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Unformatted text preview: Chapter 3 Outline Math 235 Fall 2000 3.1 The Derivative Definition of Derivative: State the formal definition of the derivative (as the limit of slopes of secant lines). Sketch a picture of what this definition means (including labelling what f ( x + h ), h , and so on are in the picture). Explain in your own words what is happening here. Interpret the derivative at a point as an instantaneous rate of change, i.e. the limit of average rates of change over smaller and smaller intervals. What is the definition of a left- or right-derivative? Also make sure you can do the above using the alternate formulation of the definition of derivative given in (3.1.5) on page 127. Calculating Derivatives using the Definition: Be able to calculate f ( x ) directly from the limit definition of derivative ( i.e. without using any differentiation rules) either at a point or as a function. Be sure you can do the algebra involved in such calculations! Tangent Lines and Normal Lines: Know how to use the derivative at a point to find the equations of the tangent and normal lines to the curve at that point. Dont just memorize the formulae in the book for these lines; be able to explain where those formulae came from! Differentiability and Continuity: Know the definition of differentiability (at a point, on an interval, or as a function). If a function is continuous, must it be dif- ferentiable? If a function is differentiable, must it be continuous? Be able to come up with examples illustrating various combinations of differentiability and continuity ( e.g. a function that is continuous but not differentiable). Determine whether a func- tion is differentiable at a point using the definition (including piecewise functions). Since this section is before the rules you should know how to do this without the rules. On the other hand, you should also know how to check for differentiability (in particular of a piecewise function) using the rules from the next section....
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