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The Companion for Finite Mathematics and Business is a dictionary-like refer- ence guide for learning and applying mathematical ideas, techniques, and formulas with the help of Mathematica, the leading computational software for students and users of mathematics in business, economics, and the life and social sciences. The material in the book is organized alphabetically for easy use and reference. It can be used as a tutorial introduction to the basics of Mathematica and touches briefly on the value of Wolfram Alpha as an extension of the material covered in the companion. Many examples illustrate the use of Mathemat- ica “Manipulations” for dynamic learning and exploration. The following excepts from selected chapters indicate the style and range of topics covered in the book. ������������������

This chapter illustrates some of the basic features of Mathematica useful for finite mathematics and business calculus. ◼ Example

�����������ⅇ The ⅇ denotes the base of the exponential . The N function can be used to produce decimal approximations.

N[ⅇ, 45] 2.71828182845904523536028747135266249775724709

Special symbols can be entered and formatting options can be invoked using special menus called “palettes.” ◼ Opening Mathematica Palettes ���������

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See also: Suppose that a function has several maxima or minima on sub-intervals of its domain. Then these maxima are referred to as “local maxima.” The absolute maximum of the function over the entire domain is the maximum the local minima, if it exists. The absolute minimum of the function on a domain is the minimum of the of “local” minima. If the is an including endpoints, then the values of the function at the endpoints are included in the set of potential absolute maxima and minima.

Mathematica Illustration

������� ◼ Solution 1

f[x_]:= Sin[x] MaxValue[f[x], x∈ Reals] 1

Solve[f[x]⩵ 1, x] π x→ ConditionalExpression +2πC[1],C[1]∈ Integers 2

Plot[f[x],{x,-20π, 20π}]

1.0

0.5

-60 -40 -20 20 40 60

-0.5

-1.0

The graph shows that the function assumes its absolute maxima and minima at infinitely many points over its domain. The graph shows that the sine function assumes its absolute maxima and minima at infinitely many points over its domain.

We can vary the interval {x, -20 π, 20 π} at will to view the graph of f[x] for other inputs and to stretch or shrink the display of the values. ◼ Solution 2

Plot[f[x],{x,-π,π}]

1.0

0.5

-3 -2 -1 1 2 3

-0.5

-1.0

This graph shows that if we restrict the domain of f to the interval -π ≤ x ≤ π, then the function has a unique global maximum and a unique global minimum on the specified domain. ���������

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See also: Discrete plots, histograms, list plots, list line plots, pie charts, trees Mathematica has an extensive repertoire of tools for visualizing data. Among them are the bar charts.

Mathematica Illustration

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BarChart[Range[10], ChartStyle→ "DarkRainbow"]

10

8

6

4

2

0 �������

BarChart[{Range[4], Reverse[Range[4]]}]

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BarChart[{{1, 2, 3, 4},{1, 4, 3, 2, 5}}, ChartLabels→{"a", "b", "c", "d", "e"}]

a b c d a b c d e

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See also: Annuities The financing of a car loan is an example of the repayment of an ordinary annuity. Mathematica Illustration

������� Suppose we are planning to buy a new car worth $35,000 and we have two choices of paying for the car: 1. 0% financing over 60 months. 2. 4.5% financing, compounded monthly, together with a rebate of $5,000. Which option should we choose? ◼ 0% financing

35 000.00 pmt= 60 583.333

◼ 4.5% financing and a $5,000 rebate We use the amortization formula

Clear[pv, i, n, pmt] i pmt[pv_, i_, n_]:= pv 1-1+i -n If we choose the rebate option, we are borrowing only $22,200 at a monthly rate of i = 0.045/12 per cent for 48 months.

pmt30 000, 0.045 12, 60 559.291

The calculations show that the 4.5% option is the way to go. The payment formula corresponds to the present value formula of an ordinary annuity. We can verify this by computing the present value of the annuity consisting of 60 monthly equal payments:

1-1+i -n pv[pmt_, i_, n_]:= pmt i

pv559.291, 0.045 12, 60 30 000.

