Calculus

A major complaint of lecturers teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2 term) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course.

SECTION I. FUNCTIONS: AN INTRODUCTION.
1. Functions are Lurking Everywhere.
2. Characterizing Functions.
3. Functions Working Together.
SECTION II. RATES OF CHANGE: AN INTRODUCTION TO THE DERIVATIVE.
4. Linearity and Local Linearity.
5. The Derivative Function.
6. The Theoretical Backbone: Limits and Continuity.
7. Fruits of Our Labor: Derivatives and Local Linearity Revisited.
SECTION III. EXPONENTIAL, POLYNOMIAL, AND RATIONAL FUNCTIONS - WITH APPLICATIONS.
8. Exponential Functions.
9. Quadratics.
10. Optimization.
11. A Portrait of Polynomials and Rational Functions.
SECTION IV. INVERSE FUNCTIONS, EXPONENTIAL AND LOGARITHMIC FUNCTIONS.
12. Inverse Functions.
13. Logarithmic Functions.
14. Differentiating Logarithmic and Exponential Functions.
15. An Interesting Limit and A Brief Introduction to Differential Equations.
SECTION V. ADDING SOPHISTICATION TO YOUR DIFFERENTIATION.
16. Taking the Derivative of Composite Functions.
17. Implicit Differentiation.
18. Additional Topics.
SECTION VI. AN EXCURSION INTO GEOMETRIC SERIES.
19. Geometric Sums, Geometric Series.
SECTION VII. TRIGONOMETRY AND THE CALCULUS OF TRIGONOMETRIC FUNCTIONS.
20. Trigonometry: Going `Round in the Circle Game.
21. Trigonometry - Circles and Triangles.
22. Differentiation of Trigonometric Functions.
SECTION VIII. THE DEFINITE INTEGRAL.
23. Introducing the Definite Integral.
24. The Area Function and Its Characteristics.
25. The Fundamental Theorem of Calculus.
26. Finding Antiderivatives - An Introduction to Indefinite Integration.
27. Numerical Methods of Approximating Definite Integrals.
28. Applying the Definite Integral: Slice and Conquer.
SECTION IX. INTEGRATION, PART II (UNWRITTEN).
29. Techniques of Integration.
30. More Applications of Integration.
SECTION X. DIFFERENTIAL EQUATIONS.
31. Differential Equations: First Order.
32. Systems of Differential Equations and Second Order Homogeneous, Constant Coeffic.
SECTION XI. SERIES.
33. Taylor Series.
APPENDICES.
Algebra.
Geometric Formulas.
Theoretical Basis.
Equal Derivatives Theorem, Increasing/Decreasing Theorem.
Proof by Induction.
Conic Sections.