not an element of 
, then we write 
.
An empty set, denoted by 
, is the set that does not contain any elements.
A set 
is called a subset of a set 
, if each element of 
is also an element of 
, we write 
. Two sets 
and 
are equal, if they have the same elements, that is 
if and only if 
and 
.
There are important operations over sets; namely the union, intersection and complement.
The union of two sets 
and 
is a set denoted by 
, which consists of elements 
such that 
belongs to either 
or 
, that is 
.
The intersection of two sets 
and 
is a set denoted by 
, which consists of elements 
such that 
belongs to both 
and 
, that is 
.
If 
is a subset of 
, then the complement of 
in 
, denoted by 
, is the set of elements which belong to 
but do not belong to 
, that is 
.
Numbers
A number is a mathematical object which is used for counting and measuring.
The set of natural numbers is denoted by 
. The integers are the collection of natural numbers, negative and zero. The set of integers is denoted by 
. The rational numbers are numbers that can be expressed as fractions with an integer numerator and a non-zero natural number denominator. The set of rational numbers is denoted by 
. The set 
of all real numbers is frequently encountered in calculus. Intuitively, the set of real numbers is the set of all points on the axis without exception. The complex numbers, 
are an extension of the real numbers, 
.
Functions
Real world phenomena can be represented by mathematical models using functions. The dependence of one quantity on another can be represented by a function. Let us consider a few examples.
(1) The temperature 
during the day at a given place depends on the time 
. For a given time there is a unique temperature. So, the temperature 
is a function of time 
. It is denoted by 
.
Here, the notation 
or 
on the right-hand side means “function of 
or dependence on 
.” However, two different temperatures cannot be associated with the same time. Whereas the same temperature can occur at two or more different times. So we call 
an independent variable, which varies independently, while we call 
a dependent variable because 
depends on time 
.
(2) The area 
of a rectangle depends on its length 
and width 
, 
or 
. Note that here the area 
depends on 
and 
. That is a function of two variables.
(3) Production cost of bread 
depends on the cost of sugar 
, the cost of wheat 
and the cost of labor 
. Thus 
.
In fact, in real world problems, a quantity may depend on several variables. Try to think some examples in everyday life.
Let 
and 
be two non empty sets and for each element 
a function 
assigns exactly one element, in a set 
known as 
, the value of 
at 
(read as “
of 
”). Here 
and 
denote the domain and co-domain of the function 
. A function may be described in many ways, algebraically by an equation or visually by a graph, or numerically in tabular form as a set of ordered n-tuples or verbally through description in words.
DEFINITION 1
A function 
is a rule that associates to each value 
in a set 
one and only one value 
in a set 
. In this case, we write 
. The set 
is called domain of the function 
, the set 
some
is called the range of 
.
DEFINITION 2
Two functions 
and 
, with domains 
and 
, respectively, are said to be equal, if 
and 
for all 
in 
.
Thus a curve in the 
-plane is the graph of a function of 
if and only if no vertical line intersects the curve more than once. For example, if each vertical line 
intersects a curve only once at 
, then exactly one functional value is defined by 
. But if a line 
intersects the curve twice at 
and 
, then the curve cannot be represented by a function because a function cannot assign two different values to 
as shown in the Figure 1.
Figure 1 http://www.mathsisfun.com/sets/images/vertical-line-test.gif
A function defined by different formulas in different parts of its domain is known as a piecewise defined function.
EXAMPLE
An example of a piecewise defined function is shown in the Figure 2. The algebraic representation is given below.
■
Figure 2
Now we illustrate the use of “Sage” which is the computer algebra system (CAS) used throughout this text book. “Sage” is a free open source software. For clarity, Sage commands are shown in bold face green. A brief introduction of how to use Sage is available at
http://matrix.skku.ac.kr/Cal-Book/
http://matrix.skku.ac.kr/2012-LAwithSage/interact/
For example, the above piecewise defined function can be plotted using the following command: http://matrix.skku.ac.kr/cal-lab/cal-1-1-Fig-1.html.
f1(x) = -x
f2(x) = x^3
f3(x) = 3
f = piecewise([[(-5, 0), f1], [(0, 2), f2], [(2, 5), f3]])
f.plot()
plot(f, -5, 5, exclude=[1], ymax=4, ymin=-1)
A function is called a step function if it jumps from one value to the next at certain values of 
. For example, the function whose graph shown in Figure 3 is an example of a step function.
Figure 3 Plot of function
g1(x) = -1
g2(x) = 1
g = piecewise([[(-5, 0), g1], [(0, 5), g2]])
g.plot()
■
1.1 EXERCISES (Functions and Graph)
http://matrix.skku.ac.kr/Cal-Book/part1/CS-Sec-1-1.htm
1.(Celsius and Fahrenheit). If the temperature is 
degrees Celsius, then the temperature is also 
degrees Fahrenheit, where 
(a) Find 
, 
, 
and 
.
(b)Suppose the outside temperature is 
degrees Celsius. What is the temperature in degrees Fahrenheit?