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See also: of composite functions The tells us how to differentiate functions constructed by “composing” two or more given functions. Wolfram Alpha Illustration

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Mathematica Illustration

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f[x_]:=3x 2 - 2; g[y_]:= Log[y];

h[x_]:= Composition[f, g][x]

h[x] -2+ 3 Log[x] 2

h[x]⩵f[y]/. y -> Log[x] True

D[h[x], x] 6 Log[x] x The function h is usually written with the composition symbol “∘” as (f∘g) and its values are denoted by f(g(x)). ������� Composition is not commutative. The function (f∘g) is rarely equal to the function (g∘f).

k[x_]:= Composition[g, f][x]

k[x] Log-2+3x 2

D[k[x], x] 6 x -2+3x 2 It is easy to verify that the function h[x] and k[x] are not equal. If we evaluate both functions at x = 1, for example, we get different results:

h[x]/. x→1 -2

k[x]/. x→1 0

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��������������� A discontinuity is a point at which a function is discontinuous. There are various reason why a in one variable is discontinuous at a point x in its domain. 1.A function may be discontinuous at x if its value f[x] is different from the of f[x] at x. In this case, the discontinuity is said to be removable. All we need to do is to redefine f[x] to be equal to the limit of f[x] at x. 2.A function may be discontinuous at x if the limit of f[x] at x does not exist. For example, the left limit may be different from the right limit. 3.A function may be discontinuous at x if either the left limit or the right limit of f[x] at x does not exist.

Mathematica Illustration

������� ◼ A removable discontinuity (x-1)x+2 f[x_]:= (x-1)

f[1] � ������ �������� ���������� ������������ � ��������� ������������� ���������� � ��������������� ������������

Indeterminate

g[x_]:= Piecewise[{{f[x],!(x⩵1)},{1, x ==1}}]

g[1] 1

������� ◼ The value f[x] is different from the limit of f[x] at x.

h[x_]:= Piecewise[{{-1, x<0},{.5, x≥0}}]

Plot[h[x],{x,-2, 2}]

0.5

-2 -1 1 2

-0.5

-1.0

{h[0], Limit[h[x], x→ 0, Direction→1], Limit[h[x], x→ 0, Direction→-1]} {0.5,-5,-5}

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See also: of the logarithmic functions Suppose that the price p and the demand x of a product are related by a price-demand equation

f[p_]:=x Then the elasticity of demand at price p is the relative rate of change of demand x, divided by the relative rate of change of price p: D[Log[f[p], p] p f'[p] elasticity[p] = ⩵ D[Log[p], p] f[p]

Mathematica Illustration

������� Suppose that x = f(p) = 1000 (40 - p), for example, then

f[p_]:= 100040-p then

DLog100040-p, p elasticity[p] = - D[Log[p], p] p 40-p

p /.{p→1} 40-p 1 39 p Manipulate ,{p, 0, 30} 40-p

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(p+a) Manipulate ,{p, 0, 30},{a, 0, 10} 40-(p+a)

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If 0 < E(p) < 1, the demand is said to be inelastic, if 1 < E(p), the demand is said to be elastic, and if E(p) = 1, a percentage change in the price of the commodity entails the same percentage change in demand. In the given example, the Manipulation shows that the demand is inelastic if the unit price of the commodity is less than $20 and is elastic if the price of the commodity exceeds $20. Furthermore, p elasticity[20] = /. p→ 20 40-p 1

Hence the percentage change in price of the commodity leads to the same percentage change in demand. �����

� ◼ Absolute equality of income, 17 ◼ See: Gini index, Lorenz ◼ Absolute maxima and minima, 17 ◼ Absolute value, 18 ◼ Absorbing Markov chains, 20 ◼ Absorbing states of a Markov chain, 21 ◼ Addition of matrices, 23 ◼ See: algebra ◼ Addition principle for counting, 23 ◼ Amortization, 24 ◼ And (∧, &&), 26 ◼ See: Boolean logic (conjunction) ◼ Angle, 26 ◼ Annuities, 27 ◼ Anti-derivative, 30 ◼ See: Anti-differentiation ◼ Anti-differentiation, 30 ◼ Area, 36 ◼ Arithmetic mean, 40 ◼ Arithmetic , 41 ◼ Arrow representation of vectors, 42 ◼ See: Vectors ◼ Asymptotes, 42 ◼ Augmented matrix, 44 ◼ Average, 45 ◼ See: Mean, median, mode ◼ Average rate of change 45

