Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Chapter 3 Review Limits and Continuity
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3.2 Differentiability
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall How f’(a) Might Fail to Exist
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall How f’(a) Might Fail to Exist
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall How f’(a) Might Fail to Exist
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall How f’(a) Might Fail to Exist
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall How f’(a) Might Fail to Exist Most of the functions we encounter in calculus are differentiable wherever they are defined, which means they will not have corners, cusps, vertical tangent lines or points of discontinuity within their domains. Their graphs will be unbroken and smooth, with a well-defined slope at each point.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Differentiability Implies Local Linearity A good way to think of differentiable functions is that they are locally linear; that is, a function that is differentiable at a closely resembles its own tangent line very close to a. In the jargon of graphing calculators, differentiable curves will “straighten out” when we zoom in on them at a point of differentiability.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Differentiability Implies Local Linearity
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Differentiability Implies Continuity The converse of Theorem 1 is false. A continuous functions might have a corner, a cusp or a vertical tangent line, and hence not be differentiable at a given point.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Intermediate Value Theorem for Derivatives Not every function can be a derivative.
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3.3 Rules for Differentiation
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Rule 1 Derivative of a Constant Function
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Rule 2 Power Rule for Positive Integer Powers of x.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Rule 3 The Constant Multiple Rule
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Rule 4 The Sum and Difference Rule
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Example Positive Integer Powers, Multiples, Sums, and Differences
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Rule 5 The Product Rule
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Rule 7 Power Rule for Negative Integer Powers of x
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Second and Higher Order Derivatives
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3.4 Velocity and Other Rates of Change
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Motion Along a Line
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Free-fall Constants (Earth)
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Sensitivity to Change When a small change in x produces a large change in the value of a function f(x), we say that the function is relatively sensitive to changes in x. The derivative f’(x) is a measure of this sensitivity.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Derivatives in Economics
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3.5 Derivatives of Trigonometric Functions
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Example Finding the Derivative of the Sine and Cosine Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Simple Harmonic Motion The motion of a weight bobbing up and down on the end of a string is an example of simple harmonic motion.
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3.6 Chain Rule
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Logarithmic Differentiation Sometimes the properties of logarithms can be used to simplify the differentiation process, even if logarithms themselves must be introduced as a step in the process. The process of introducing logarithms before differentiating is called logarithmic differentiation.
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Quick Quiz Sections 3.7 – 3.9
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Quick Quiz Sections 3.7 – 3.9
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Chapter Test Solutions