Download Mathematical Function Illustration using Wolfram Alpha and Mathematica and more Lecture notes Logic in PDF only on Docsity!
The Mathematics Companion for Finite Mathematics and Business Calculus 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
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
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 ]
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" ]
0
2
4
6
8
10
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
See also: Annuities
The financing of a car loan is an example of the repayment of an ordinary annuity.
Wolfram Alpha Illustration
^ ����� � ����
����� ��������������� ����� ���� (������������ �������) ���������� �������������� � ( � )�������������������������������� � ���������������� � ( � )�������������������������������� � ( � )� ��������������������� � ◦ � ���������������������� � � ����������������� � � ( � ( � ))����� � � ( � )� �����ⅆ � /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 ]
- 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 + 3 x 2 D [ k [ x ] , x ] 6 x
- 2 + 3 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 → 1 0
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
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]
⩵ p f'[p] f[p]
Mathematica Illustration
Suppose that x = f(p) = 1000 (40 - p), for example, then f [ p _] : = 1000 40 - p then
elasticity [ p ] = - D Log ^1000 ^40 -^ p , p D [ Log [ p ] , p ] p 40 - p p 40 - p
/. {p → 1 } 1 39
Manipulate p 40 - p
, {p, 0, 30}
�
Manipulate (p^ +^ a) 40 - (p + a)
, {p, 0, 30}, {a, 0, 10}
� �
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, elasticity [ 20 ] = p 40 - p
/. p → 20 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 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
◼ 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
◼ See: Boolean logic, or
◼ 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
�
◼ 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
◼ See: Differentiation
- ◼ Augmented matrix,
- ◼ Average,
- ◼ Average rate of change
- ◼ Bar charts, �
- ◼ Bayes' formula,
- ◼ Bernoulli trial,
- ◼ Binomial coefficients,
- ◼ Binomial distribution,
- ◼ Binomial probability experiments,
- ◼ Binomial probability distribution, ◼ See: Bernoulli trials
- ◼ Binomial theorem,
- ◼ Boolean logic,
- ◼ Break-even point,
- ◼ Business functions, ◼ See: Business functions
- ◼ Cost functions,
- ◼ Demand functions,
- ◼ Profit functions,
- ◼ Revenue functions,
- ◼ Break-even point,
- ◼ Car loans, �
- ◼ Chain rule for differentiation,
- ◼ Characteristic polynomial,
- ◼ Codomain of a function,
- ◼ Column vectors,
- ◼ Combinations,
- ◼ Common logarithms,
- ◼ Complement of a set,
- ◼ Complex numbers,
- ◼ Derivatives of a product function,
- ◼ Derivatives of a quotient functions,
- ◼ Derivatives of trigonometric functions,
- ◼ Determinant of a matrix,
- ◼ Diagonal matrix,
- ◼ Diagonal of a matrix,
- ◼ Difference quotient,
- ◼ Differentiable function,
- ◼ Differential equation,
- ◼ Differentials,
- ◼ Differentiation,
- ◼ Dimensions of a matrix,
- ◼ Discontinuities,
- ◼ Discrete probability distribution,
- ◼ Disjoint sets,
- ◼ Disjoint union of sets,
- ◼ Disjunction,
- ◼ Domain of a function,
- ◼ Dot product,
- ◼ E, �
- ◼ Elasticity of demand for a product,
- ◼ Empty set,
- ◼ Equally likely assumption,
- ◼ Equivalent matrices,
- ◼ Events,
- ◼ Expected value of a random variable,
- ◼ Exponential functions,
- ◼ Exponential growth and decay,
- ◼ Factorial function, �
- ◼ Factoring polynomials,
- ◼ Feasible regions,
- ◼ Finance,
- ◼ First derivatives and the graphs of functions, ◼ See: Mathematics of finance
- ◼ First derivative test,
- ◼ First-order differential equations,
- ◼ Functions of one variable,
- ◼ Functions of several variables,
- ◼ Fundamental theorem of algebra,
- ◼ Fundamental theorem of calculus,
- ◼ Future value of an annuity,
- ◼ Gauss-Jordan elimination, �
- ◼ Geometric sequence,
- ◼ Gini index,
- ◼ Graphing of functions and relations,
- ◼ Histograms, �
- ◼ Homogeneous linear systems,
- ◼ Horizontal asymptotes,
- ◼ Horizontal tangents,
- ◼ Identity matrices, �
- ◼ If and only if,
- ◼ If then, ◼ See: Boolean logic (implication), logical equivalence, tautologies
- ◼ Implication, ◼ See: Boolean logic (implication)
- ◼ Implicit differentiation, ◼ See: Boolean logic
- ◼ Inconsistent linear systems,
- ◼ Increasing functions, ◼ See: Systems of linear equations
- ◼ Indefinite and definite integrals,
- ◼ Inelastic demand for a product,
- ◼ Inequalities, ◼ See: Elasticity of demand for a product
- ◼ Infinite geometric series,
- ◼ Infinite limits,
- ◼ Infinite sets,
- ◼ Inflection points of a function,
- ◼ Instantaneous rate of change,
- ◼ Integer exponents,
- ◼ Integers,
- ◼ Integral of a function,
- ◼ Integration by parts,
- ◼ Integration by substitution,
- ◼ Interest,
- ◼ Intersection and union of events, ◼ See: Compound interest, continuous compound interest, simple interest
- ◼ Intersection of sets, ◼ See: Empty set, intersection of sets, sample spaces, union of sets
- ◼ Inverse functions,
- ◼ Inverse of a matrix,
- ◼ Leontief input-output analysis, �
- ◼ L' Hôpital's rule,
- ◼ Limits,
- ◼ Limiting matrix,
- ◼ Limits and derivatives, ◼ See: Markov chains
- ◼ Limits at infinity, ◼ See: Difference quotient, limits
- ◼ Limits of integration,
- ◼ Quotient rule for differentiation,
- ◼ Radians, �
- ◼ Radicals,
- ◼ Random variables,
- ◼ Range of a function,
- ◼ Rates of change,
- ◼ Rational exponents,
- ◼ Rational expressions,
- ◼ Rational functions,
- ◼ Rational numbers,
- ◼ Real exponents, ◼ See also: Integers, real numbers, complex numbers
- ◼ Real numbers,
- ◼ Reduced form of an augmented matrix,
- ◼ Regression analysis,
- ◼ Regular Markov chains,
- ◼ Related rates, ◼ See: Markov chains
- ◼ Relative maxima and minima,
- ◼ Relative rate of change, ◼ See: Second derivative test for local extrema
- ◼ Revenue functions,
- ◼ Row operations, ◼ See: Business Functions
- ◼ Row-reduced matrix, ◼ See: Gauss-Jordan elimination
- ◼ Row vectors,
- ◼ Saddle points, �
- ◼ Sample spaces,
- ◼ Scalar multiplication of matrices,
