Course 4 Unit 1 - Rates of Change
1st Edition

In Course 4, the mathematical strands in the Contemporary Mathematics in Context program become increasingly blended within units. For example, the content in Unit 3, Logarithmic Functions and Data Models, is primarily from the algebra and functions strand, but also includes content from the statistics and probability strand. Another example is Unit 7, Functions and Symbolic Reasoning. This unit includes content from both the algebra and functions and geometry and trigonometry strands.

Course 4 of the Contemporary Mathematics in Context program consists of mathematical content that allows considerable flexibility in tailoring a course to best prepare students for various undergraduate mathematics programs. (See the descriptions of Course 4 Units.)

Unit Overview

Rates of Change develops student understanding of the fundamental concepts underlying calculus and their applications.

Unit Objectives
  • To estimate the rate of change for a variety of quantities using tables of numerical data, graphical representations, and symbolic rules and to develop student ability to relate the rate of change in a quantity to the graph of that quantity
  • To recognize that the graphs of many nonlinear functions "look" linear when zoomed in around a point; and thus the rate of change at a point for a nonlinear function can be approximated with the rate of change for a linear function
  • To estimate the net change in a quantity whose rate function is given in graphical, tabular, and symbolic forms using systematic approximations to its rate-of-change function and geometric considerations
  • To estimate net change in a quantity by systematically approximating areas or using integrals in conjunction with a calculator or computer integration tool

Sample Overview

This sample material is the first investigation from Lesson 2, "Rates of Change of Familiar Functions." In this investigation, students use methods of estimating instantaneous rates of change learned in Lesson 1 to estimate rates of change at specific points for linear, quadratic, exponential, and trigonometric functions. In the remainder of this lesson, students discover ways to use a function rule to find the rule of its derivative and investigate how the shape of the graph of a function can be used to describe and sketch the graph of its derivative.

Instructional Design

Throughout the curriculum, interesting problem contexts serve as the foundation for instruction. As lessons unfold around these problem situations, classroom instruction tends to follow a common pattern as elaborated under Instructional Design.

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How the Algebra and Functions Strand Continues

In the remainder of Course 4, students develop understanding of logarithmic functions and their use in modeling and analyzing problem situations.

Students intending to pursue college majors in the mathematical, physical, and biological sciences and engineering extend their ability to use polynomial and rational functions to solve problems and extend their ability to manipulate symbolic representations of exponential, common and natural logarithmic, and trigonometric functions.

A unit that develops understanding and skill in the use of standard spreadsheet operations while reviewing and extending many of the basic algebra topics from Courses 1-3 is included for students intending to pursue college programs in social, management, and some of the health sciences or humanities.

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