ISBN:
9781468464849
Language:
English
Pages:
Online-Ressource
,
online resource
Edition:
Springer eBook Collection. Humanities, Social Sciences and Law
Parallel Title:
Erscheint auch als
Parallel Title:
Erscheint auch als
Keywords:
Science (General)
;
Mathematical analysis.
;
Social sciences.
;
Humanities.
Abstract:
1 Squares, Square Roots, and the Quadratic Formula -- The Definition -- Example: ?67.89 -- The Algorithm -- Example: ?100 -- Exercises -- Problems -- 2 More Functions and Graphs -- Definition: Limits of Sequences -- Example: x3-3x-1=0 -- Finding z3 with another Algorithm -- Finding z3 with Synthetic Division -- Example: 4x3+3x2-2x-1=0 -- Exercises -- Problems -- 3 Limits and Continuity -- Example: ƒ(x)=3x+4 -- Examples: Theorems for Sums and Products -- Examples: Limits of Quotients -- Exercises -- Problems -- 4 Differentiation, Derivatives, and Differentials -- Example: ƒ(x)=x2 -- Example: ƒ(x)=1/x -- Rules for Differentiation -- Derivatives for Polynomials -- Example: The Derivative of ?x -- Differentials -- Example: ?103, Example: ?142.3 -- Example: Painting a Cube -- Composites and Inverses -- Exercises -- Problems -- 5 Maxima, Minima, and the Mean Value Theorem -- Example: A Minimal Fence -- The Mean Value Theorem -- Example: Car Speed -- Example: Painting a Cube -- Exercises -- Problems -- 6 Trigonometric Functions -- Angles -- Trig Functions -- Triangles -- Example: The Derivative for sin x -- Derivatives for Trig Functions -- Example: ƒ(x)=x sin x-1 -- Inverse Trig Functions -- Example: ƒ(x)=2 arcsin x-3 -- Exercises -- Problems -- 7 Definite Integrals -- Example: ? and the Area of a Disc -- Riemann Sums and the Integral -- Example: The Area under ƒ(x)=x sin x -- Average Values -- Fundamental Theorems -- Trapezoidal Sums -- Example: The Sine Integral -- Exercises -- Problems -- 8 Logarithms and Exponentials -- The Definition of Logarithm -- Example: In 2 -- The Graph of In x -- Exponentials -- Example: A Calculation of e -- Example: Compound Interest and Growth -- Example: Carbon Dating and Decay -- Exercises -- Problems -- 9 Volumes -- Example: The Slab Method for a Cone -- Example: The Slab Method for a Ball -- Example: The Shell Method for a Cone -- Exercises -- Problems -- 10 Curves and Polar Coordinates -- Example: ƒ(x)=2?x -- Example: g(x)=x2/4 -- Example: Parametric Equations and the Exponential Spiral -- Polar Coordinates -- Example: The Spiral of Archimedes -- Exercises -- Problems -- 11 Sequences and Series -- The Definitions -- Example: The Harmonic Series -- Example: p-Series -- Geometric Series -- Example: An Alternating Series -- Example: Estimation of Remainders by Integrals -- Example: Estimation of Remainders for Alternating Series -- Example: Remainders Compared to Geometric Series -- Round-off -- Exercises -- Problems -- 12 Power Series -- The Theorems -- Example: ex -- Taylor Polynomials -- The Remainder Function -- Example: The Calculation of ex -- Example: Alternative Methods for ex -- Exercises -- Problems -- 13 Taylor Series -- Taylor’s Theorem -- Example: In x -- Newton’s Method -- Example: 2x+1= eX -- Example: ƒ(x)=(x-l)/x2 -- Example: Integrating the Sine Integral with Series -- Example: The Fresnel Integral -- The Error in Series Integration -- Example: l/(l-x2) -- Exercises -- Problems -- 14 Differential Equations -- Example: y’=ky and Exponential Growth -- Some Definitions -- Separable Variables -- Example: The Rumor DE -- Example: Series Solution by Computed Coefficients for y’ = 2xy -- Example: Series Solution by Undetermined Coefficients for y’-x-y -- Example: A Stepwise Process -- Exercises -- Problems -- Appendix: Some Calculation Techniques and Machine Tricks -- Invisible Registers -- Program Records -- Rewriting Formulas -- Constant Arithmetic -- Factoring Integers -- Integer Parts and Conversion of Decimals -- Polynomial Evaluation and Synthetic Division -- Taylor Series Evaluation -- Artificial Scientific Notation -- Round-off, Overflow, and Underflow -- Handling Large Exponents -- Machine Damage and Error -- Reference data and Formulas -- Greek Alphabet -- Mathematical Constants -- Conversion of Units -- Algebra -- Geometry -- Ellipse; Center at Origin -- Hyperbola; Center at Origin -- Trigonometric Functions -- Exponential and Logarithmic Functions -- Differentiation -- Integration Formulas -- Indefinite Integrals.
