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Precalc0110to0111-page12 - • We will now define...

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(Section 1.11: Limits and Derivatives in Calculus) 1.11.1 SECTION 1.11: LIMITS AND DERIVATIVES IN CALCULUS LEARNING OBJECTIVES • Be able to develop limit definitions of derivatives and use them to find derivatives. • Understand the graphical interpretation of a derivative as the slope of a tangent line. • Understand the practical interpretation of a derivative as an instantaneous rate of change, possibly velocity. • Be able to find equations of tangent lines to graphs. • Relate derivatives to local linearization of, and marginal change in, a function. • Use derivatives to determine where a function is increasing, decreasing, or constant. PART A: DISCUSSION • In Section 1.5, we discussed limits . In Section 1.10, we saw how difference quotients represented slopes of secant lines and average rates of change
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Unformatted text preview: . • We will now define derivatives as limits of difference quotients . Derivatives represent slopes of tangent lines (which model or approximate graphs locally) and instantaneous rates of change . • Difference quotients on “small” intervals might be used to approximate derivatives. Conversely, derivatives might be used to approximate marginal change in a function, an idea used in economics. • In calculus, we will use derivatives to help us graph functions by finding where they are increasing , decreasing , or constant . (See Section 1.2.) • Limits, differentiation (the process of taking derivatives), and integration (which reverses differentiation; see Section 9.8) are the three key topics you will find in the first half of a calculus book, and they will be key themes throughout....
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