Basic Operations on Numbers, Expressions, and Functions

Abstract

Chapter 2 introduces the essential commands of Mathematica. Basic operations on numbers, expressions, functions, two- and three-dimensional graphics are introduced and discussed. Mathematica's extensive image processing capabilities are introduced.

Keywords

Operations on numbers; Operations on functions; Two-dimensional graphics; Three-dimensional graphics; Image processing

2.1 Numerical Calculations and Built-In Functions

2.1.1 Numerical Calculations

The basic arithmetic operations (addition, subtraction, multiplication, division, and exponentiation) are performed in the natural way with Mathematica. Whenever possible, Mathematica gives an exact answer and reduces fractions.

1. “a plus b,” a+b, is entered as a+b;

2. “a minus b,” a−b, is entered as a-b;

3. “a times b,” ab, is entered as either a*b or a b (note the space between the symbols a and b);

4. “a divided by b,” a/b, is entered as a/b. Executing the command a/b results in a fraction reduced to lowest terms; and

5. “a raised to the bth power,” ab, is entered as a^b.

Example 2.1

Calculate (a) 121+542; (b) 3231−9876; (c) (−23)(76); (d) (22341)(832748)(387281); and (e) 46731.

Solution

These calculations are carried out in the following screen shot. In each case, the input is typed and then evaluated by pressing Enter. In the last case, the Basic Math template is used to enter the fraction.

□

The term an/m=anm=(am)n is entered as a^(n/m). For n/m=1/2, the command Sqrt[a] can be used instead. Usually, the result is returned in unevaluated form but N can be used to obtain numerical approximations to virtually any degree of accuracy. With N[expr,n], Mathematica yields a numerical approximation of expr to n digits of precision, if possible. At other times, Simplify can be used to produce the expected results.

Remark 2.1

If the expression b in ab contains more than one symbol, be sure that the exponent is included in parentheses. Entering a^n/m computes an/m=1man while entering a^(n/m) computes an/m.

Example 2.2

Compute (a) 27 and (b) 823=82/3.

Solution

(a) Mathematica automatically simplifies 27=33. We use N to obtain an approximation of 27. (b) Mathematica automatically simplifies 82/3.

N[number] and number//N return numerical approximations of number.

□

When computing odd roots of negative numbers, Mathematica's results are surprising to the novice. Namely, Mathematica returns a complex number. We will see that this has important consequences when graphing certain functions.

Example 2.3

Calculate (a) 13(−2764)2 and (b) (−2764)2/3.

Solution

(a) Because Mathematica follows the order of operations, (-27/64)^2/3 first computes (−27/64)2 and then divides the result by 3.

(b) On the other hand, (-27/64)^(2/3) raises −27/64 to the 2/3 power. Mathematica does not automatically simplify (−2764)2/3.

However, when we use N, Mathematica returns the numerical version of the principal root of (−2764)2/3.

To obtain the result

( − 27 64 ) 2 / 3 = ( − 27 64 3 ) 2 = ( − 3 4 ) 2 = 9 16 ,

which would be expected by most algebra and calculus students, we first square −27/64 and then take the third root.

The Surd function automatically returns the real-valued nth root of x that most beginning users of Mathematica expect.

Then,

returns the result 9/16. □

2.1.2 Built-In Constants

Mathematica has built-in definitions of nearly all commonly used mathematical constants and functions. To list a few, e≈2.71828 is denoted by E, π≈3.14159 is denoted by Pi, and i=−1 is denoted by I. Usually, Mathematica performs complex arithmetic automatically.

Other built-in constants include ∞, denoted by Infinity, Euler's constant, γ≈0.577216, denoted by EulerGamma, Catalan's constant, approximately 0.915966, denoted by Catalan, and the golden ratio, 12(1+5)≈1.61803, denoted by GoldenRatio.

Example 2.4

Entering

N [ E , 50 ]

2.7182818284590452353602874713526624977572470937000

returns a 50-digit approximation of e. Entering

N [ π , 25 ]

3.141592653589793238462643

returns a 25-digit approximation of π. Entering

3 + I 4 − I

11 17 + 7i 17

performs the division (3+i)/(4−i) and writes the result in standard form.

2.1.3 Built-In Functions

Functions frequently encountered by beginning users include the exponential function, Exp[x]; the natural logarithm, Log[x]; the absolute value function, Abs[x]; the trigonometric functions Sin[x], Cos[x], Tan[x], Sec[x], Csc[x], and Cot[x]; the inverse trigonometric functions ArcSin[x], ArcCos[x], ArcTan[x], ArcSec[x], ArcCsc[x], and ArcCot[x]; the hyperbolic trigonometric functions Sinh[x], Cosh[x], and Tanh[x]; and their inverses ArcSinh[x], ArcCosh[x], and ArcTanh[x]. Generally, Mathematica tries to return an exact value unless otherwise specified with N.

Several examples of the natural logarithm and the exponential functions are given next. Mathematica often recognizes the properties associated with these functions and simplifies expressions accordingly.

Example 2.5

Entering

E − 5

1 e 5

E − 5 // N

0.00673795

returns an approximation of e−5=1/e5. Entering

N[number] or number//N return approximations of number.

Exp[x] computes ex. Enter E to compute e≈2.718.

Log [ Exp [ 3 ] ]

computes ln⁡e3=3. Entering

Log[x] computes ln⁡x. ln⁡x and ex are inverse functions (ln⁡ex=x and eln⁡x=x) and Mathematica uses these properties when simplifying expressions involving these functions.

Exp [ Log [ Pi ] ]

computes eln⁡π=π. Entering

Abs [ − 5 ]

computes |−5|=5. Entering

Abs[x] returns the absolute value of x, |x|.

Abs [ ( 3 + 2 I ) / ( 2 − 9 I ) ]

13 85

computes |(3+2i)/(2−9i)|. Entering

Cos [ Pi / 12 ]

1 + 3 2 2

N [ Cos [ Pi / 12 ] ]

0.965926

computes the exact value of cos⁡(π/12) and then an approximation. Although Mathematica cannot compute the exact value of tan⁡1000, entering

N[number] and number//N return approximations of number.

N [ Tan [ 1000 ] ]

1.47032

returns an approximation of tan⁡1000. Similarly, entering

N[ArcSin[1/3]]

0.339837

returns an approximation of sin−1⁡(1/3) and entering

ArcCos [ 2 / 3 ] // N

0.841069

returns an approximation of cos−1⁡(2/3).

Mathematica is able to apply many identities that relate the trigonometric and exponential functions using the functions TrigExpand, TrigFactor, TrigReduce, TrigToExp, and ExpToTrig.

Example 2.6

Mathematica does not automatically apply the identity sin2⁡x+cos2⁡x=1.

Cos [ x ] ∧ 2 + Sin [ x ] ∧ 2

Cos [ x ] 2 + Sin [ x ] 2

To apply the identity, we use Simplify. Generally, Simplify[expression] attempts to simplify expression.

Simplify [ Cos [ x ] ∧ 2 + Sin [ x ] ∧ 2 ]

Use TrigExpand to multiply expressions or to rewrite trigonometric functions. In this case, entering

Many of the algebraic manipulation commands can be accessed from the AlgebraicManipulation palette.

TrigExpand [ Cos [ 3 x ] ]

Cos [ x ] 3 − 3 Cos [ x ] Sin [ x ] 2

writes cos⁡3x in terms of trigonometric functions with argument x. We use the TrigReduce function to convert products to sums.

