CalcNotes03A

CalcNotes03A - CHAPTER 3 Derivatives 3.1 Derivatives...

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CHAPTER 3: Derivatives 3.1: Derivatives, Tangent Lines, and Rates of Change 3.2: Derivative Functions and Differentiability 3.3: Techniques of Differentiation 3.4: Derivatives of Trigonometric Functions 3.5: Differentials and Linearization of Functions 3.6: Chain Rule 3.7: Implicit Differentiation 3.8: Related Rates • Derivatives represent slopes of tangent lines and rates of change (such as velocity). • In this chapter, we will define derivatives and derivative functions using limits. • We will develop short cut techniques for finding derivatives. • Tangent lines correspond to local linear approximations of functions. • Implicit differentiation is a technique used in applied related rates problems.
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(Section 3.1: Derivatives, Tangent Lines, and Rates of Change) 3.1.1 SECTION 3.1: DERIVATIVES, TANGENT LINES, AND RATES OF CHANGE LEARNING OBJECTIVES • Relate difference quotients to slopes of secant lines and average rates of change. • Know, understand, and apply the Limit Definition of the Derivative at a Point. • Relate derivatives to slopes of tangent lines and instantaneous rates of change. • Relate opposite reciprocals of derivatives to slopes of normal lines. PART A: SECANT LINES • For now, assume that f is a polynomial function of x . (We will relax this assumption in Part B.) Assume that a is a constant. • Temporarily fix an arbitrary real value of x . (By “ arbitrary ,” we mean that any real value will do). Later, instead of thinking of x as a fixed (or single) value, we will think of it as a “moving” or “varying” variable that can take on different values. The secant line to the graph of f on the interval a , x [] , where a < x , is the line that passes through the points a , fa () and x , fx . secare is Latin for “to cut.” The slope of this secant line is given by: rise run = ± x ± a . • We call this a difference quotient , because it has the form: difference of outputs difference of inputs .
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(Section 3.1: Derivatives, Tangent Lines, and Rates of Change) 3.1.2 PART B: TANGENT LINES and DERIVATIVES If we now treat x as a variable and let x ± a , the corresponding secant lines approach the red tangent line below. tangere is Latin for “to touch.” A secant line to the graph of f must intersect it in at least two distinct points. A tangent line only need intersect the graph in one point, where the line might “just touch” the graph. (There could be other intersection points). • This “limiting process” makes the tangent line a creature of calculus , not just precalculus. Below, we let x approach a Below, we let x approach a from the right x ± a + () . from the left x ± a ² . ( S e e F o o t n o t e 1 . ) • We define the slope of the tangent line to be the (two-sided) limit of the difference quotient as x ± a , if that limit exists.
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This note was uploaded on 09/08/2011 for the course MATH 150 taught by Professor Bart during the Spring '06 term at Mesa CC.

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CalcNotes03A - CHAPTER 3 Derivatives 3.1 Derivatives...

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