(c)What temperature is the same in both degrees Fahrenheit and in degrees Celsius?
http://matrix.skku.ac.kr/cal-lab/cal-1-1-1.html
Sol. (a) 
(b) 
(c) Let 
. Then 
.
That is 
. Hence 
.
Therefore the temperature that the same in both degrees Fahrenheit and in degrees Celsius is 
.
[Practice Math & Code in http://sage.skku.edu/ or https://sagecell.sagemath.org/ or https://cocalc.com/]
F(x)=9/5*x + 32
G(x)=x
show(plot(F(x), (x, -50, 30)) + plot(G(x), (x, -50, 30), linestyle='--'))
2. (Brain Weight Problem) The weight 
of a human’s brain is directly proportional to his/her body weight 
.
(a) It is known that a person whose weight is 
has a brain that weighs 
. Find an equation of variation expressing 
as a function of 
.
(b) Express the variation constant as a percent and interpret the resulting equation.
(c) What is the weight of the brain of a person who weighs 
?
Sol. (a) 
.
Since 
, 
.
Hence the required equation is 
.
(b)
.
That is, 
.
Hence the variation constant is 
and brain weight is 
of body weight.
(c) Since 
,
3. (Muscle Weight) The weight 
of the muscles in a human is directly proportional to his/her body weight 
.
(a)It is known that a person who weighs 200lb has 80lb of muscles. Find an equation of variation expressing 
as a function of 
.
(b) Express the variation constant in percentage and interpret the resulting equation.
Sol. (a) Since 
, 
.
So, we know that 
.
Hence 
.
(b)
.
That is, 
.
Hence the variation constant is 
and muscle weight is 
of body weight.
4.(Estimating Heights) An anthropologist can use certain linear functions to estimate the height of a male or female, given the length of certain bones. The humerus is the bone from the elbow to the shoulder. Let 
be the length of the humerus, in centimeters. Then the height, in centimeters, of a male with a humerus of length 
is given by
.
The height in centimeters of a female with a humerus of length 
is given by
3
A 26cm humerus was uncovered in some ruins.
(a) If we assume it was from a male, how tall was he?
(b) If we assume it was from a female, how tall was she?
Sol. (a) 
(b) 
5. (Urban Population) The population of a town is 
. After a growth of 
, its new population is 
.
(a) Assuming that 
is directly proportional to 
, find an equation of variation.
(b) Find 
when 
.
(c) Find 
when 
.
Sol. (a) 
(b) 
(c) 
.
That is 
.
6. (Average Age of Women at First Marriage) In general, our society is marrying at a later age. The average age of women at first marriage can be approximated by the linear function
,
where 
is the average age of women at 1st marriage 
years after 
. Thus, 
is the average age of women at 1st marriage in the year 
, 
is the mean age in 
, and so on.
(a) Find 
, 
, 
, 
and 
.
(b)What will be the mean age of women at first marriage in 
?
(c) Graph 
.
Sol. (a)
(b)
(c) In http://sage.skku.edu/ or https://sagecell.sagemath.org/ or https://cocalc.com/
A(x) = 0.08*x+19.7
A.plot()
* Your tools to draw graphs (in Chrome or Firefox browser) :
Calculus-Grapher :
http://matrix.skku.ac.kr/cal-lab/sage-grapher.html
Parametric Equation Grapher :
http://matrix.skku.ac.kr/cal-lab/sage-grapher-para.html
Polar Equation Grapher :
http://matrix.skku.ac.kr/cal-lab/sage-grapher-polar.html
Implicit Function Grapher :
http://matrix.skku.ac.kr/cal-lab/sage-grapher-imp.html
1.2 Symmetry
http://matrix.skku.ac.kr/Cal-Book/part1/CS-Sec-1-2.htm
Symmetry
The points 
, 
, 
and 
form the corners of a rectangle. The points 
and 
are said to be symmetric about the 
-axis, the 
and 
are said to be symmetric about the 
-axis, and the 
and 
are said to be symmetric about the origin,
In http://sage.skku.edu/ or https://sagecell.sagemath.org/ or https://cocalc.com/
sage: P = plot(x^2,x,-2.5*pi,2.5*pi,linestyle="--", color='red')
sage: Q = plot(x^3,x,-2.5*pi,2.5*pi)
sage: R = text("2. y= x^3 ",(-3.0,-3))
sage: S = text("1. y= x^2",(4,4), color='red')
sage: show(P+Q+R+S, ymax=10, ymin=-10)
Figure 1
Even and Odd function
A function 
is said to be an even function if 
for all 
in the domain of 
. Geometrically, the graph of an even function is symmetric about the 
-axis.
For example, the function 
is even because
.
Its graph is symmetric about the 
-axis (see Figure 2).
sage: P = plot(cos(x),x,-2.5*pi,2.5*pi,linestyle="--", color='red')
sage: Q = plot(sin(x),x,-2.5*pi,2.5*pi)
sage: R = text("2. y=sin(x)",(4.0,0.6))
sage: S = text("1. y=cos(x)",(-2.3,0.7), color='red')
sage: show(P+Q+R+S)
Figure 2 Symmetry (-axis) Even function
Similarly 
and 
are even functions.