Abstract:
How THIS BOOK DIFFERS This book is about the calculus. What distinguishes it, however, from other books is that it uses the pocket calculator to illustrate the theory. A computation that requires hours of labor when done by hand with tables is quite inappropriate as an example or exercise in a beginning calculus course. But that same computation can become a delicate illustration of the theory when the student does it in seconds on his calculator. t Furthermore, the student's own personal involvement and easy accomplishment give hi~ reassurance and en couragement. The machine is like a microscope, and its magnification is a hundred millionfold. We shall be interested in limits, and no stage of numerical approximation proves anything about the limit. However, the derivative of fex) = 67.SgX, for instance, acquires real meaning when a student first appreciates its values as numbers, as limits of 10 100 1000 t A quick example is 1.1 , 1.01 , 1.001 , •••• Another example is t = 0.1, 0.01, in the function e/3t+9-3)/t. ix difference quotients of numbers, rather than as values of a function that is itself the result of abstract manipulation.
Description / Table of Contents:
1 Squares, Square Roots, and the Quadratic FormulaThe Definition -- Example: ?67.89 -- The Algorithm -- Example: ?100 -- Exercises -- Problems -- 2 More Functions and Graphs -- Definition: Limits of Sequences -- Example: x3-3x-1=0 -- Finding z3 with another Algorithm -- Finding z3 with Synthetic Division -- Example: 4x3+3x2-2x-1=0 -- Exercises -- Problems -- 3 Limits and Continuity -- Example: ƒ(x)=3x+4 -- Examples: Theorems for Sums and Products -- Examples: Limits of Quotients -- Exercises -- Problems -- 4 Differentiation, Derivatives, and Differentials -- Example: ƒ(x)=x2 -- Example: ƒ(x)=1/x -- Rules for Differentiation -- Derivatives for Polynomials -- Example: The Derivative of ?x -- Differentials -- Example: ?103, Example: ?142.3 -- Example: Painting a Cube -- Composites and Inverses -- Exercises -- Problems -- 5 Maxima, Minima, and the Mean Value Theorem -- Example: A Minimal Fence -- The Mean Value Theorem -- Example: Car Speed -- Example: Painting a Cube -- Exercises -- Problems -- 6 Trigonometric Functions -- Angles -- Trig Functions -- Triangles -- Example: The Derivative for sin x -- Derivatives for Trig Functions -- Example: ƒ(x)=x sin x-1 -- Inverse Trig Functions -- Example: ƒ(x)=2 arcsin x-3 -- Exercises -- Problems -- 7 Definite Integrals -- Example: ? and the Area of a Disc -- Riemann Sums and the Integral -- Example: The Area under ƒ(x)=x sin x -- Average Values -- Fundamental Theorems -- Trapezoidal Sums -- Example: The Sine Integral -- Exercises -- Problems -- 8 Logarithms and Exponentials -- The Definition of Logarithm -- Example: In 2 -- The Graph of In x -- Exponentials -- Example: A Calculation of e -- Example: Compound Interest and Growth -- Example: Carbon Dating and Decay -- Exercises -- Problems -- 9 Volumes -- Example: The Slab Method for a Cone -- Example: The Slab Method for a Ball -- Example: The Shell Method for a Cone -- Exercises -- Problems -- 10 Curves and Polar Coordinates -- Example: ƒ(x)=2?x -- Example: g(x)=x2/4 -- Example: Parametric Equations and the Exponential Spiral -- Polar Coordinates -- Example: The Spiral of Archimedes -- Exercises -- Problems -- 11 Sequences and Series -- The Definitions -- Example: The Harmonic Series -- Example: p-Series -- Geometric Series -- Example: An Alternating Series -- Example: Estimation of Remainders by Integrals -- Example: Estimation of Remainders for Alternating Series -- Example: Remainders Compared to Geometric Series -- Round-off -- Exercises -- Problems -- 12 Power Series -- The Theorems -- Example: ex -- Taylor Polynomials -- The Remainder Function -- Example: The Calculation of ex -- Example: Alternative Methods for ex -- Exercises -- Problems -- 13 Taylor Series -- Taylor’s Theorem -- Example: In x -- Newton’s Method -- Example: 2x+1= eX -- Example: ƒ(x)=(x-l)/x2 -- Example: Integrating the Sine Integral with Series -- Example: The Fresnel Integral -- The Error in Series Integration -- Example: l/(l-x2) -- Exercises -- Problems -- 14 Differential Equations -- Example: y’=ky and Exponential Growth -- Some Definitions -- Separable Variables -- Example: The Rumor DE -- Example: Series Solution by Computed Coefficients for y’ = 2xy -- Example: Series Solution by Undetermined Coefficients for y’-x-y -- Example: A Stepwise Process -- Exercises -- Problems -- Appendix: Some Calculation Techniques and Machine Tricks -- Invisible Registers -- Program Records -- Rewriting Formulas -- Constant Arithmetic -- Factoring Integers -- Integer Parts and Conversion of Decimals -- Polynomial Evaluation and Synthetic Division -- Taylor Series Evaluation -- Artificial Scientific Notation -- Round-off, Overflow, and Underflow -- Handling Large Exponents -- Machine Damage and Error -- Reference data and Formulas -- Greek Alphabet -- Mathematical Constants -- Conversion of Units -- Algebra -- Geometry -- Ellipse; Center at Origin -- Hyperbola; Center at Origin -- Trigonometric Functions -- Exponential and Logarithmic Functions -- Differentiation -- Integration Formulas -- Indefinite Integrals.
DOI:
10.1007/978-1-4684-6484-9
URL:
Volltext
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