TrigReduce [ Sin [ 3 x ] Cos [ 4 x ] ]

1 2 ( − Sin [ x ] + Sin [ 7 x ] )

We use TrigExpand to write

TrigExpand [ Sin [ 3 x ] Cos [ 4 x ] ]

TrigExpand [ Cos [ 2 x ] ]

− Sin [ x ] 2 + 7 2 Cos [ x ] 6 Sin [ x ] − 35 2 Cos [ x ] 4 Sin [ x ] 3 + 21 2 Cos [ x ] 2 Sin [ x ] 5 − Sin [ x ] 7 2

Cos [ x ] 2 − Sin [ x ] 2

in terms of trigonometric functions with argument x. We use ExpToTrig to convert exponential expressions to trigonometric expressions.

ExpToTrig [ 1 / 2 ( Exp [ x ] + Exp [ − x ] ) ]

Cosh [ x ]

Similarly, we use TrigToExp to convert trigonometric expressions to exponential expressions.

TrigToExp [ Sin [ x ] ]

1 2 i e − i x − 1 2 i e ix

Usually, you can use Simplify to apply elementary identities.

Simplify [ Tan [ x ] ∧ 2 + 1 ]

Sec [ x ] 2

A Word of Caution

Remember that there are certain ambiguities in traditional mathematical notation. For example, the expression sin2⁡(π/6) is usually interpreted to mean “compute sin⁡(π/6) and square the result.” That is, sin2⁡(π/6)=[sin⁡(π/6)]2. The symbol sin is not being squared; the number sin⁡(π/6) is squared. With Mathematica, we must be especially careful and follow the standard order of operations exactly, especially when using InputForm. We see that entering

computes sin2⁡(π/6)=[sin⁡(π/6)]2 while

raises the symbol Sin to the power 2[π6]. Mathematica interprets sin2⁡(π6) to be the product of the symbols sin² and π/6. However, using TraditionalForm we are able to evaluate sin2⁡(π/6)=[sin⁡(π/6)]2 with Mathematica using conventional mathematical notation.

Be aware, however, that traditional mathematical notation does contain certain ambiguities and Mathematica may not return the result you expect if you enter input using TraditionalForm unless you are especially careful to follow the standard order of operations, as the following warning message indicates.

Example 2.7

As stated, Mathematica follows the order of operations exactly. To see how Mathematica performs a calculation, TreeForm presents the sequence graphically.

For example, for the calculation (a+b)2(c+d)3x+y−z, TreeForm gives us the results shown in Fig. 2.1.

Figure 2.1 Visualizing the order in which Mathematica carries out a sequence of operations.

Clear[a,b,c,d,x,y,z]TreeForm[(a+b)∧2(c+d)∧3/(x+y−z)]

2.2 Expressions and Functions: Elementary Algebra

2.2.1 Basic Algebraic Operations on Expressions

Expressions involving unknowns are entered in the same way as numbers. Mathematica performs standard algebraic operations on mathematical expressions. For example, the commands

1. Factor[expression] factors expression;

2. Expand[expression] multiplies expression;

3. Together[expression] writes expression as a single fraction; and

4. Simplify[expression] performs basic algebraic manipulations on expression and returns the simplest form it finds.

For basic information about any of these commands (or any other) enter ?command as we do here for Factor.

Many of the algebraic manipulation commands can be accessed from the AlgebraicManipulation palette.

or access the Documentation Center as we do here for Factor.

When entering expressions, be sure to include a space or * between variables to denote multiplication to avoid any ambiguity or confusion. For example, Mathematica interprets 4x to be the product of x and 4. On the other hand, Mathematica interprets x4 to be a new symbol with the name “x4.” Similarly, xy is interpreted as a symbol with the name “xy” while x y and x*y are interpreted as the product of x and y, xy.

Example 2.8

(a) Factor the polynomial 12x2+27xy−84y2. (b) Expand the expression (x+y)2(3x−y)3. (c) Write the sum 2x2−x22 as a single fraction.

Solution

The result obtained with Factor indicates that 12x2+27xy−84y2=3(4x−7y)(x+4y). When typing the command, be sure to include a space, or *, between the x and y terms to denote multiplication. xy represents an expression while x y or x*y denotes x multiplied by y.

Factor [ 12 x 2 + 27 x y − 84 y 2 ]

3 ( 4 x − 7 y ) ( x + 4 y )

We use Expand to compute the product (x+y)2(3x−y)3 and Together to express 2x2−x22 as a single fraction.

Expand [ ( x + y ) 2 ( 3 x − y ) 3 ]

27 x 5 + 27 x 4 y − 18 x 3 y 2 − 10 x 2 y 3 + 7 x y 4 − y 5

Together [ 2 x 2 − x 2 2 ]

4−x42x2 □

To factor an expression like x2−3=x2−(3)2=(x−3)(x+3), use Factor with the Extension option.

Factor[x̂2-3] returns x2−3.

Factor [ x ∧ 2 − 3 , Extension → { Sqrt [ 3 ] } ]

− ( 3 − x ) ( 3 + x )

Similarly, use Factor with the Extension option to factor expressions like x2+1=x2−i2=(x+i)(x−i).

Factor [ x ∧ 2 + 1 ]

1 + x 2

Factor [ x ∧ 2 + 1 , Extension → { I } ]

( − i + x ) ( i + x )

Mathematica does not automatically simplify x2, to the expression x

Sqrt [ x ∧ 2 ]

because without restrictions on x, x2=|x|. The command PowerExpand[expression] simplifies expression assuming that all variables are positive. Alternatively, you can use Assumptions to tell Mathematica to assume that x>0 or to assume that x<0.

Simplify [ Sqrt [ x ∧ 2 ] , Assumptions → x > 0 ]

Simplify [ Sqrt [ x ∧ 2 ] , Assumptions → x < 0 ]

−x

Alternatively, PowerExpand will automatically make “appropriate” assumptions when simplifying expressions.

PowerExpand [ Sqrt [ x ∧ 2 ] ]

Thus, entering

Simplify [ Sqrt [ a ∧ 2 b ∧ 4 ] ]

a 2 b 4

returns a2b4 but entering

PowerExpand [ Sqrt [ a ∧ 2 b ∧ 4 ] ]

a b 2

Simplify [ Sqrt [ a ∧ 2 b ∧ 4 ] , Assumptions →

{ a > 0 , b > 0 } ]

a b 2

returns ab2. However, if a<0 and b>0, a2b4=−ab2.

Simplify [ Sqrt [ a ∧ 2 b ∧ 4 ] , Assumptions →

{ a < 0 , b > 0 } ]

− a b 2

In general, a space is not needed between a number and a symbol to denote multiplication when a symbol follows a number. That is, 3dog means “3 times variable dog; dog3 is a variable with name dog3. Mathematica interprets 3 dog, dog*3, and dog 3 as “3 times variable dog. However, when multiplying two variables, either include a space or * between the variables.

1. cat dog means “variable cat times variable dog.”

2. cat*dog means “variable cat times variable dog.”

3. But, catdog is interpreted as a variable catdog.

The command Apart[expression] computes the partial fraction decomposition of expression; Cancel[expression] factors the numerator and denominator of expression then reduces expression to lowest terms.

Example 2.9

(a) Determine the partial fraction decomposition of 1(x−3)(x−1). (b) Simplify x2−1x2−2x+1.

Solution

Apart is used to see that 1(x−3)(x−1)=12(x−3)−12(x−1). Then, Cancel is used to find that x2−1x2−2x+1=(x−1)(x+1)(x−1)2=x+1x−1. In this calculation, we have assumed that x≠1, an assumption made by Cancel but not by Simplify.

