ERIC Number: EJ1192856
Record Type: Journal
Publication Date: 2018-Oct
Introducing Roots and Extrema in Calculus
Adams, Caleb L.
Mathematics Teacher, v112 n2 p132-135 Oct 2018
Polynomials with rational roots and extrema may be difficult to create. Although techniques for solving cubic polynomials exist, students struggle with solutions that are in a complicated format. Presented in this article is a way instructors may wish to introduce the topics of roots and critical numbers of polynomial functions in calculus. In a typical calculus course, one application of the derivative that students investigate is finding relative extrema of a function after learning methods to find derivatives of functions. Zeros of the function's derivative are deemed critical numbers. Through a variety of methods, students are able to determine whether the critical numbers are indicative of relative minima or maxima of the function (i.e., sign tables and both first and second derivative tests, etc.). Although irrational values are acceptable to mathematicians--and are often common when working with polynomial functions--in the author's experience, many students are uncomfortable with such values. Substitution of an irrational value into a polynomial function is done to check the validity of the solution; however, students may struggle simplifying the cube of 1 minus the square root of 2. To facilitate the ease and comfort of students learning these concepts, the author suggests that instructors of a first course in calculus initially use cubic polynomial functions that have both rational roots and rational relative extrema. Once students demonstrate the ability to determine the values of the roots and extrema of the cubic, the instructor is then free to increase the level of difficulty (e.g., rational roots with irrational extrema or irrational roots and rational extrema). This article will provide calculus teachers with information about the construction and use of such polynomial functions.
Descriptors: Mathematics Instruction, Calculus, Mathematical Concepts, Concept Formation, Teaching Methods, Algebra, Mathematical Formulas, Secondary School Mathematics, College Mathematics, High Schools
National Council of Teachers of Mathematics. 1906 Association Drive, Reston, VA 20191. Tel: 800-235-7566; Tel: 703-620-9840; Fax: 703-476-2570; e-mail: NCTM@nctm.org; Web site: http://www.nctm.org/publications/mathematics-teacher/
Publication Type: Journal Articles; Reports - Descriptive
Education Level: Secondary Education
Authoring Institution: N/A