A function 
is called an odd function if 
for all 
in the domain of 
. Geometrically, the graph of an odd function is symmetric about the origin.
For example, the function 
is odd because
Its graph is symmetric about the origin (see Figure 3).
Figure 3 Symmetry (origin) Odd function
Similarly 
and 
are odd functions.
However, a function may be neither even nor odd. The graph of such a function is neither symmetric about the 
-axis, nor about the origin. For example, 
is neither even nor odd (see Figure 4).
Figure 4 
Increasing and Decreasing Function
Let 
be defined on an interval 
.
(a) 
is increasing on 
if, whenever 
in 
, and 
, we have 
.
(b) 
is decreasing on 
if, whenever 
in 
, and 
, we have 
.
For example, the function in Example 1 in section 1.1 is decreasing in the interval (
, 0) and increasing in the interval (0, 2) and is neither increasing nor decreasing in the interval (2, ∞).
If the inequality in the above definition is strict, we call the function (a) strictly increasing and (b) strictly decreasing, respectively. The function 
is an example of strictly increasing function in Figure 5.
f(x) = x^3
f.plot()
Figure 5 
Try to find a strictly decreasing function on an interval.
A graph of function which is neither even nor odd : y = sin(x)+cos(x)
var('x,y')
P = plot(cos(x),x,-pi,2.5*pi,linestyle="-")
Q = plot(sin(x),x,-pi,2.5*pi)
p1=plot(sin(x)+cos(x), x, -pi,2.5*pi, color='red',linestyle="--");
S = text("y=sin(x)+cos(x)",(4.8,1.4), color='red')
show(P+Q+p1+S, ymax=1.7, ymin=-2.0)
1.2 EXERCISES (Symmetry)
http://matrix.skku.ac.kr/Cal-Book/part1/CS-Sec-1-2-Sol.htm
1-4. Graph the following functions.
1. 
Sol
2. 
Sol In http://sage.skku.edu/ or https://sagecell.sagemath.org/ or https://cocalc.com/
f1(x) = 1-x
f2(x) = 2-x
f = piecewise([[(0, 1), f1], [(1, 2), f2]])
f.plot()
3. 
Sol
f1(x) = 4-x^2
f2(x) = x^2 +2*x
f = piecewise([[(-3, 1), f1], [(1, 3), f2]])
f.plot()
4. 
Sol
5. Find the formula for graph of funstion.
Sol
6-10. Graph the functions in Exercises 6-10. What symmetry, if any, do the graphs have? Specify the intervals where the function is increasing and where it is decreasing.
var('x')
f1(x) = -x^3
f1.plot()
6. 
Sol
(i) symmetric about the origin, and
(ii) decreasing on 
7. 
Consider manipulating the code to see changes in the graph
http://matrix.skku.ac.kr/cal-lab/cal-1-2-7.html
Sol
plot(-1/x, -2, 2, exclude=[0], ymax=5, ymin=-5)
(i) symmetric about the origin,
(ii) increasing on 
,
(iii) increasing on 
8. 
Sol
plot(sqrt(abs(x)), -2, 2, exclude=[0], ymax=5, ymin=-5)
(i) symmetric with respect to the 
-axis
(ii) decreasing on 
,
(iii) increasing on 
9. 
(similar to 6)
Sol
plot((x^3)/8, -2, 2, ymax=5, ymin=-5)
(i) symmetric about the origin
(ii) increasing on ℝ
10. 
Sol
plot((-x^3)/8, -2, 2, ymax=5, ymin=-5)
(i) it has no symmetries
(ii) decreasing on 
11-16. Determine whether the given function is even, odd, or neither. Give reasons for your answer.
11. 
Sol Since 
for all 
, 
is even function.
plot(3, -2, 2, ymax=5, ymin=-5)
12. 
Sol Since 
for all 
, 
is an even function.
plot(x^2 +1, -2, 2, ymax=5, ymin=-5)
13. 
Sol Since 
for all 
, 
is an odd function.
plot(x^3 + x, -2, 2, ymax=5, ymin=-5)
14. 
Sol Since 
for all 
, 
is an even function.
plot(1/(x^2 -1), -2, 2, exclude=[1], ymax=5, ymin=-5)
15. 
Sol 
.
This means 
and also 
. Hence 
is neither an odd nor even function.
plot(1/(x -1), -2, 2, exclude=[1], ymax=5, ymin=-5)
16. 
Sol
plot(2*x +1, -2, 2, ymax=5, ymin=-5)
.
is neither an odd nor even function.
http://matrix.skku.ac.kr/cal-lab/cal-1-2-Rational-Fts.html
# [Note] Sage code for Graph of rational functions:
plot((1+x^2)/((x-1)*(x+1)*(x-2)), -3, 4, exclude= [-1, 1, 2], ymax=20, ymin=-20, figsize=6)
1.3 Common Functions
http://matrix.skku.ac.kr/Cal-Book/part1/CS-Sec-1-3.htm
Historically, calculus developed from theoretical investigations of physical laws of motion. Nowadays, mathematics is used not only in the natural sciences (physics, chemistry, biology, earth science, etc.) and engineering, but also in some social sciences such as economics.
