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Unformatted text preview: Chapter 2: The Derivative Summary: Chapter 2 builds upon the ideas of limits and continuity discussed in the previous chapter. By using limits, the instantaneous rate at which a function changes with respect to its inputs can be investigated. This leads to the ability to find tangent lines to a function at a given point. These two concepts are en- compassed by the derivative which is a new function that can be used to find the slope of a curve at a given point. A progression from continuous functions to those which have tangent lines to those which are differentiable can be seen. The concept of the derivative also provides some valuable tools for determining the shape of a given function and how a function behaves. These tools are a direct result of a derivative’s ability to describe the slope of a function at a point. Many tools for finding the derivatives of functions are also discussed in this chapter so that derivatives of more complicated functions can be found. OBJECTIVES: After reading and working through this chapter you should be able to do the following: 1. Understand the definition of the derivative and its implications (§2 . 1 , 2 . 2) 2. Be able to read and understand different notations for the derivative (§2 . 2) 3. Find derivatives of simple functions like polynomials (§2 . 3) or trigonomet- ric functions (§2 . 5) 4. Use the product and/or quotient rules to find derivatives of functions involv- ing products or quotients of other functions (§2 . 4) 5. Use the chain rule (either form) to find the derivatives of functions which involve the compositions of other functions (§2 . 6) 2.1 Tangent Lines and Rates of Change PURPOSE: To use limits and continuity to investigate the slope of a function at a point. 33 34 This section attempts to bridge the gap between limits and continuity and the slope of a graph. The process is one where the slope of a secant line can be seen as slope of a secant line the average rate of change of a function. This becomes the instantaneous rate of change as the two points defining the secant line approach each other in the limit. instantaneous rate of change The slope of the tangent line at a point x is defined as a limit: m tan = lim h → f ( x + h ) − f ( x ) h This slope can then be used to construct the tangent line to a function f ( x ) at the point ( x , f ( x )) : y − f ( x ) = m tan ( x − x ) This is nothing more than the point-slope form of a line. In this case, the slope m tan is determined by a limiting process. The big idea in this section (and the next) is the equivalence of the following three things: 1. The slope of a tangent line to y = f ( x ) at the point x = x . 2. The instantaneous rate of change of y = f ( x ) at the point x = x ....
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- Spring '10