Calculus

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

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.

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

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)

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?

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)

A(x) = 0.08*x+19.7

A.plot()

* Your tools to draw graphs (in Chrome or Firefox browser) :

Calculus-Grapher :

Parametric Equation Grapher :

Polar Equation Grapher :

Implicit Function Grapher :

1.2 Symmetry

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,

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.

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

Sol

plot(-1/x, -2, 2, exclude=[0], ymax=5, ymin=-5)

(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)

(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.

# [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

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:

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

1.3 EXERCISES (Common Functions)

http://matrix.skku.ac.kr/Cal-Book/part1/CS-Sec-1-3-Sol.htm

http://youtu.be/x0E0ZMxZ3Og

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)

6. ,

Sol

plot(exp(-1/x), x, 0, 10, color='blue')

1.4 Translation, Stretching and Rotation of Functions

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

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

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-book/part1/CS-Sec-1-4.htm

EXAMPLE

Sketch the graph of the function .

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

http://youtu.be/vx7GCWY68Zw

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')

8.

Sol

var('x')

plot(abs(x-2), x, 0, 4, color='blue')

9.

Sol

var('x')

plot(1+sqrt(x-1), x, 1, 4, color='blue')

10.

Sol

var('x')

plot((x+1)^(2/3), x, -1, 4, color='blue')

11.

Sol

var('x')

plot(1-x^(2/3), x, 0, 4, color='blue')

12.

Sol

var('x');

plot((x-1)^(1/3)-1, x, 1, 4, color='blue')

13.

Sol

var('x')

show(plot(1/(x-2) -1, x, 1, 4, color='blue'), ymax=10, ymin=-10)

14.

Sol

var('x')

show(plot(1/x +2, x, -2, 2, color='blue'), ymax=10, ymin=-10)

15.

Sol

var('x')

show(plot(1/(x-1)^2, x, -1, 1, color='blue'), ymax=5, ymin=-1)

16.

Sol

var('x')

show(plot(1/(x-1)^2, x, -2, 4, color='blue'), ymax=10, ymin=-2)