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

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1 The Mathematics Companion for Finite Mathematics and Business Calculus is a dictionary-like reference 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 Mathematica Manipulations for dynamic learning and exploration. The following excepts from selected chapters indicate the style and range of topics covered in the book. 1 This chapter illustrates some of the basic features of Mathematica useful for finite mathematics and business calculus. Example 11e The symbol e denotes the base of the exponential function. The N function can be used to produce decimal approximations. N[e, 45] Special symbols can be entered and formatting options can be invoked using special menus called palettes. Opening Mathematica Palettes

2 See also: Maxima and minima 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 set of local minima. If the domain of a function is an interval 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]

3 MaxValue[f[x], x Reals] 1 Solve[f[x] 1, x] x ConditionalExpression π + 2 π C[1], C[1] Integers 2 Plot[f[x], {x, -20 π, 20 π}] The graph shows that the sine 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 real number inputs and to stretch or shrink the display of the functional values. Solution 2 Plot[f[x], {x, -π, π}] 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.

4 1 1 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 BarChart[Range[10], ChartStyle "DarkRainbow"]

5 BarChart[{Range[4], Reverse[Range[4]]}] 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 1 See also: Annuities 1 The financing of a car loan is an example of the repayment of an ordinary annuity.

6 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 % financing, compounded monthly, together with a rebate of $5,000. Which option should we choose? 0% financing pmt = % financing and a $5,000 rebate We use the amortization formula Clear[pv, i, n, pmt] i pmt[pv_, i_, n_] := pv 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. pmt30 000, , 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: pv[pmt_, i_, n_] := pmt i-n i pv , , See also: Derivatives of composite functions The chain rule tells us how to differentiate functions constructed by composing two or more given functions.

7 Wolfram Alpha Illustration 1 ( ) 111() () () (())11 () 1d/d d/d d/d Mathematica Illustration f[x_] := 3 x 2-2; g[y_] := Log[y]; h[x_] := Composition[f, g][x] h[x] 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)).

8 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 x 2 D[k[x], x] 6 x x 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 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 limit 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

9 (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}] {h[0], Limit[h[x], x 0, Direction 1], Limit[h[x], x 0, Direction -1]} {0.5, -5, -5} See also: Derivative 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

10 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: elasticity[p] = D[Log[f[p], p] D[Log[p], p] Mathematica Illustration p f'[p] f[p] Suppose that x = f(p) = 1000 (40 - p), for example, then f[p_] := p then DLog p, p elasticity[p] = - D[Log[p], p] p 40 - p p 40 - p 1 39 /. {p 1} p Manipulate, {p, 0, 30} 40 - p (p + a) Manipulate, {p, 0, 30}, {a, 0, 10} 40 - (p + a) 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

11 the commodity is less than $20 and is elastic if the price of the commodity exceeds $20. Furthermore, elasticity[20] = 1 p 40 - p /. p 20 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 curve Absolute maxima and minima, 17 Absolute value, 18 Absorbing Markov chains, 20 Absorbing states of a Markov chain, 21 Addition of matrices, 23 See: Matrix 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 sequence, 41 Arrow representation of vectors, 42 See: Vectors Asymptotes, 42

12 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 theorem, 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 polynomial, 68 Codomain of a function, 70 Column vectors, 71 Combinations, 72 Common logarithms, 73 Complement of a set, 74 Complex numbers, 76

13 Composite function, 77 See: Functions, chain rule Compound event, 77 Compound interest, 77 Concavity of the graph of a function, 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: Trigonometric functions 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 integral, 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 exponential function, 102 Derivatives of a logarithmic function, 103

14 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 Differentiable function, 112 Differential equation, 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 Dot product,133 E, 135 Elasticity of demand for a product, 137 Empty set, 138 Equally likely assumption, 140 Equivalent matrices, 140 Events, 141 Expected value of a random variable, 142 Exponential functions, 144 Exponential growth and decay, 146 Factorial function, 148 Factoring polynomials, 149

15 Feasible regions, 150 Finance, 152 See: Mathematics of finance First derivatives and the graphs of functions, 152 First derivative test, 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 tangents, 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

16 See: Systems of linear equations Increasing functions, 191 Indefinite and definite integrals, 193 Inelastic demand for a product, 194 See: Elasticity of demand for a product Inequalities, 194 Infinite geometric series, 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 Integration by parts, 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

17 Linear equations, 221 Linear functions, 225 Linear inequalities, 227 Linear programming, 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

18 Mean, 265 See: Arithmetic mean Measures of central tendency, 265 Measures of dispersion, 266 Median, 268 Method of least squares, 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, Trapezoidal 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)

19 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-slope form of a line equation, 298 Polynomial equations, 299 Polynomials, 301 Polynomials and rational functions, 302 Power series, 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 Product rule 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

20 Quotient rule for differentiation, 320 See: Differentiation Radians, 321 Radicals, 321 Random variables, 322 Range of a function, 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 Related rates, 335 Relative maxima and minima, 337 See: Second derivative 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

21 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 Sequences, 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 Slopes 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 geometric series, 370 Systems of linear equations, 371 Tangent function, 373 See: Trigonometric functions Tangent lines, 373 Tangent planes, 376 Tautologies, 377 See: Boolean logic

22 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: Feasible region 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