Calculator Calculus

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: 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.