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Plot[F[x],{x,0,4}]. This plots F[x] from 0 to 4 using many points. Every time the value of F is computed, the integral is evaluated from scratch applying integration by parts each time. The result is a lengthy delay in displaying the graph. Using = causes the graph to be displayed more quickly.
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F[x_]: =
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t Exp[t] Sin[t] t
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F[x_]= t Exp[t] Sin[t] t;
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Plot[F[x], {x,0,1}]//Timing
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Plot[F[x], {x,0,1}]//Timing
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{16.203, 0.3
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Note the significant difference in time required to plot this function.
Often, you will want to evaluate an expression without assigning a value to a symbol. This can be done with the ReplaceAll (/.) replacement operator.
expression /. rule applies a rule or list of rules to each subpart of expression.
EXAMPLE 41 Suppose we want to evaluate x2 + 5x + 6 when x = 3, but do not want to assign a value to x.
Clear[x] x2 + 5x + 6 /. x 3 30 x
Global `x
(x is left undefined)
/. can also be used to replace an expression by another expression. Several replacements can be made at the same time if braces are used.
EXAMPLE 42
2 x + 3 + (2 x + 3)2 /. 2 x + 3 3 y + 5
2 3 y + 5 + (3 y + 5)
EXAMPLE 43
x2 + y /. {y x, x y} x + y2
SOLVED PROBLEMS
2.41 The Mathematica command Expand[expression], which is discussed in 7, expands expression algebraically. Define two symbols, a and b, as Expand[(x + 1)^3], using = and , respectively. Then let x = u + v and compute a and b.
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SOLUTION
a = Expand[(x + 1)^3] 1 + 3 x + 3 x2 + x3 b Expand[(x + 1)^3] x = u + v; a
2 1 + 3 (u + v)+3 (u + v) + (u + v)3
Expansion occurs immediately. Expansion does not occur until b is called.
u + v replaces x after expansion.
b 1 + 3 u + 3 u2 + u3 + 3 v + 6 u v + 3 u2 v + 3 v2 + 3 u v2 + v3 u + v replaces x before expansion.
2.42 The command Together, which is discussed in 7, combines the sum or difference of two or more fractions into one fraction. Define two symbols, y and z, as Together[a+b] using, respectively, = and . Then let a = 1/x and b = 1/(x+1)and compute y and z.
SOLUTION
y = Together[a + b] a+b a = 1/x; b = 1/(x + 1); y 1+ 1 x 1+ x z 1+2x x(1+ x)
Together is executed after the fractions are introduced so the fractions are combined into one. Since Together was executed prior to the introduction of the fractions, the result is the sum of a and b. At this point a and b are not fractions so Together does nothing.
z Together[a + b]
2.43 The Mathematica command Factor[expression] attempts to factor the algebraic expression, expression. Type a = Factor[poly] and b Factor[poly]. Then let poly = x2 + 2x + 1. Compute a and b and explain the difference in output.
SOLUTION
a = Factor[poly]; b Factor[poly]; poly = x2 + 2 x + 1; a 1 + 2 x + x2 b (1 + x)2
Since a is computed before poly is defined, its value is the factored form of the symbol poly, which is just poly. Then poly is replaced by x2 + 2 x + 1. On the other hand, b is not evaluated until called in the next to last line, so Mathematica factors the polynomial. 2.44 Replace x with x2 + 2x + 3 in the expression x2 + 5x + 6.
SOLUTION
x 2 + 5 x + 6 /. x x 2 + 2 x + 3 6 +5(3 + 2 x + x 2 )+(3 + 2 x + x 2 )2
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2.45 Replace y with x + 1 and z with x + 2 in the expression (x + y + z)2.
SOLUTION
(x + y + z)2 /. {y x + 1, z x + 2} (3 + 3 x)2
2.6 Logical Relations
Do not confuse = with , a logical equality. lhs rhs is True if and only if lhs and rhs have the same value; otherwise it is False. Logical equalities are used extensively in connection with equation solving ( 6). Other logical relations are available. The following list summarizes them.
Equal[x, y] or x y is True if and only if x and y have the same value. Unequal[x, y] or x!= y or x y is True if and only if x and y have different values. Less[x, y] or x < y is True if and only if x is numerically less than y. Greater[x, y] or x > y is True if and only if x is numerically greater than y. LessEqual[x, y] or x <= y or x y is True if and only if x is numerically less than y or equal to y. GreaterEqual[x, y] or x >= y or x y is True if and only if x is numerically greater than y or equal to y.
Note that Equal and Unequal can be used for comparing both numerical and certain non-numerical quantities, while Less, Greater, LessEqual, and GreaterEqual are strictly numerical comparisons.
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