Calculus: Dynamic Mathematics, Single and Multivariable

Calculus: Dynamic Mathematics, Single and Multivariable presents a complete calculus course for high school students. The thread that runs through the book is a justification of why we need calculus. The book begins with attempts to make predictions based on rates of change, and from there the idea of the derivative is developed. The initial attempts at using tables of data, graphs, and algebraic processes show the limitations of using rates of change without calculus. This gives the student a reason for knowing and studying limits and derivatives. Similarly, notions of area approximations are used to develop the idea of the definite integral, and notions of polynomial approximations stimulate the need for convergence tests of infinite series. Concepts of linear algebra are introduced before any multivariable calculus concepts. The ideas of determinants, eigenvalues, and matrices justify some techniques in multivariable calculus, such as the Jacobian, the Lagrange multiplier, predator-prey differential equations, and substitution of variables in double integrals. Rather than rules being given without a reason why the rule exsits, everything is justified and all concepts are explained at the level of a senior high school student. These justifications appear in the text as inductive arguments, or more often as formal proofs. The first nine chapters of the book constitute all the material required for a course in AP Calculus BC. For a free preview, please go to my website, www.drchuckgarner.com.