Apart [ 1 ( x − 3 ) ( x − 1 ) ]

1 2 ( − 3 + x ) − 1 2 ( − 1 + x )

Cancel [ x 2 − 1 x 2 − 2 x + 1 ]

1+x−1+x □

In addition, Mathematica has several built-in functions for manipulating parts of fractions.

1. Numerator[fraction] yields the numerator of fraction.

2. ExpandNumerator[fraction] expands the numerator of fraction.

3. Denominator[fraction] yields the denominator of fraction.

4. ExpandDenominator[fraction] expands the denominator of fraction.

Example 2.10

Given x3+2x2−x−2x3+x2−4x−4, (a) factor both the numerator and denominator; (b) reduce x3+2x2−x−2x3+x2−4x−4 to lowest terms; and (c) find the partial fraction decomposition of x3+2x2−x−2x3+x2−4x−4.

Solution

The numerator of x3+2x2−x−2x3+x2−4x−4 is extracted with Numerator. We then use Factor together with %, which is used to refer to the most recent output, to factor the result of executing the Numerator command.

Numerator [ x 3 + 2 x 2 − x − 2 x 3 + x 2 − 4 x − 4 ]

− 2 − x + 2 x 2 + x 3

Factor [ % ]

( − 1 + x ) ( 1 + x ) ( 2 + x )

Similarly, we use Denominator to extract the denominator of the fraction. Again, Factor together with % is used to factor the previous result, which corresponds to the denominator of the fraction.

Denominator [ x 3 + 2 x 2 − x − 2 x 3 + x 2 − 4 x − 4 ]

−4−4x+x2+x3

Factor [ % ]

( − 2 + x ) ( 1 + x ) ( 2 + x )

Cancel is used to reduce the fraction to lowest terms.

Cancel [ x 3 + 2 x 2 − x − 2 x 3 + x 2 − 4 x − 4 ]

− 1 + x − 2 + x

Finally, Apart is used to find its partial fraction decomposition.

Apart [ x 3 + 2 x 2 − x − 2 x 3 + x 2 − 4 x − 4 ]

1+1−2+x □

You can also take advantage of the AlgebraicManipulation palette, which is accessed by going to Palettes under the Mathematica menu, followed by AlgebraicManipulation, to evaluate expressions.

Example 2.11

Simplify 2(x−3)2(x+1)3(x+1)4/3+2(x−3)(x+1)2/3.

Solution

First, we type the expression.

Then, select the expression.

Move the cursor to the palette and click on Simplify. Mathematica simplifies the expression.

□

2.2.2 Naming and Evaluating Expressions

In Mathematica, objects can be named. Naming objects is convenient: we can avoid typing the same mathematical expression repeatedly (as we did in Example 2.10) and named expressions can be referenced throughout a notebook or Mathematica session. Every Mathematica object can be named – expressions, functions, graphics, and so on can be named with Mathematica. Objects are named by using a single equals sign (=).

Because every built-in Mathematica function begins with a capital letter, we adopt the convention that every mathematical object we name in this text will begin with a lowercase letter. Consequently, we will be certain to avoid any possible ambiguity with any built-in Mathematica objects.

With Mathematica 11, the default option is to display known objects in black and unknown objects in blue. Thus, in the following screen shot,

Fraction, x, y, apart, pi, and e are in blue; Apart, 2, Pi, π, ?, Plot, Expand, Cancel, Exp, and E are in black.

To automatically update named variables, Dynamic[x] returns the current value of x.

Thus, Dynamic[x] returns dog.

However, when we enter x=7 afterwards, Dynamic[x] is automatically updated to the new value of x.

Expressions are easily evaluated using ReplaceAll, which is abbreviated with /. and obtained by typing a backslash (/) followed by a period (.), together with Rule, which is abbreviated with -> and obtained by typing a dash (minus sign) (-) followed by a greater than sign (>). For example, entering the command

x^2 /. x->3

returns the value of the expression x2 if x=3. Note, however, this does not assign the symbol x the value 3: entering x=3 assigns x the value 3.

Example 2.12

Evaluate x3+2x2−x−2x3+x2−4x−4 if x=4, x=−3, and x=2.

Solution

To avoid retyping x3+2x2−x−2x3+x2−4x−4, we define fraction to be x3+2x2−x−2x3+x2−4x−4.

Of course, you can simply copy and paste this expression if you neither want to name it nor retype it.

Clear [ x ]

fraction = x 3 + 2 x 2 − x − 2 x 3 + x 2 − 4 x − 4

− 2 − x + 2 x 2 + x 3 − 4 − 4 x + x 2 + x 3

/. is used to evaluate fraction if x=4 and then if x=−3.

If you include a semi-colon (;) at the end of the command, the resulting output is suppressed.

fraction /. x - > 4

3 2

fraction /. x - > − 3

4 5

When we try to replace each x in fraction by 2, we see that the result is undefined: division by 0 is always undefined.

However, when we use Cancel to first simplify and then use ReplaceAll to evaluate,

we see that the result is 3/4. The result indicates that limx→−2⁡x3+2x2−x−2x3+x2−4x−4=34. We confirm this result with Limit.

Generally, Limit[f[x],x->a] attempts to compute limx→a⁡f(x). The Limit function is discussed in more detail in the next chapter. □

2.2.3 Defining and Evaluating Functions

It is important to remember that functions, expressions, and graphics can be named anything that is not the name of a built-in Mathematica function or command. As previously indicated, every built-in Mathematica object begins with a capital letter so every user-defined function, expression, or other object in this text will be assigned a name using lowercase letters, exclusively. This way, the possibility of conflicting with a built-in Mathematica command or function is completely eliminated. Because definitions of functions and names of objects are frequently modified, we introduce the command Clear. Clear[expression] clears all definitions of expression, if any. You can see if a particular symbol has a definition by entering ?symbol.

In Mathematica, an elementary function of a single variable, y=f(x)=expression inx, is typically defined using the form

f[x_]=expression in x or f[x_]:=expression in x.

Notice that when you first define a function, you must always enclose the argument in square brackets ([...]) and place an underline (or blank) “_” after the argument on the left-hand side of the equals sign in the definition of the function.

Example 2.13

Entering

f [ x _ ] = x / ( x ∧ 2 + 1 )

x 1 + x 2

defines and computes f(x)=x/(x2+1). Entering

f [ 3 ]

3 10

computes f(3)=3/(32+1)=3/10. Entering

f [ a ]

a 1 + a 2

computes f(a)=a/(a2+1). Entering

f [ 3 + h ]

3 + h 1 + ( 3 + h ) 2

computes f(3+h)=(3+h)/((3+h)2+1). Entering

n1 = Simplify [ ( f [ 3 + h ] − f [ 3 ] ) / h ]

− 8 + 3 h 10 ( 10 + 6 h + h 2 )

computes and simplifies f(3+h)−f(3)h and names the result n1. Entering

n1 /. h → 0

− 2 25

evaluates n1 if h=0. Entering

n2 = Together [ ( f [ a + h ] − f [ a ] ) / h ]

1 − a 2 − a h ( 1 + a 2 ) ( 1 + a 2 + 2 a h + h 2 )

computes and simplifies f(a+h)−f(a)h and names the result n2. Entering

n2 /. h → 0

1 − a 2 ( 1 + a 2 ) 2

evaluates n2 if h=0.

Often, you will need to evaluate a function for the values in a list,

list = { a 1 , a 2 , a 3 , … , a n } .