� ◼ Bar charts, 47 ◼ Bayes' formula, 49 ◼ Bernoulli trial, 49 ◼ Binomial coefficients, 50 ◼ Binomial distribution, 51 ◼ Binomial probability experiments, 53 ◼ See: Bernoulli trials ◼ Binomial probability distribution, 53 ◼ Binomial , 54 ◼ Boolean logic, 55 ◼ Break-even point, 58 ◼ See: Business functions ◼ Business functions, 58 ◼ Cost functions, 58 ◼ Demand functions, 60 ◼ Profit functions, 61 ◼ Revenue functions, 62 ◼ Break-even point, 63

� ◼ Car loans, 65 ◼ Chain rule for differentiation, 66 ◼ Characteristic , 68 ◼ of a function, 70 ◼ Column vectors, 71 ◼ Combinations, 72 ◼ Common logarithms, 73 ◼ Complement of a set, 74 ◼ Complex numbers, 76 ◼ Composite function, 77 ◼ See: Functions, chain rule ◼ Compound event, 77 ◼ Compound interest, 77 ◼ Concavity of the , 78 ◼ Conditional, 79 ◼ See: Boolean logic, if-then ◼ Conditional probability, 79 ◼ Conjunction, 81 ◼ See: And, Boolean logic ◼ Consistent linear systems, 81 ◼ Constant functions, 82 ◼ See: Functions ◼ Continuous compound interest, 82 ◼ Continuous functions, 83 ◼ Cosine function, 86 ◼ See: ◼ Cost function, 86 ◼ See: Business functions ◼ Counting principles, 86 ◼ Critical points of a function, 88 ◼ Curve fitting, 89 ◼ See: Regression analysis

� ◼ Decreasing function, 90 ◼ Definite , 92 ◼ Degree measure of an angle, 96 ◼ Degree of a polynomial, 97 ◼ Demand function, 98 ◼ See: Business functions ◼ Derivatives of a function, 98 ◼ Derivatives of a composite function, 101 ◼ Derivatives of an , 102 ◼ Derivatives of a logarithmic function, 103 ◼ Derivatives of a product function, 104 ◼ Derivatives of a quotient functions, 105 ◼ Derivatives of trigonometric functions, 106 ◼ Determinant of a matrix, 108 ◼ Diagonal matrix, 109 ◼ Diagonal of a matrix, 110 ◼ Difference quotient, 110 ◼ , 112 ◼ , 114 ◼ Differentials, 116 ◼ Differentiation, 122 ◼ Dimensions of a matrix, 126 ◼ Discontinuities, 126 ◼ Discrete probability distribution, 128 ◼ Disjoint sets, 129 ◼ Disjoint union of sets, 129 ◼ Disjunction, 131 ◼ See: Boolean logic, or ◼ Domain of a function, 131 ◼ ,133

� ◼ E, 135 ◼ Elasticity of demand for a product, 137 ◼ , 138 ◼ Equally likely assumption, 140 ◼ Equivalent matrices, 140 ◼ Events, 141 ◼ Expected value of a , 142 ◼ Exponential functions, 144 ◼ Exponential growth and decay, 146

� ◼ function, 148 ◼ Factoring , 149 ◼ Feasible regions, 150 ◼ Finance, 152 ◼ See: Mathematics of finance ◼ First derivatives and the graphs of functions, 152 ◼ First , 155 ◼ First-order differential equations, 158 ◼ Functions of one variable, 159 ◼ Functions of several variables, 160 ◼ Fundamental theorem of algebra, 163 ◼ Fundamental theorem of calculus, 164 ◼ Future value of an annuity, 165

� ◼ Gauss-Jordan elimination, 167 ◼ Geometric sequence, 170 ◼ Gini index, 171 ◼ Graphing of functions and relations, 172

� ◼ Histograms, 175 ◼ Homogeneous linear systems, 180 ◼ Horizontal asymptotes, 181 ◼ Horizontal , 182

