What is calculus [electronic resource] : from simple algebra to deep analysis / R Michael Range.

Includes index.

Tangents to circles Tangents to parabolas Motion with variable speed Tangents to graphs of polynomials Rules for differentiation More general algebraic functions Beyond algebraic functions The cast : functions of a real variable Real numbers Functions Simple periodic functions Exponential functions Natural operations on functions Algebraic operations and functions Derivatives : how to measure change Algebraic derivatives by approximation Derivatives of exponential functions Differentiability and local linear approximation Properties of continuous functions Derivatives of trigonometric functions Simple differentiation rules Product and quotient rules Some applications of derivatives Exponential models The inverse problem and antiderivatives "Explosive growth" models Acceleration and motion with constant acceleration Periodic motions Geometric properties of graphs An algorithm for solving equations Applications to optimization Higher order approximations and Taylor polynomials The definite integral The area problem More applications of definite integrals Properties of definite integrals The fundamental theorem of calculus Existence of definite integrals Reversing the chain rule : substitution Reversing the product rule : integration by parts Higher order approximations, part 2: Taylor's theorem Excursion into complex numbers and the Euler identity.

This unique book provides a new and well-motivated introduction to calculus and analysis that are historically significant fundamental areas of mathematics and widely used in many disciplines.