Mathematical modeling is a process of expressing real world problems or phenomena in terms of functions.
Now we introduce some useful functions.
Linear Function
Figure 1 
[Practice Math & Code in http://sage.skku.edu/ or https://sagecell.sagemath.org/ or https://cocalc.com/]
var('x,y')
p1=plot(-3*x+5, x, -2,2, color='blue');
p2 = text("$y=-x+ 5 $", (1,6), fontsize=20, color='blue')
show(p1+p2, ymax=10, ymin=-1)
If 
is a linear function of 
, then the graph of 
is a straight line. In slope-intercept form the equation of a straight line can be written as
,
where 
is the slope of the line and 
is the 
-intercept. One characteristic feature of a linear function is that it increases/decreases at a constant rate.
For example, Figure 1 shows a graph of the linear function 
.
Polynomials
A polynomial 
over 
is a function defined as
,
where 
is a non-negative integer and the real numbers 
, 
, 
, ..., 
are constants called the coefficients of the polynomial. The domain of any polynomial is 
. If the leading coefficient 
, then the degree of the polynomial is 
. For example, the function 
is a polynomial of degree 5.
A polynomial of degree 1 is a linear function of the form 
. A polynomial of the form 
, 
, is of degree 2 and is called a quadratic function. The graph of a quadratic function can be obtained by shifting the parabola 
, because we can write 
for some 
, 
, 
. The parabola opens upward if 
(e.g. 
) and downward if 
(e.g. 
). Figure 2 
Figure 3 
var('x,y')
p1=plot(2*x^2 +4*x-5, x, -5,5, color='blue');
p2 = text("$y=2x^2 +4x-5 $", (2,45), fontsize=15, color='blue')
show(p1+p2, ymax=60, ymin=-10)
var('x,y')
p1=plot(5*x^3 + 3*x^2 -4*x - 1, x, -2.5,2, color='blue');
p2 = text("$y=5x^3 +3x^2 -4x-1 $", (-1.4,15), fontsize=15, color='blue')
show(p1+p2, ymax=20, ymin=-10)
A cubic function is a polynomial of degree 3
, 
.
Power Functions
A power function is a function of the form 
, defined on an appropriate domain where 
is a constant. We consider the following cases where 
, 
, 
, 
.
CaseⅠ: 
, where 
is a positive integer.
The graphs of 
for 
1, 2, 3, 4, and 5, are shown in Figure 4, which are polynomials with only one term.
Figure 4 Graphs of 
for 
var('x,y')
p1=plot(x, x, -2,2, color='blue');
p2=plot(1/x, x, -2,2, color='purple');
p7=plot(1/(x^2), x, -2,2, linestyle="--", color='purple');
p3=plot(x^2, x, -2,2, color='red');
p4=plot((1/2)*x^3, x, -2,2, linestyle="--", color='green');
p5=plot((1/4)*x^4, x, -2,2, linestyle="--", color='black');
p6 = text("$y=x^n $", (0.4,1.2), fontsize=15, color='blue')
show(p1+p2+p3+p4+p5+p6+p7, ymax=2, ymin=-2)
The general shape of the graph of 
depends on whether 
is even or odd. If 
is even, then 
is an even function and its graph is similar to the graph of 
. If 
is odd, then 
is an odd function and its graph is similar to graph of 
.
Case Ⅱ: 
, where 
is a positive integer.
The function 
(
) is a root function.
For 
it is the square root function 
that has domain 
and whose graph is the upper half of the parabola 
. For other even values of 
, the graph of 
is similar to the graph of 
. For 
we have the cube root function 
whose domain is ℝ. When 
is odd 
, the graph of 
is similar to that of 
.
var('x,y')
p1=plot(sqrt(50*x), x, -5,5, color='blue');
p2 = text("$y=sqrt(50*x) $", (2,45), fontsize=15, color='blue')
show(p1+p2, ymax=25, ymin=-3)
var('x,y')
p1=plot(sqrt(1- x^2), x, -5,5, color='blue');
show(p1, ymax=2, ymin=-1)
var('x,y')
p1=plot((x-1)^(1/3), x, -5,5, color='blue');
show(p1, ymax=2, ymin=-1)
Figure 5 Graphs of root functions
Case Ⅲ: 
, then the function 
is known as a reciprocal function, whose graph is shown in Figure 6.
var('x,y')
p1=plot((x)^(-1), x, -5,5, color='blue');
show(p1, ymax=2, ymin=-2)
Figure 6 The reciprocal function
For example, Boyle’s Law is represented by the reciprocal function below. It states that, when the temperature is constant, the volume 
of gas is inversely proportional to the pressure 
:
, where 
is a constant.
Rational Functions
A rational function 
is a function of the form
where 
and 
are polynomial functions. The domain of 
consists of all values of 
for which 
. The reciprocal function 
, is a rational function, whose domain is 
.