Once f(x) has been defined, Map[f,list] returns the list

{ f ( a 1 ) , f ( a 2 ) , f ( a 3 ) , … , f ( a n ) }

Also,

1. Table[f[n],{n,n1,n2}] returns the list

{ f ( n 1 ) , f ( n 1 + 1 ) , f ( n 1 + 2 ) , … , f ( n 2 ) }

2. Table[(n,f[n]),{n,n1,n2}] returns the list of ordered pairs

{ ( n 1 , f ( n 1 ) ) , ( n 1 + 1 , f ( n 1 + 1 ) ) , ( n 1 + 2 , f ( n 1 + 2 ) ) , … , ( n 2 , f ( n 2 ) ) }

The Table function will be discussed in more detail as needed as well as in Chapters 4 and 5.

Example 2.14

Entering

Clear [ h ]

h [ t _ ] = ( 1 + t ) ∧ ( 1 / t )

h [ 1 ]

( 1 + t ) 1 t

defines h(t)=(1+t)1/t and then computes h(1)=2. Because division by 0 is always undefined, h(0) is undefined.

However, h(t) is defined for all t>0. In the following, we use RandomReal together with Table to generate 6 random numbers “close” to 0 and name the resulting list t1. Because we are using RandomReal, your results will almost certainly differ from those here.

RandomReal[{a,b}] returns a random real number between a and b; RandomReal[{a,b},n] returns n random real numbers between a and b.

t1=Table[RandomReal[{0,10∧(−n)}],{n,0,5}]

{ 0.399958 , 0.0392505 , 0.00674767 , 0.00018339 , 0.0000760418 , 4.3836670778376435 ` * ∧ -6 }

We then use Map to compute h(t) for each of the values in the list t1.

From the result, we might correctly deduce that limt→0+⁡(1+t)1/t=e. We can compute the limit with the Limit function, which will be discussed in more detail in Chapter 3. For now, we remark that Limit[f[x],x->a] attempts to compute limx→a⁡f(x).

Limit [ h [ t ] , t → 0 ]

In each of these cases, do not forget to include the “blank” (or underline) (_) on the left-hand side of the equals sign and right bracket in the definition of each function. Remember to always include arguments of functions in square brackets.

Example 2.15

Entering

Including a semi-colon at the end of a command suppresses the resulting output because Mathematica names (or, assigns) the function a name but does not evaluate the function after the assignment.

Clear [ f ]

f [ 0 ] = 1 ;

f [ 1 ] = 1 ;

f [ n _ ] := f [ n − 1 ] + f [ n − 2 ]

defines the recursively-defined function defined by f(0)=1, f(1)=1, and f(n)=f(n−1)+f(n−2). For example, f(2)=f(1)+f(0)=1+1=2; f(3)=f(2)+f(1)=2+1=3. We use Table to create a list of ordered pairs (n,f(n)) for n=0,1,…,10.

The fn we have defined here return the Fibonacci number Fn. Fibonacci[n] also returns the nth Fibonacci number.

Table [ { n , f [ n ] } , { n , 0 , 10 } ]

{ { 0 , 1 } , { 1 , 1 } , { 2 , 2 } , { 3 , 3 } , { 4 , 5 } , { 5 , 8 } , { 6 , 13 } , { 7 , 21 } , { 8 , 34 } , { 9 , 55 } , { 10 , 89 } }

In the preceding examples, the functions were defined using each of the forms f[x_]:=... and f[x_]=.... As a practical matter, when defining “routine” functions with domains consisting of sets of real numbers and ranges consisting of sets of real numbers, either form can be used. Defining a function using the form f[x_]=... instructs Mathematica to define f and then compute and return f[x] (immediate assignment); defining a function using the form f[x_]:=... instructs Mathematica to define f. In this case, f[x] is not computed and, thus, Mathematica returns no output (delayed assignment). The form f[x_]:=... should be used when Mathematica cannot evaluate f[x] unless x is a particular value, as with recursively-defined functions or piecewise-defined functions which we will discuss shortly.

Generally, if attempting to define a function using the form f[x_]=... produces one or more error messages, use the form f[x_]:=... instead.

To define piecewise-defined functions, we usually use Condition (/;) as illustrated in the following example. In simple situations we take advantage of Piecewise.

Example 2.16

Entering

Clear [ f ]

f [ t _ ] := Sin [ 1 / t ] /; t > 0

defines f(t)=sin⁡(1/t) for t>0. Entering

f [ 1 / ( 10 Pi ) ]

is evaluated because 1/(10π)>0. However, both of the following commands are returned unevaluated. In the first case, −1 is not greater than 0 (f(t) is not defined for t⩽0). In the second case, Mathematica does not know the value of a so it cannot determine if it is or is not greater than 0.

f [ − 1 ]

f [ a ]

Entering

f[t_]:=−t/;t⩽0

defines f(t)=−t for t⩽0. Now, the domain of f(t) is all real numbers. That is, we have defined the piecewise-defined function

f ( t ) = { sin ⁡ ( 1 / t ) , t > 0 − t , t ⩽ 0 .

We can now evaluate f(t) for any real number t.

f [ 2 / ( 5 Pi ) ]

f [ 0 ]

f [ − 10 ]

However, f(a) still returns unevaluated because Mathematica does not know if a⩽0 or if a>0.

f [ a ]

Recursively-defined functions are handled in the same way. The following example shows how to define a periodic function.

Example 2.17

Entering

Clear [ g ]

g [ x _ ] := x /; 0 ⩽ x < 1

g [ x _ ] := 1 /; 1 ⩽ x < 2

g [ x _ ] := 3 − x /; 2 ⩽ x < 4

g [ x _ ] := g [ x − 3 ] /; x ⩾ 3

defines the recursively-defined function g(x). For 0⩽x<3, g(x) is defined by

g ( x ) = { x , 0 ⩽ x < 1 1 , 1 ⩽ x < 2 3 − x , 2 ⩽ x < 3 .

For x⩾3, g(x)=g(x−3). Entering

g [ 7 ]

computes g(7)=g(4)=g(1)=1. We use Table to create a list of ordered pairs (x,g(x)) for 25 equally spaced values of x between 0 and 6.

Table [ { x , g [ x ] } , { x , 0 , 6 , 6 / 24 } ]

{ { 0 , 0 } , { 1 4 , 1 4 } , { 1 2 , 1 2 } , { 3 4 , 3 4 } , { 1 , 1 } , { 5 4 , 1 } ,

{ 3 2 , 1 } , { 7 4 , 1 } , { 2 , 1 } , { 9 4 , 3 4 } , { 5 2 , 1 2 } , { 11 4 , 1 4 } , { 3 , 0 } ,

{ 13 4 , 1 4 } , { 7 2 , 1 2 } , { 15 4 , 3 4 } , { 4 , 1 } , { 17 4 , 1 } , { 9 2 , 1 } ,

{ 19 4 , 1 } , { 5 , 1 } , { 21 4 , 3 4 } , { 11 2 , 1 2 } , { 23 4 , 1 4 } , { 6 , 0 } }

We will discuss additional ways to define, manipulate, and evaluate functions as needed. However, Mathematica's extensive programming language allows a great deal of flexibility in defining functions, many of which are beyond the scope of this text. These powerful techniques are discussed in detail in texts like Wellin's Programming with Mathematica: An Introduction [18], Gray's Mastering Mathematica: Programming Methods and Applications [9], Wellin's Essentials of Programming in Mathematica [19] and Maeder's The Mathematica Programmer II and Programming in Mathematica [12,13].

2.3 Graphing Functions, Expressions, and Equations

One of the best features of Mathematica is its graphics capabilities. In this section, we discuss methods of graphing functions, expressions, and equations and several of the options available to help graph functions.

