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…mehr
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.
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: (x)=3x+4.- Examples: Theorems for Sums and Products.- Examples: Limits of Quotients.- Exercises.- Problems.- 4 Differentiation, Derivatives, and Differentials.- Example: (x)=x2.- Example: (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: (x)=x sin x-1.- Inverse Trig Functions.- Example: (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 (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: (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: (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.
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: (x)=3x+4.- Examples: Theorems for Sums and Products.- Examples: Limits of Quotients.- Exercises.- Problems.- 4 Differentiation, Derivatives, and Differentials.- Example: (x)=x2.- Example: (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: (x)=x sin x-1.- Inverse Trig Functions.- Example: (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 (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: (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: (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.
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