Figure 7 shows a graph of the rational function 
, whose domain is 
.
var('x,y')
p1=plot((x)^(-2), x, -5,5, color='blue');
show(p1, ymax=2, ymin=-2)
Figure 7 Graph of a rational function
Algebraic Functions
An algebraic function is a function constructed by using the algebraic operations (such as addition, subtraction, multiplication, division and taking roots) on polynomials.
For example, 
is an algebraic function. See Figure 8 for the graph of 
.
var('x,y')
p1=plot(sqrt(1-x^2 ), x, -5,5, color='blue');
show(p1, ymax=1, ymin=-0.5)
Figure 8 
Trigonometric and Periodic Functions
Trigonometric functions (sometimes called circular functions) are functions of an angle, which is usually measured in radians. The most familiar trigonometric functions are the sine, cosine and tangent functions. A periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which are periodic over intervals of length 
radians. They are important in the study of triangles and many periodic motions that occur in nature. The graphs of the sine and cosine functions are shown Figure 9 and 10.
var('x,y')
p1=plot(sin(x), x, -2*pi,2*pi, color='blue');
#p2=plot(1/x, x, -2,2, color='purple');
p6 = text("$y=sin(x) $", (3,2), fontsize=15, color='blue')
show(p1+p6, ymax=pi, ymin=-pi)
Figure 9 Graph of 
var('x,y')
p1=plot(cos(x), x, -2*pi,2*pi, color='blue');
#p2=plot(1/x, x, -2,2, color='purple');
p6 = text("$y=cos(x) $", (3,2), fontsize=15, color='blue')
show(p1+p6, ymax=pi, ymin=-pi)
Figure 10 Graph of 
We have 
, 
(for all 
). The domain of 
and 
is 
. The range of 
and 
is 
.
Since
and 
for all values of 
, both sine and cosine functions are periodic functions with period 
.
The tangent function is defined as 
, except for when 
, and its graph is shown in Figure 11. When 
, 
, it is undefined. Its range is 
. Since 
for all 
it is periodic with period 
.
var('x,y')
p1=plot(tan(x), x, -2*pi,2*pi, color='blue');
#p2=plot(1/x, x, -2,2, color='purple');
p6 = text("$y=tan(x) $", (3,2), fontsize=15, color='blue')
show(p1+p6, ymax=pi, ymin=-pi)
Figure 11 Graph of 
Cosecant, secant, and cotangent are the reciprocals of the sine, cosine, and tangent functions, respectively and their graph are shown in Figurs 12-14.
Figure 12 Graph of 
var('x,y')
p1=plot(sec(x), x, -2*pi,2*pi, color='blue');
#p2=plot(1/x, x, -2,2, color='purple');
p6 = text("$y=sec(x) $", (-3,2), fontsize=15, color='blue')
show(p1+p6, ymax=pi, ymin=-pi)
Figure 13 Graph of 
var('x,y')
p1=plot(cot(x), x, -2*pi,2*pi, color='blue');
#p2=plot(1/x, x, -2,2, color='purple');
p6 = text("$y=cot(x) $", (1.8,2), fontsize=15, color='blue')
show(p1+p6, ymax=pi, ymin=-pi)
Figure 14 Graph of 
Exponential Functions
DEFINITION 1
If 
and 
, then the function
is called an exponential function.
The number 
is called the base and 
is called the exponent.
Properties of Exponential Functions
The following list summarizes some of the important properties of the exponential function 
.
The domain is 
and the range is 
.
The 
-intercept of 
is 
. The graph of 
has no 
-intercept.
The function 
is increasing on 
for 
. (see Figure 15.)
The function 
is decreasing on 
for 
(see Figure 16.)
The 
-axis is a horizontal asymptote for the graph of 
.
The function 
is one-to-one.
var('x,y')
p1=plot(2^x, x, -2,2, color='blue');
p2=plot(4^x, x, -2,2, color='purple');
p7=plot(8^x, x, -2,2, linestyle="--", color='purple');
p3=plot(10^x, x, -2,2, color='red');
p4=plot(12^x, x, -2,2, linestyle="--",color='red');
p5=plot(14^x, x, -2,2, color='green');
p6 = text("$y=a^x , a>0 $", (-1,3), fontsize=15, color='blue')
show(p1+p2+p3+p4+p5+p6+p7, ymax=4, ymin=-0.5)
Figure 15 Exponential functions of 
Figure 16 Exponential functions of 
Logarithmic Functions
DEFINITION 2
The logarithmic function with base 
and 
, is defined by
if and only if 
The logarithmic equation 
and the exponential equation 
are equivalent.
Properties of Logarithmic Functions
The following list summarizes some of the important properties of logarithmic function 
.
The domain is 
and the range is 
.
The 
-intercept of 
is 
. The function of 
has no 
-intercept.
The function 
is increasing on 
for 
(see Figure 17.)
The function 
is decreasing on 
for 
. (Sketch)
The 
-axis is a vertical asymptote for the graph of 
.