2.3.1 Functions of a Single Variable

The commands

Plot[f[x],{x,a,b}] and Plot[f[x],{x,a,x1,x2,...,xn,b}]

graph the function y=f(x) on the intervals [a,b] and [a,x1)∪(x1,x2)∪⋯∪(xn,b], respectively. Mathematica returns information about the basic syntax of the Plot command with ?Plot or use the Documentation Center to obtain detailed information regarding Plot.

Remember that every Mathematica object can be assigned a name, including graphics. Use Show to combine graphics together. Show[p1,p2,...,pn] displays the graphics p1, p2, ..., pn together. Show[GraphicsRow[{p1,p2,...,pn}]] displays the graphics p1, p2, ..., pn in a row. Show[GraphicsColumn[{p1,p2,...,pn}]] displays the graphics p1, p2, ..., pn in a column.

Show[GraphicsGrid[{{p11,p12,...,p1m},{p21,p22,...,p2m},...,
    {pn1,pn2,...,pnm}}]]

displays the graphics p11, pp12, ..., pnm in an n-row by m-column array.

Example 2.18

Graph y=sin⁡x and y=cos⁡x for −2π⩽x⩽2π.

Solution

Entering

p1 = Plot [ Sin [ x ] , { x , − 2 Pi , 2 Pi } ]

graphs y=sin⁡x for −2π⩽x⩽2π and names the result p1. The plot is shown in Fig. 2.2 (a). With

Figure 2.2 (a) y=sin⁡x for −2π ⩽ x ⩽ 2π. (b) A “reddish” plot of y=cos⁡x for −2π ⩽ x ⩽ 2π. (c) Combining two graphics with Epilog and Inset.

p1b = Plot [ Cos [ x ] , { x , − 2 Pi , 2 Pi } ,

ColorFunction → “ValentineTones” ,

PlotStyle → Thickness [ . 025 ] ]

we create a slightly thicker plot of y=cos⁡x and shade the plot using the ValentineTones color gradient. See Fig. 2.2 (b).

Show[p1,p2,...,pn] shows the graphics p1, ..., pn. You can also use Show to rerender graphics. Using Show with the Epilog option together with Inset, we place a small version of the cosine plot in the sine plot. See Fig. 2.2 (c).

p1c = Show [ p1 ,

Epilog → Inset [ p1b , { 0 , 0 } , Automatic , 5 ] ]

Multiple graphics can be shown in rows, columns, or grids using GraphicsRow (to display several graphics as a row), GraphicsColumn (to display several graphics as a column) or GraphicsGrid (to display several graphics as a grid, array or matrix), respectively. Thus,

Show [ GraphicsRow [ { p1 , p1b , p1c } ] ]

generates Fig. 2.2.

A different way of approaching the problem would be to graph the two functions to scale and then show them together. Include a semi-colon (;) at the end of a command to suppress the result. In the following, we graph y=sin⁡x in blue and y=cos⁡x in red, naming the plots p1 and p2 respectively. Neither is displayed because we include a semi-colon at the end of each command. We then combine the graphics with Show. The result is shown to scale because the option AspectRatio->Automatic is included in the Show command (see Fig. 2.3).

Figure 2.3 y=sin⁡x in blue and y=cos⁡x in red graphed to scale on the interval [−2π,2π]. (For interpretation of the colors in this figure, the reader is referred to the web version of this chapter.)

p1 = Plot [ Sin [ x ] , { x , − 2 Pi , 2 Pi } , PlotStyle → Blue ] ;

p2 = Plot [ Cos [ x ] , { x , − 2 Pi , 2 Pi } , PlotStyle → Red ] ;

Show [ p1 , p2 , PlotRange → { { − 2 Pi , 2 Pi } , Automatic } ,

AxesLabel → { x , y } ,

AspectRatio→Automatic] □

Be careful when graphing functions with discontinuities. Often Mathematica will catch discontinuities. In other cases, it doesn't and you might need to use the Exclusions option to generate a more accurate plot.

Example 2.19

Graph s(t) for 0⩽t⩽5 where s(t)=1 for 0⩽t<1 and s(t)=1+s(t−1) for t⩾1.

Solution

After defining s(t),

Clear [ s ]

s [ t _ ] := 1 /; 0 ⩽ t < 1 ;

s [ t _ ] := 1 + s [ t − 1 ] /; t ⩾ 1 ;

we use Plot to graph s(t) for 0⩽t⩽5 in Fig. 2.4 (a).

Figure 2.4 (a) s(t)=1 + s(t − 1), 0 ⩽ t ⩽ 5. (b) Catching the discontinuities.

p1 = Plot [ s [ t ] , { t , 0 , 5 } , AspectRatio → Automatic ,

AxesLabel→{t,s}]

Of course, Fig. 2.4 (a) is not completely precise: vertical lines are never the graphs of functions. In this case, discontinuities occur at t=1,2,3,4, and 5. If we were to redraw the figure by hand, we would erase the vertical line segments, and then for emphasis place open dots at (1,1), (2,2), (3,3), (4,4), and (5,5) and then closed dots at (1,2), (2,3), (3,4), (4,5), and (5,6). In cases like this where Plot does not automatically detect discontinuities you can specify them with Exclusions. See Fig. 2.4 (b).

p2 = Plot [ s [ t ] , { t , 0 , 5 } , AspectRatio → Automatic ,

Exclusions → { 1 , 2 , 3 , 4 } , AxesLabel → { t , s } ]

Show [ GraphicsRow [ { p1 , p2 } ] ]

To fine-tune graphics, use the Drawing Tools and Graphics Inspector palettes, which are accessed under Graphics in the menu.

□

Entering Options[Plot] lists all Plot options and their default values. The most frequently used options include PlotStyle, DisplayFunction, AspectRatio, PlotRange, PlotLabel, and AxesLabel.

1. PlotStyle controls the color and thickness of a plot. PlotStyle-> GrayLevel[w], where 0⩽w⩽1 instructs Mathematica to generate the plot in GrayLevel[w]. GrayLevel[0] corresponds to black and GrayLevel[1] corresponds to white. Color plots can be generated using RGBColor. RGBColor[1,0,0] corresponds to red, RGBColor[0,1,0] corresponds to green, and RGBColor[0,0,1] corresponds to blue. You can also use any of the named colors listed on the Color Schemes palette. If you prefer CMYK color coding, use CMYKColor[c,m,y,k] to help achieve the desired effects in your graphics.

PlotStyle->Dashing[{a1,a2,...,an}] indicates that successive segments be dashed with repeating lengths of a1, a2, …, an. The thickness of the plot is controlled with PlotStyle->Thickness[w], where w is the fraction of the total width of the graphic. For a single plot, the PlotStyle options are combined with PlotStyle->{{option1, option2, ... , optionn}].

2. A plot is not displayed when the option DisplayFunction-> Identity is included or when a semi-colon (;) is included at the end of the command. Including the option DisplayFunction->$ DisplayFunction in Show or Plot commands instructs Mathematica to display graphics.

3. The ratio of height to width of a plot is controlled by AspectRatio. The default is 1/GoldenRatio. Generally, a plot is drawn to scale when the option AspectRatio-> Automatic is included in the Plot or Show command.

4. PlotRange controls the horizontal and vertical axes. PlotRange->{c,d} specifies that the vertical axis displayed corresponds to the interval c⩽y⩽d while PlotRange->{{a,b}, {c,d}} specifes that the horizontal axis displayed corresponds to the interval a⩽x⩽b and that the vertical axis displayed corresponds to the interval c⩽y⩽d.

5. PlotLabel->"titleofplot" labels the plot titleofplot.

6. AxesLabel->{"xaxislabel","yaxislabel"} labels the x-axis with xaxislabel and the y-axis with yaxislabel.

Example 2.20

Graph y=sin⁡x, y=cos⁡x, and y=tan⁡x together with their inverse functions.