� ◼ Identity matrices, 184 ◼ If and only if, 185 ◼ See: Boolean logic (implication), logical equivalence, tautologies ◼ If then, 185 ◼ See: Boolean logic (implication) ◼ Implication, 185 ◼ See: Boolean logic ◼ Implicit differentiation, 185 ◼ Inconsistent linear systems, 190 ◼ See: Systems of linear equations ◼ Increasing functions, 191 ◼ Indefinite and definite , 193 ◼ Inelastic demand for a product, 194 ◼ See: Elasticity of demand for a product ◼ Inequalities, 194 ◼ Infinite geometric , 194 ◼ Infinite limits, 196 ◼ Infinite sets, 197 ◼ Inflection points of a function, 198 ◼ Instantaneous rate of change, 199 ◼ Integer exponents, 200 ◼ Integers, 200 ◼ Integral of a function, 201 ◼ , 204 ◼ Integration by substitution, 205 ◼ Interest, 207 ◼ See: Compound interest, continuous compound interest, simple interest ◼ Intersection and union of events, 207 ◼ See: Empty set, intersection of sets, sample spaces, union of sets ◼ Intersection of sets, 207 ◼ Inverse functions, 208 ◼ Inverse of a matrix, 208

� ◼ Leontief input-output analysis, 211 ◼ L' Hôpital's rule, 214 ◼ Limits, 215 ◼ Limiting matrix, 218 ◼ See: Markov chains ◼ Limits and derivatives, 218 ◼ See: Difference quotient, limits ◼ Limits at infinity, 218 ◼ Limits of integration, 221 ◼ See: Lower limits of integration, upper limits of integration ◼ Linear equations, 221 ◼ Linear functions, 225 ◼ Linear inequalities, 227 ◼ , 229 ◼ Linear regression, 232 ◼ See: Regression analysis ◼ Linear systems, 232 ◼ See: Systems of linear equations, matrix equations ◼ List line plots, 232 ◼ List plots, 233 ◼ Local extrema, 235 ◼ See: Maxima and minima ◼ Logarithmic functions, 235 ◼ Logarithms, 237 ◼ See: Common logarithms, logarithmic functions, natural logarithms ◼ Logic, 238 ◼ See: Boolean logic ◼ Logical equivalence, 238 ◼ Logical implication, 239 ◼ Lorenz curve, 240 ◼ Lower limits of integration, 240 ◼ See: Integration, limits of integration

� ◼ Marginal analysis, 241 ◼ Markov chains, 249 ◼ Mathematica domains, 252 ◼ Mathematics of finance, 254 ◼ See: Annuities, arithmetic sequence, compound interest, continuous interest, future value, geometric sequence, interest, present value, rate of interest, simple interest ◼ Matrices, 254 ◼ Matrix algebra, 256 ◼ Matrix equations, 261 ◼ Matrix multiplication, 262 ◼ See: Matrix algebra ◼ Maxima and minima, 262 ◼ Mean, 265 ◼ See: Arithmetic mean ◼ Measures of central tendency, 265 ◼ Measures of dispersion, 266 ◼ Median, 268 ◼ Method of , 269 ◼ Mode, 271 ◼ Mortgages, 272 ◼ See: Annuities ◼ Multiplication of matrices, 272 ◼ See: Matrix algebra ◼ Multiplication principle for counting, 272 ◼ Mutually exclusive events, 273

� ◼ Natural logarithms, 276 ◼ Negation, 277 ◼ See: Boolean logic, not ◼ Non-differentiable function, 277 ◼ Normal probability distribution, 278 ◼ Not, 283 ◼ See: Boolean logic (negation) ◼ Numerical integration methods, 283 ◼ See: Simpson's rule,

