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# ch04 - 66 Chapter 4 The Derivative in Graphing and...

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Chapter 4: The Derivative in Graphing and Applications Summary: The main purpose of this chapter is to use the derivative as a tool to assist in the graphing of functions and for solving optimization problems. The most prominent use of the derivative is to help determine the overall behavior of a function. Types of behavior that can be described by using the derivative are when the function is increasing or decreasing and where the function is concave up or down. From these features, conclusions can be obtained about the extrema and inflection points of a function. This chapter, then, intends to show some of the many uses of derivatives. For example, derivatives can be used to find the roots of functions. This is the pri- mary motivation behind Newton’s Method. In motion problems where a position function describes the location of an object or a particle, the derivatives of the position function have special meanings such as the velocity and acceleration of the particle. Derivatives also find their way into many application problems where certain quantities either need to be maximized or minimized. Because derivatives can be used to find the extreme points of a function, they can be instrumental in optimizing quantities in many application problems. OBJECTIVES: After reading and working through this chapter you should be able to do the following: 1. Determine where a function is increasing or decreasing (§4 . 1 , 4 . 2). 2. Determine the concavity of a function (§4 . 1). 3. Locate critical points (§4 . 2), relative extrema (§4 . 2), and points of inflection of a function (§4 . 1). 4. Sketch the curve of a function based upon information from its derivatives together with information about asymptotes and intercepts (§4 . 2 , 4 . 3). 5. Find absolute extrema of a continuous function on a closed interval (§4 . 4). 6. Find the maximizing or minimizing value in various application problems (§4 . 5). 67
68 7. Use derivatives to discuss the motion of a particle that has a position func- tion for its location along a line (§4 . 6). 8. Apply Newton’s Method to find roots of functions (§4 . 7). 9. Draw conclusions about the value of a functions derivative at a point by using the Mean-Value Theorem and Rolle’s Theorem (§4 . 8). 4.1 Analysis of Functions I: Increase, Decrease, and Concavity PURPOSE: To relate the derivative of a function to the ideas of increasing, decreasing and concavity of functions. The primary focus of this section is to introduce the ideas of the increase and decrease of functions, the concavity of functions and how all of these relate to the derivative of a function. As it turns out, there are some convenient ways to remember these concepts that depend upon the tangent line of a function at any given point. Since the derivative can be used to find the slope of a tangent line, it becomes an important method for discussing the increase, decrease and concavity of a function.

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ch04 - 66 Chapter 4 The Derivative in Graphing and...

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