Solution

In p2 and p3, we use Plot to graph y=sin−1⁡x and y=x, respectively. Neither plot is displayed because we include a semi-colon at the end of the Plot commands. p1, p2, and p3 are displayed together with Show in Fig. 2.5 (a). The plot is shown to scale; the graph of y=sin⁡x is in black, y=sin−1⁡x is in gray, and y=x is dashed.

Figure 2.5 (a) y=sin⁡x, y=sin−1⁡x, and y = x. (b) y=cos⁡x, y=cos−1⁡x, and y = x.

p1 = Plot [ Sin [ x ] , { x , − 2 Pi , 2 Pi } ,

PlotStyle → Black ] ;

p2 = Plot [ ArcSin [ x ] , { x , − 1 , 1 } ,

PlotStyle → Gray ] ;

p3 = Plot [ x , { x , − 2 Pi , 2 Pi } ,

PlotStyle → { { Black , Dashing [ { 0.01 } ] } } ] ;

t1a = Show [ p1 , p2 , p3 , PlotRange → { − 2 Pi , 2 Pi } ,

AspectRatio → Automatic , AxesLabel → { x , y } ]

The command Plot[{f1[x],f2[x],...,fn[x]},{x,a,b}] plots f1(x), f2(x), …, fn(x) together for a⩽x⩽b. Simple PlotStyle options are incorporated with

PlotStyle->{option1,option2,...,optionn

where optioni corresponds to the plot of fi(x). Multiple options are incorporated using

PlotStyle->{{options1},{options2},...,{optionsn}}

where optionsi are the options corresponding to the plot of fi(x).

In the following, we use Plot to graph y=cos⁡x, y=cos−1⁡x, and y=x together. The plot in Fig. 2.5 (b) is shown to scale; the graph of y=cos⁡x is in black, y=cos−1⁡x is in gray, and y=x is dashed.

p1 = Plot [ Cos [ x ] , { x , 0 , 2 Pi } ,

PlotStyle → Black ] ;

p2 = Plot [ ArcCos [ x ] , { x , − 1 , 1 } ,

PlotStyle → Gray ] ;

p3 = Plot [ x , { x , − 2 Pi , 2 Pi } ,

PlotStyle → { { Black , Dashing [ { 0.01 } ] } } ] ;

t1b = Show [ p1 , p2 , p3 , PlotRange → { { − Pi , 2 Pi } , { − Pi , 2 Pi } } ,

AspectRatio → Automatic , AxesLabel → { x , y } ]

We use the same approach to graph y=tan⁡x, y=tan−1⁡x, and y=x in Fig. 2.6.

Figure 2.6 y=tan⁡x, y=tan−1⁡x, and y = x.

p1 = Plot [ Tan [ x ] , { x , − Pi / 2 , Pi / 2 } ,

PlotStyle → Black ] ;

p2 = Plot [ ArcTan [ x ] , { x , − 2 Pi , 2 Pi } ,

PlotStyle → Gray ] ;

p3 = Plot [ x , { x , − 2 Pi , 2 Pi } ,

PlotStyle → { { Black , Dashing [ { 0.01 } ] } } ] ;

t1c = Show [ p1 , p2 , p3 , PlotRange → { { − 2 Pi , 2 Pi } , { − 2 Pi , 2 Pi } } ,

AspectRatio → Automatic , AxesLabel → { x , y } ]

Use Show together with GraphicsRow to display graphics in rectangular arrays. Entering

Show [ GraphicsRow [ { p2 , p3 , t1c } ] ]

shows the three plots p2, p3, and t1c in a row as shown in Fig. 2.7. □

Figure 2.7 The elementary trigonometric functions and their inverses.

Example 2.21

Inverse Functions

f(x) and g(x) are inverse functions if

f ( g ( x ) ) = g ( f ( x ) ) = x .

If f(x) and g(x) are inverse functions, their graphs are symmetric about the line y=x. The command

Composition[f1,f2,f3,...,fn,x]

computes the composition

( f 1 ∘ f 2 ∘ ⋯ ∘ f n ) ( x ) = f 1 ( f 2 ( ⋯ ( f n ( x ) ) ) ) .

For two functions f(x) and g(x), it is usually easiest to compute the composition f(g(x)) with f[g[x]] or f[x]//g.

Show that

f ( x ) = e x and g ( x ) = ln ⁡ x

are inverse functions.

Solution

When using Mathematica, Exp[x] represents the function y=ex. Similarly, Log[x] represents the natural logarithm function, y=ln⁡x.

After defining f(x) and g(x),

f(x) and g(x) are not returned because a semi-colon is included at the end of each command.

f [ x _ ] = Exp [ x ] ;

g [ x _ ] = Log [ x ] ;

f [ g [ x ] ]

Simplify [ g [ f [ x ] ] ]

Log [ e x ]

we compute and simplify the compositions f(g(x)) and g(f(x)). Because both results simplify to f(g(x))=g(f(x))=x, f(x) and g(x) are inverse functions. Observe that Simplify does not automatically reduce ln⁡(ex)=x. To do so, we use PowerExpand.

PowerExpand [ g [ f [ x ] ] ]

To see that the graphs of f(x) and g(x) are symmetric about the line y=x, we use Plot to graph f(x), g(x), and y=x together in Fig. 2.8. Because Tooltip is being applied to the set of functions being plotted, you can identify each curve by sliding the cursor over the curve: when the cursor is placed over a curve, Mathematica displays its definition. In the Plot command, observe how we are able to graph all three functions, f(x)=ex, g(x)=ln⁡x, and y=x together as well as control how each graph is plotted using the PlotStyle option. The range (y-axis) displayed is controlled using the PlotRange option. The option AspectRatio->1 instructs Mathematica to display the axes in a 1−1 (square) ratio. In this case, because the domains and ranges are the same, the resulting plot is drawn to scale. If the domains and ranges are not the same and you wish to have the plot displayed to scale, use the option AspectRatio->Automatic.

Figure 2.8 f(x)=e^x in black, g(x)=ln⁡x in gray, and y = x dashed in light gray.

Plot [ Tooltip [ { f [ x ] , g [ x ] , x } ] , { x , − 10 , 10 } ,

PlotStyle → { { Black } , { Gray } , { LightGray , Dashing [ . 01 ] } } ,

PlotRange → { − 10 , 10 } , AspectRatio → 1 ]

In the plot, observe that the graphs of f(x)=ex and g(x)=ln⁡x are symmetric about the line y=x. The plot also illustrates that the domain and range of a function and its inverse are interchanged: f(x) has domain (−∞,∞) and range (0,∞); g(x) has domain (0,∞) and range (−∞,∞). □

For repeated compositions of a function with itself, Nest[f,x,n] computes the composition

( f ∘ f ∘ f ∘ ⋯ ∘ f ) ︸ n times ( x ) = ( f ( f ( f ⋯ ) ) ) ︸ n times ( x ) = f n ( x ) .

Example 2.22

Graph f(x), f10(x), f20(x), f30(x), f40(x), and f50(x) if f(x)=sin⁡x for 0⩽x⩽2π.

Solution

After defining f(x)=sin⁡x,

f [ x _ ] = Sin [ x ] ;

we graph f(x) in p1 with Plot

p1 = Plot [ f [ x ] , { x , 0 , 2 Pi } ] ;

and then illustrate the use of Nest by computing f5(x).

Nest [ f , x , 5 ]

Sin [ Sin [ Sin [ Sin [ Sin [ x ] ] ] ] ]

Next, we use Table together with Nest to create the list of functions

{ f 10 ( x ) , f 20 ( x ) , f 30 ( x ) , f 40 ( x ) , f 50 ( x ) } .

Because the resulting output is rather long, we include a semi-colon at the end of the Table command to suppress the resulting output.

Table[f[i],{i,a,b,istep}] computes f(i) for i values from a to b using increments of istep.

toplot = Table [ Nest [ f , x , n ] , { n , 10 , 50 , 10 } ] ;

We then graph the functions in toplot on the interval [0,2π] with Plot, applying the Tooltip function to the list being plotted so they can easily be identified. Last, we use Show to display p1 and p2 together in Fig. 2.9.

Figure 2.9 f(x) in blue; the graphs of f¹⁰(x), f²⁰(x), f³⁰(x), f⁴⁰(x), and f⁵⁰(x) are in different colors. You can use Tooltip to identify each curve. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this chapter.)

p2 = Plot [ Tooltip [ Evaluate [ toplot ] ] , { x , 0 , 2 Pi } ] ;

Show[p1,p2]

In the plot, we see that repeatedly composing sine with itself has a flattening effect on y=sin⁡x. □

When graphing functions involving odd roots, Mathematica's results may be surprising to the beginner. The key is to use the Surd function or remember that Mathematica follows the order of operations exactly and understand that without restrictions on x, x2=|x|.

Example 2.23

Graph y=x1/3(x−2)2/3(x+1)4/3.

Solution

Entering

p1 = Plot [ x ∧ ( 1 / 3 ) ( x − 2 ) ∧ ( 2 / 3 ) ( x + 1 ) ∧ ( 4 / 3 ) ,

{ x , − 3 , 5 } , PlotRange → { − 4 , 4 } ,

AspectRatio → Automatic , PlotStyle → Black ,

PlotLabel → “(a)” ]

does not produce the graph we expect (see Fig. 2.10 (a)) because many of us consider y=x1/3(x−2)2/3(x+1)4/3 to be a real-valued function with domain (−∞,∞). Generally, Mathematica does return a real number when computing the odd root of a negative number. For example, x3=−1 has three solutions

Figure 2.10 (a), (b) and (c) three plots of y = x^1/3(x−2)^2/3(x+1)^4/3.

Solve is discussed in more detail in the next section.

Clear [ x ]

s1 = Solve [ x ∧ 3 == − 1 ]

{ { x → − 1 } , { x → ( − 1 ) 1 / 3 } , { x → − ( − 1 ) 2 / 3 } }

N [ s1 ]

{ { x → − 1 . } , { x → 0.5 + 0.866025 i } , { x → 0.5 − 0.866025 i } }

When computing an odd root of a negative number, Mathematica has many choices (as illustrated above) and chooses a root with positive imaginary part – the result is not a real number.

( − 1 ) ∧ ( 1 / 3 ) // N

0.5 + 0.866025 i

To obtain real values when computing odd roots of negative numbers, first let sign(x)={x/|x|, if x≠0,0, if x=0. Sign[x] returns sign(x). Then, for the reduced fraction n/m with m odd, xn/m={sign(x)|x|n/m, if n is odd |x|n/m, if n is even . See Fig. 2.10 (b).

p2 = Plot [ Sign [ x ] Abs [ x ] ∧ ( 1 / 3 ) Abs [ x − 2 ] ∧ ( 2 / 3 ) Abs [ x + 1 ] ∧ ( 4 / 3 ) ,

{ x , − 3 , 5 } , PlotRange → { − 4 , 4 } , PlotStyle → Black ,

AspectRatio → Automatic , PlotLabel → “(b)” ]

Alternatively, use the Surd function to obtain the desired results. Surd[x,n] returns the real-valued nth root of x.

p3 = Plot [ Surd [ x , 3 ] Surd [ ( x − 2 ) ∧ 2 , 3 ] Surd [ ( x + 1 ) ∧ 4 , 3 ] ,

{ x , − 3 , 5 } , PlotRange → { − 4 , 4 } ,

AspectRatio → Automatic , PlotStyle → Black ,

PlotLabel → “(c)” ]

All three graphics are shown in Fig. 2.10 using Show together with GraphicsRow.

Show[GraphicsRow[{p1,p2,p3}]] □

The command

ListPlot[{{x1,y1},{x2,y2},...,{xn,yn}}]

plots the list of points {(x1,y1),(x2,y2),…,(xn,yn)}. The size of the points in the resulting plot is controlled with the option PlotStyle->PointSize[w], where w is the fraction of the total width of the graphic. For two-dimensional graphics, the default value is 0.008. The command

ListPlot[{y1,y2,..,yn}]

plots the list of points {(1,y1),(2,y2),…,(n,yn)}. To connect successive points with line segments, use the command ListLinePlot.

ListLinePlot[{x1,y1},{x2,y2},...,{xn,yn}}]

connects the consecutive points (x1,y1), (x2,y2), …, (xn,yn) with line segments.

Example 2.24

Plot the “principal trigonometric values.”

Solution

The “principal trigonometric values” are usually interpreted to mean the values of the functions sin⁡x and cos⁡x at the angles θ=0, π/6, π/4, π/3, π/2, 2π/3, 3π/4, 5π/6, π, 7Pi/6, 5π/4, 4π/3, 3π/2, 5π/3, 7π/4, 11π/6, and 2π.

We begin by defining these points in prinvals. These points are then plotted in lp1 with ListPlot. We illustrate the use of the PlotStyle, AspectRatio, and PlotLabel options.

prinvals = { { 1 , 0 } , { 1 / 2 , Sqrt [ 3 ] / 2 } , { 1 / Sqrt [ 2 ] , 1 / Sqrt [ 2 ] } , { Sqrt [ 3 ] / 2 , 1 / 2 } ,

{ 0 , 1 } , { − 1 / 2 , Sqrt [ 3 ] / 2 } , { − 1 / Sqrt [ 2 ] , 1 / Sqrt [ 2 ] } , { − Sqrt [ 3 ] / 2 , 1 / 2 } ,

{ − 1 , 0 } , { − 1 / 2 , − Sqrt [ 3 ] / 2 } , { − 1 / Sqrt [ 2 ] , − 1 / Sqrt [ 2 ] } , { − Sqrt [ 3 ] / 2 , − 1 / 2 } ,

{ 0 , − 1 } , { 1 / 2 , − Sqrt [ 3 ] / 2 } , { 1 / Sqrt [ 2 ] , − 1 / Sqrt [ 2 ] } , { Sqrt [ 3 ] / 2 , − 1 / 2 } } ;

lp1 = ListPlot [ prinvals , AspectRatio → Automatic ,

PlotStyle → { Black , PointSize [ . 025 ] } ,

PlotLabel → “(a)” ]

Next, we define prinlines to be a set of ordered pairs that will draw line segments between connected points. These lines are graphed by using Map to apply ListLinePlot to each ordered pair in prinlines. Generally, Map[f,list] applies the function f to each element of list. We use Show and illustrate the use of the AspectRatio and PlotLabel options to graph these line segments in lp2.

prinlines = { { { 1 , 0 } , { − 1 , 0 } } , { { 1 / 2 , Sqrt [ 3 ] / 2 } , { − 1 / 2 , − Sqrt [ 3 ] / 2 } } ,

{ { 1 / Sqrt [ 2 ] , 1 / Sqrt [ 2 ] } , { − 1 / Sqrt [ 2 ] , − 1 / Sqrt [ 2 ] } } ,

{ { Sqrt [ 3 ] / 2 , 1 / 2 } , { − Sqrt [ 3 ] / 2 , − 1 / 2 } } ,

{ { 0 , 1 } , { 0 , − 1 } } , { { − 1 / 2 , Sqrt [ 3 ] / 2 } , { 1 / 2 , − Sqrt [ 3 ] / 2 } } ,

{ { − 1 / Sqrt [ 2 ] , 1 / Sqrt [ 2 ] } , { 1 / Sqrt [ 2 ] , − 1 / Sqrt [ 2 ] } } ,

{ { − Sqrt [ 3 ] / 2 , 1 / 2 } , { Sqrt [ 3 ] / 2 , − 1 / 2 } } } ;

prinlines2 = Map [ ListLinePlot [ # ,

PlotStyle → { { GrayLevel [ . 5 ] , Dashing [ . 01 ] } } ] & , prinlines ] ;

lp2 = Show [ prinlines2 , AspectRatio → Automatic ,

PlotLabel → “(b)” ]

In lp3, we use Plot to graph the functions y=1−x2 and y=−1−x2. In the color version of this text, the CMYKColor code corresponds to a deep red. The graph is drawn to scale because we include the option AspectRatio->Automatic.

lp3 = Plot [ { Sqrt [ 1 − x ∧ 2 ] , − Sqrt [ 1 − x ∧ 2 ] } , { x , − 1 , 1 } ,

PlotStyle → CMYKColor [ 0 , 0.89 , 0.94 , 0.28 ] ,

AspectRatio → Automatic , PlotLabel → “(c)” ]

Finally, we use Show to display all four graphics together in lp4.

lp4 = Show [ lp1 , lp2 , lp3 , PlotLabel → “(d)” ]

The four images are displayed as a graphics grid using Show together with GraphicsGrid in Fig. 2.11.

Figure 2.11 (from left to right) (a)–(d) The four plots lp1, lp2, lp3, and lp4 combined into a single graphic.

Show[GraphicsGrid[{{lp1,lp2},{lp3,lp4}}]] □

A comprehensive discussion of Mathematica's extensive graphics capabilities cannot be reasonably covered in a single section of a text so our approach is to address issues that might of interest or present a different point of view to the beginning user of Mathematica. In the previous example, we saw that x3+1=0 has three solutions, two of which are complex. To visualize this graphically, observe that the zeros of z3+1=0 are the level curves of f(x,y)=|(x+iy)3+1| (x, y real) corresponding to 0. In a plot of f(x,y), the solutions are the zeros. Shortly we will discuss ContourPlot and Plot3D. For now, we remark that

cp1 = ContourPlot [ Abs [ ( x + I y ) ∧ 3 + 1 ] , { x , − 2 , 1 } , { y , − 3 / 2 , 3 / 2 } ,

Contours → 30 , Axes → True ]

p13d = Plot3D [ Abs [ ( x + I y ) ∧ 3 + 1 ] , { x , − 2 , 1 } , { y , − 3 / 2 , 3 / 2 } ,

Axes → True , PlotRange → { 0 , 15 } , MeshFunctions → ( # 3 & ) ,

Mesh → 35 ]

Show [ GraphicsRow [ { cp1 , p13d } ] ]

generates several level curves of f(x,y) (Fig. 2.12 (a)) and a 3-d plot of f(x,y) (Fig. 2.12 (b)) that help us see the zeros of the original equation. In the 3-d plot, note how we use the MeshFunctions option to generate contours on the 3-d plot.

Figure 2.12 (a) Contour plot of f(x,y). (b) 3-d plot of f(x,y).

2.3.2 Parametric and Polar Plots in Two Dimensions

For parametrically defined functions, x=x(t), y=y(t), a⩽t⩽b, use ParametricPlot. ParametricPlot[{x[t],y[t]},{t,a,b}] attempts to graph the parametrically defined functions x=x(t), y=y(t), a⩽t⩽b.

To graph the polar function, r=f(θ), use PolarPlot.

PolarPlot[f[theta],{theta,alpha,beta}]

attempts to graph r=f(θ) for α⩽θ⩽β.

To obtain the θ symbol with Mathematica on a Mac, press the esc key, type “theta”, and then press the esc key again. Use the same process to obtain other symbols, such as α, β, and so on.

Example 2.25

The Unit Circle

The unit circle is the set of points (x,y) exactly 1 unit from the origin, (0,0), and, in rectangular coordinates, has equation x2+y2=1. The unit circle is the classic example of a relation that is neither a function of x nor a function of y. The top half of the unit circle is given by y=1−x2 and the bottom half is given by y=−1−x2.

p1 = Plot [ { Sqrt [ 1 − x ∧ 2 ] , − Sqrt [ 1 − x ∧ 2 ] } , { x , − 1 , 1 } ,

PlotRange → { { − 3 / 2 , 3 / 2 } , { − 3 / 2 , 3 / 2 } } ,

AspectRatio → Automatic , PlotStyle → CMYKColor [ 1.00 , . 70 , 0 , . 75 ] ,

PlotLabel → “Georgia Southern Blue” ]

Each point (x,y) on the unit circle is a function of the angle, t, that subtends the x-axis, which leads to a parametric representation of the unit circle, {x=cos⁡t,y=sin⁡t,0⩽t⩽2π, which we graph with ParametricPlot.

p2 = ParametricPlot [ { Cos [ t ] , Sin [ t ] } , { t , 0 , 2 Pi } ,

PlotRange → { { − 3 / 2 , 3 / 2 } , { − 3 / 2 , 3 / 2 } } ,

AspectRatio → Automatic , PlotStyle → CMYKColor [ . 40 , . 43 , . 84 , . 08 ] ,

PlotLabel → “Georgia Southern Gold” ]

Using the change of variables x=rcos⁡t and y=rsin⁡t to convert from rectangular to polar coordinates, a polar equation for the unit circle is r=1. We use PolarPlot to graph r=1.

p3 = PolarPlot [ 1 , { t , 0 , 2 Pi } , PlotRange → { { − 3 / 2 , 3 / 2 } , { − 3 / 2 , 3 / 2 } } ,

AspectRatio → Automatic , PlotStyle → CMYKColor [ . 97 , . 33 , . 78 , . 24 ] ,

PlotLabel → “Ohio University Green” ]

We display p1, p2, and p3 side-by-side using Show together with GraphicsRow in Fig. 2.13. Of course, they all look the same, just in different colors (or shades of gray if you are reading the printed version of this text).

Figure 2.13 The unit circle generated with Plot, ParametricPlot, and PolarPlot.

Show [ GraphicsRow [ { p1 , p2 , p3 } ] ]

To illustrate several other features of Mathematica's extensive graphics capability, we elaborate on the example and construct a unit circle that can be used as a handout in a trigonometry, pre-calculus, or calculus class.

In p1, we graph the unit circle.

p1 = ParametricPlot [ { Cos [ t ] , Sin [ t ] } , { t , 0 , 2 Pi } ,

PlotStyle → Black ]

Next, we use ParametricPlot to connect the principal value points on the unit circle with line segments by observing that the parametric equations x=(1−t)a+tc, y=(1−t)b+td, 0⩽t⩽1 graphs a line segment connecting (a,b) and (c,d).

p2a = ParametricPlot [ { { 1 / 2 , Sqrt [ 3 ] / 2 } ( t − 1 ) ,

{ − 1 / 2 , − Sqrt [ 3 ] / 2 } ( t − 1 ) } ,

{ t , 0 , 1 } , PlotStyle → { { Black , Dashing [ { . 01 } ] } } ,

PlotLabel→“(b)”]

p2b = ParametricPlot [ { { Sqrt [ 3 ] / 2 , 1 / 2 } ( t − 1 ) ,

{ − Sqrt [ 3 ] / 2 , − 1 / 2 } ( t − 1 ) } ,