� ◼ Odds, 284 ◼ One-to-one correspondence, 285 ◼ See: One-to-one functions ◼ One-to-one functions, 285 ◼ Operations on polynomials, 287 ◼ Operations on rational functions, 288 ◼ Optimization in business and economics, 290 ◼ Or, 291 ◼ See: Boolean logic (disjunction) � ◼ Parabolas, 292 ◼ See: Quadratic equations, quadratic formula, quadratic functions ◼ Percentage rate of change, 292 ◼ Permutations, 292 ◼ Pie charts, 293 ◼ Piecewise defined functions, 295 ◼ Pivot of a matrix, 296 ◼ See: Row-reduced matrix ◼ Points of diminishing returns, 296 ◼ Point- form of a line equation, 298 ◼ Polynomial equations, 299 ◼ Polynomials, 301 ◼ Polynomials and rational functions, 302 ◼ , 303 ◼ Present value of an annuity, 304 ◼ Price-demand functions, 306 ◼ See: Business functions ◼ Probability, 306 ◼ Probability distribution of a random variable, 311 ◼ Probability spaces, 312 ◼ for differentiation, 313 ◼ See: Differentiation ◼ Profit functions, 314 ◼ See: Business functions ◼ Profit-loss analysis, 314 ◼ See: Break-even point

� ◼ Quadratic equations, 315 ◼ Quadratic formula, 317 ◼ Quadratic functions, 319 ◼ Quadratic regression, 320 ◼ See: Break-even point ◼ for differentiation, 320 ◼ See: Differentiation

� ◼ , 321 ◼ Radicals, 321 ◼ Random variables, 322 ◼ , 325 ◼ Rates of change, 326 ◼ Rational exponents, 327 ◼ Rational expressions, 327 ◼ Rational functions, 329 ◼ Rational numbers, 329 ◼ See also: Integers, real numbers, complex numbers ◼ Real exponents, 330 ◼ Real numbers, 331 ◼ Reduced form of an augmented matrix, 333 ◼ Regression analysis, 333 ◼ Regular Markov chains, 335 ◼ See: Markov chains ◼ , 335 ◼ Relative maxima and minima, 337 ◼ See: test for local extrema ◼ Relative rate of change, 338 ◼ Revenue functions, 338 ◼ See: Business Functions ◼ Row operations, 338 ◼ See: Gauss-Jordan elimination ◼ Row-reduced matrix, 338 ◼ Row vectors, 340

� ◼ Saddle points, 343 ◼ Sample spaces, 344 ◼ Scalar multiplication of matrices, 345 ◼ See: Matrix algebra ◼ Scatter diagram, 345 ◼ See: List plot ◼ Scientific notation, 345 ◼ Second derivatives and graphs of functions, 346 ◼ Second derivative test for local extrema, 349 ◼ , 349 ◼ Series, 351 ◼ Sets, 352 ◼ Sigma notation, 355 ◼ Simple events, 357 ◼ See: Sample spaces ◼ Simple interest, 357 ◼ Simpson's rule for approximating definite integrals, 357 ◼ Sine function, 360 ◼ See: Trigonometric functions ◼ Sinking funds, 360 ◼ Slope-intercept equation of a straight line, 361 ◼ of tangents to the graph of a function, 362 ◼ Slope of a straight line, 364 ◼ Slope-point equation of a straight line, 365 ◼ Solving linear systems, 365 ◼ Standard deviation, 367 ◼ Sums of arithmetic sequences, 368 ◼ Sums of finite geometric sequences, 369 ◼ Sums of infinite , 370 ◼ Systems of linear equations, 371

� ◼ function, 373 ◼ See: Trigonometric functions ◼ Tangent lines, 373 ◼ Tangent planes, 376 ◼ Tautologies, 377 ◼ See: Boolean logic ◼ Technology matrices, 377 ◼ Total income, 380 ◼ See: Income ◼ Transition matrix, 380 ◼ Transpose of a matrix, 381 ◼ See: Matrix algebra ◼ Trapezoidal rule for approximating definite integrals, 381 ◼ Trigonometric functions, 384 ◼ Truth-tables, 387 ◼ See: Boolean logic

� ◼ Unbounded solution region, 388 ◼ See: ◼ Union of alphanumeric lists, 388 ◼ Union of sets, 389 ◼ Upper limits of integration, 390 ◼ See: Integration, lower limits of integration

� ◼ Variables and constants, 391 ◼ Variance, 393 ◼ Vectors, 393 ◼ Venn diagrams, 395 ◼ Vertex form of a quadratic function, 398 ◼ Vertical asymptotes, 399 ◼ Vertical lines, 401