The function 
is one-to-one.
var('x,y')
p1=plot(log(x), x, .01,2, color='blue');
p2=plot(log(x)/2, x, .01,2, color='purple');
p7=plot(log(x)/3, x, .01,2, linestyle="--", color='purple');
p3=plot(log(x)/4, x, .01,2, color='red');
p4=plot(log(x)/5, x, .01,2, linestyle="--",color='red');
p5=plot(log(x)/6, x, .01,2, color='green');
p6 = text("$y=ln(x)/ln(a) $", (1,0.5), fontsize=15, color='blue')
show(p1+p2+p3+p4+p5+p6+p7, ymax=1, ymin=-2)
Figure 17 Logarithmic functions, 
Transcendental Functions
Functions that are not algebraic are called transcendental functions. For example, trigonometric, exponential and logarithmic functions are transcendental functions.
The following functions are transcendental:
http://matrix.skku.ac.kr/cal-book/part1/CS-Sec-1-3.htm
var('x,y')
p1=plot(x^pi, x, -0.1,2, color='blue');
p2=plot(sin(x), x, -2,2, linestyle="--", color='purple');
p5 = text("y=x^(pi)", (0.5,1), fontsize=15, color='blue')
p6 = text("$y=sin(x) $", (-1, 1), fontsize=15, color='purple')
show(p1+p2+p5+p6, ymax=2, ymin=-1.5)
Figure 18 Cosine and Sine functions
http://matrix.skku.ac.kr/cal-lab/sage-grapher.html
1.3 EXERCISES (Common Functions)
http://matrix.skku.ac.kr/Cal-Book/part1/CS-Sec-1-3-Sol.htm
1.If
,find
and
.
Sol 
var('x');
A=plot(3*x^2-x+2, x, -3, 3, color='blue')
S = text("$y=3x^2-x+2$",(1,20),fontsize=15, color='red')
show (A+S)
2. Find the domain of the following functions.
(a) 
(b) 
(c) 
(d) 
(e) 
Sol (a) 
(b) 
(c) 
(d) 
(e) 
var('x');
A=plot(3/(3*x-1), x, 0.5, 3, color='blue')
B=plot((6*x+4)/(x^2+3*x+2), x, 0.5, 3, color='red')
C=plot(sqrt(x)+x^(1/3), x, 0.5, 3, color='green')
D=plot(sqrt(x)+sqrt(4-x), x, 0.5, 3, linestyle='--')
E=plot((x^2-5*x)^(-1/4), x, -3, -1, linestyle='--', color='red')
show (A+B+C+D+E)
3. Find the domain and sketch the graph of the following functions.
(a) 
(b) 
(c) 
Sol (a) 
so 
. So it’s
Domain: 
var('x,y');
p1=plot((1/(5*x-1)^(1/2)), x, 0.3, 5, color='blue');
show(p1, aspect_ratio=1, ymax=2)
(b) 
so 
. So it’s domain:
var('x, y');
p1=plot((x^2+4*x+3)/(5*x+3), x, -4, 2, color='blue');
show(p1, aspect_ratio=1, ymax=3, ymin=-3)
(c) Domain : 
plot(piecewise([[(-3, 0), x], [(0, 3), x+1]]))
CAS 4. Determine whether 
is even, odd or neither. If 
is even or odd, use symmetry to sketch its graph.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
Sol (a) even function (b) odd function (c) neither
(d) even function (e) odd function (f) neither
var('x');
A=plot(x^(-2), x, -3, 3, color='blue')
B=plot(x^(-3), x, -3, 3, color='red')
C=plot(x^2+x, x, -3, 3, color='green')
D=plot(x^4-4*x^2, x, -3, 3, linestyle='--')
E=plot(x^3-x, x, -3, 3, linestyle='--', color='red')
F=plot(3*x^3+2*x+1, x, -3, 3, linestyle='--', color='green')
show (A+B+C+D+E+F, ymin=-20,ymax=20)
5-6. Draw the graph of given functions.
5. 
Sol
p1=plot(exp(-(x^2)/2), x, -3, 3, color='blue')
show(p1, aspect_ratio=1, ymax=2)
http://matrix.skku.ac.kr/cal-lab/cal-1-3-5.html
6. 
, 
Sol
plot(exp(-1/x), x, 0, 10, color='blue')
1.4 Translation, Stretching and Rotation of Functions
http://matrix.skku.ac.kr/Cal-Book/part1/CS-Sec-1-4.htm
If we apply certain translations to the graph of a given function, the graphs of certain related functions can be obtained. This will enable us to write equations for the graphs and to sketch the graphs of many functions quickly.
Translation: Vertical and Horizontal Shifts:
If c is a positive number, then the graph of 
translates the graph of the function 
parallel to the vertical axis; addition translates the graph in the positive direction and subtraction translates it in the negative direction. Similarly, if we replace the independent variable 
by its translation (
by 
). The graphs of the new function is obtained from the graph of the original function by a horizontal translation by 
units.
Thus for 
, we have the following:
(ⅰ) 
, translates the graph of 
up 
units.
(ⅱ) 
, translates the graph of 
down 
units.
(ⅲ) 
, translates the graph of 
to the left 
units.
(ⅳ) 
, translates the graph of 
to the right 
units.
