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Learn how to apply the Power rule in finding derivatives and generalize functions for all x values. Understand the significance of derivatives and the behavior of functions. Get ready to ace Lesson 3.2 with in-depth insights and practical examples. Explore how derivatives reveal crucial information about functions, including slope and tangential behavior. Enhance your calculus skills and tackle complex problems with confidence.
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Wednesday, october 25th “Consider the postage stamp: its usefulness consists in the ability to stick to one thing till it gets there.” ~Josh Billings • Score 2.8 • Lesson 3.2 • Reminders
Lesson 3.2 The Derivative as a Function
Generalizing for all x … Section 3.1, Figure 3 Page 102
Ready for a shortcut? The Power rule:
11. Let Complete the table below for y’. -1.5 -1.5 6 -2 2.5 6 2.5 0 0 6 -6
The value of the derivative and what it tells me about f(x) f’(x) is zero f’(x) is positive f’(x) is negative Slope of the tangent line to f is positive Slope of the tangent line to f is zero Slope of the tangent line to f is negative f has a horizontal tangent line at that point f is increasing at that point f is decreasing at that point
The value of the derivative and what it tells me about f(x) f’(x) is zero f’(x) is positive f’(x) is negative Slope of the tangent line to f is positive Slope of the tangent line to f is zero Slope of the tangent line to f is negative f has a horizontal tangent line at that point f is increasing at that point f is decreasing at that point
Not all functions have a derivative at every single point! When the limit exists, we say that the function is differentiable at a.
A function is NOT DIFFERENTIABLE if the graph has these characteristics:
