ABSTRACT
This chapter explores the consequences of the derivatives. One can approximate a function at a point using polynomials of various degrees. One can first find the constant that best approximates a function at a point. This is called the zeroth order Taylor polynomial. The chapter discusses Critical points, Rolle’s theorem and the Mean value theorem. The Rolle’s theorem is easy to understand. The Mean value theorem is a bit more technical. It is also sort of common sense. The chapter discusses the Zeroth Order Taylor Polynomials, the Order One Taylor Polynomial, and second order approximations. The Tangent line approximation is the best linear function. One can find higher order Taylor Polynomial approximations by applying Rolle’s theorem repeatedly. Sometimes it is easier to approximate the differences between two close functions. Approximating two close cosines, graphing the error for two close cosines, code fragment, two close exponentials, and approximations in a cancer model are described in detail.