# Section 1.4 (Translation: vertical and horizontal shifts)
f(x)=cos(x)*sin(x^2)
p1=f.plot(-1.5,1.9,label="$f(x)$")
g1=f(x)+1
p2=g1.plot(-1.5,1.5,color='red',label="$f(x)+1$")
p1+p2
g2=f(x)-1
p3=g2.plot(-1.5,2,color='green',label="$f(x)-1$")
p1+p2+p3
h1=f(x-1)
h2=f(x+1)
p4=h1.plot(-1.5,2,color='purple',linestyle="--",label="$f(x-1)$")
p5=h2.plot(-1.5,2.5,color='black',linestyle=":",label="$f(x+1)$")
p1+p2+p3+p4+p5
Figure 1 
and its translations
http://matrix.skku.ac.kr/cal-lab/cal-1-4-Fig-A.html
Reflection:
Another way of transforming the graph of a function is by reflecting about a coordinate axis.
Suppose 
is a function. Then the graph of
(ⅰ) 
is the graph of 
reflected about the 
-axis.
(ⅱ) 
is the graph of 
reflected about the 
-axis.
## Section 1.4, Reflection
f(x)=cos(x^2)*exp(-x)
p1=f.plot(-1,1,legend_label="graph of $f(x)$")
g1=-f(x)
p2=g1.plot(-1,1,color='red',legend_label="graph of $-f(x)$")
g2=f(-x)
p3=g2.plot(-1,1,color='green',legend_label="graph of $f(-x)$")
p1+p2+p3
Figure 2 Reflections of
http://matrix.skku.ac.kr/cal-lab/cal-1-4-Fig-B.html
Vertical and Horizontal Stretching:
If a function 
is multiplied by a constant 
, the shape of the graph is changed but retains roughly its original shape. The graph of 
is the graph of 
distorted vertically; the graph 
is either stretched or compressed vertically depending on the value of 
. In other words, a vertical stretch is a stretch of the graph of 
toward or away from the 
-axis. The graph of the function 
is distorted horizontally, either by stretching the graph 
away from the 
-axis or by squeezing the graph of 
toward the 
-axis.
Suppose 
is a function and 
is a positive constant. Then
(ⅰ) 
is the graph of 
vertically stretched by a factor 
.
(ⅱ)
is the graph of 
vertically compressed by a factor 
.
(ⅲ)
is the graph of 
horizontally compressed by a factor 
.
(ⅳ)
is the graph of 
horizontally stretched by a factor 
.
Thus for 
, the graph is scaled.
## Stretching (Vertical and Horizontal Stretching)
f(x)=x^2
c=2
p1=f.plot(-2,3,label="$f(x)$")
g1=c*f(x)
p2=g1.plot(-2,2,color='red',label="$cf(x)$")
p1+p2
g2=1/c*f(x)
p3=g2.plot(-2,2,color='green',label="$f(x)/c$")
p1+p2+p3
h1=f(c*x)
h2=f(x/c)
p4=h1.plot(-2,2,color='purple',linestyle="--",label="$f(cx)$")
p5=h2.plot(-2,2,color='black',linestyle=":",label="$f(x/c)$")
p1+p2+p3+p4+p5
Figure 3 Vertical and Horizontal Stretching of 
http://matrix.skku.ac.kr/cal-lab/cal-1-4-Fig-C.html
http://matrix.skku.ac.kr/cal-book/part1/CS-Sec-1-4.htm
EXAMPLE
Sketch the graph of the function 
.
http://matrix.skku.ac.kr/cal-lab/cal-1-4-Exm-1.html
Sol Rewriting
Let us plot the graph of 
. Start with the graph of the parabola 
, compress it vertically by a factor 2, followed by a shift of 4 units to the left, and then finally shift 8 units downward. This is shown in Figures 5-8. If 
, then 
. So the graph lies below the 
-axis. To draw the graph of the function 
, we reflect the negative part of the graph about the 
-axis. This is shown in Figure 4.
f(x) = abs((x^2 + 8*x)/2)
plot(f, (x, -13, 5))
Figure 4 
f(x) = x^2
plot(f, (x, -13, 5))
Figure 5 
f(x) = (1/2)*x^2
plot(f, (x, -13, 13))
Figure 6 
f(x) = (1/2)*(x+4)^2
plot(f, (x, -13, 5))
Figure 7 
f(x) = (1/2)*(x+4)^2-8
plot(f, (x, -13, 5))
Figure 8 
f(x) = abs((1/2)*(x+4)^2-8)
plot(f, (x, -13, 5))
Figure 9 
Arithmetic Combinations:
If 
and 
are two functions, then the sum 
, the difference 
, the product 
, and the quotient 
are defined as follows:
(a) 
(b) 
(c) 
(d) 
,
If the domain of 
is the set 
and the domain of 
is the set 
, then the domain of 
, 
, and 
is the set 
, and the domain of 
is the set
.
EXAMPLE
If 
and 
, find the functions 
and 
(and their domains).
Sol The domain of 
is 
. The domain of 
consists of all numbers 
such that 
, or 
so 
, or 
, Thus the domain of g is the interval 
. The intersection of the domains of 
and 
is 
.
Thus, according to the above definitions, we have the following
(
)
(
)
(
)
(
)
Note that the interval 
is the domain of 
and should not include 
because 
. ■
f(x) = sin(x)
g(x) = tan(x)
h(x) = f/(g-2)
k(x) = x^(20)
s(x) = 3*x-5
F(x) = h(k(s(x)))
plot(F, x, -0.0000001, 0.0000001, ymax=2, ymin=-2)
Composite Functions
Given any two functions 
and 
, if we find the image 
of a number 
in the domain of 
. If this value 
is in the domain of 
, then we find the value of 
. The new function 
, is known as the composite function 
. It is obtained by substituting 
into 
, and is known as the composition (or composite) of 
and 
. It is denoted by 
(“
circle 
”).
In general, 
. For example, let 
and 
. Then 
and 
. Clearly 
.
Figure 10 
, 
The domain of 
is the set of all 
in the domain of 
such that 
is in the domain of 
. In other words, 
is defined whenever both 
and 
are defined.
The composition of three or more functions say, the composite function 
is found by first applying, 
then 
, and then 
:
.
http://matrix.skku.ac.kr/cal-lab/sage-grapher.html
EXAMPLE Given 
, 
. Find the composite functions 
and 
.
Sol We have 
. ■
EXAMPLE
Given 
and 
, Find for (a)-(d) each function and its domain.
(a) 
(b) 
(c) 
(d) 
Sol (a) 
The domain of 
is 
.
(b) 
For 
to be defined, we must have 
. For 
to be defined, we must have 
, that is, 
, or 
. Thus, we have 
, so the domain of 
is the closed interval 
.
(c) 
The domain of is 
is 
.
(d) 
This expression is defined when 
, that is, 
and 
. This latter inequality is equivalent to 
, or 
, that is, 
. Thus, 
, so the domain of 
is the closed interval 
.
■
EXAMPLE
Find 
, where 
, 
, and 
.
Sol 
■
EXAMPLE
Given 
, find functions 
, 
and 
such that 
.
Sol Since 
the formula for 
says: First add 8, then take the sine of the result, and finally take cube. So we let 
, 
, 
.
Then
. ■
1.4 EXERCISES (Translation,StretchingandRotationofFunctions)
http://matrix.skku.ac.kr/Cal-Book/part1/CS-Sec-1-4-Sol.htm
1-6. Exercises 1-6 show how many units and in what directions the graphs of the given equations are to be shifted. Find an equation for the shifted graph.
1. 
. Up 
.
Sol 
2. 
. Left 
, Down 
.
Sol 
3. 
. Left 
.
Sol 
4. 
. Up 
, Right 
.
Sol 
5. 
. Down 
, Right 
.
Sol 
6. 
. Down 
, Left 
.
Sol 
CAS 7-16. Graph the functions in Exercises 7-16.
7. 
Sol
var('x')
plot(sqrt(x+4), x, -4, 2, color='blue')
·http://matrix.skku.ac.kr/cal-lab/cal-1-4-7.html
8. 
Sol
var('x')
plot(abs(x-2), x, 0, 4, color='blue')
http://matrix.skku.ac.kr/cal-lab/cal-1-4-8.html
9. 
Sol
var('x')
plot(1+sqrt(x-1), x, 1, 4, color='blue')
http://matrix.skku.ac.kr/cal-lab/cal-1-4-9.html
10. 
Sol
var('x')
plot((x+1)^(2/3), x, -1, 4, color='blue')
http://matrix.skku.ac.kr/cal-lab/cal-1-4-10.html
11. 
Sol
var('x')
plot(1-x^(2/3), x, 0, 4, color='blue')
http://matrix.skku.ac.kr/cal-lab/cal-1-4-11.html
12. 
Sol
var('x');
plot((x-1)^(1/3)-1, x, 1, 4, color='blue')
http://matrix.skku.ac.kr/cal-lab/cal-1-4-12.html
13. 
Sol
var('x')
show(plot(1/(x-2) -1, x, 1, 4, color='blue'), ymax=10, ymin=-10)
http://matrix.skku.ac.kr/cal-lab/cal-1-4-13.html
14. 
Sol
var('x')
show(plot(1/x +2, x, -2, 2, color='blue'), ymax=10, ymin=-10)
http://matrix.skku.ac.kr/cal-lab/cal-1-4-14.html
15. 
Sol
var('x')
show(plot(1/(x-1)^2, x, -1, 1, color='blue'), ymax=5, ymin=-1)
http://matrix.skku.ac.kr/cal-lab/cal-1-4-15.html
16. 
Sol
var('x')
show(plot(1/(x-1)^2, x, -2, 4, color='blue'), ymax=10, ymin=-2)
http://matrix.skku.ac.kr/cal-lab/cal-1-4-16.html
Copyright @ 2019 SKKU Matrix Lab. All rights reserved.
Made by Manager: Prof. Sang-Gu Lee and Dr. Jae Hwa Lee
http://matrix.skku.ac.kr/sglee/ and http://matrix.skku.ac.kr/cal-book/
*This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B03035865).
