Intro+to+the+derivative+notes.pdf

For x 5 we draw the tangent at p and estimate its

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For x = 5 we draw the tangent at P and estimate its slope to be about 3 / 2, so f 0 (5) 1 . 5. This allows us to plot the point P 0 (5 , 1 . 5) on the graph of f 0 directly beneath P . Repeating this procedure at several points, we can get a better estimation of the graph of f 0 .
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34 J. S´ anchez-Ortega Graphing a function given its derivative graph Note that the procedure above also allows us to sketch the graph of f provided the graph of f 0 . Notice that if f 0 > 0 then f is increasing, and if f 0 < 0 then f is decreasing. Also, f 0 ( a ) = 0 means that f has an horizontal tangent at ( a, f ( a )). As we will discuss later on, in such a case f might have a maximum or minimum at x = a . Example 3.34. The function below is f 0 ( x ). Which of the following graphs is f ( x ) ?
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3. Introduction to the derivative: Limits and Continuity 35 Solution: Let us start by noticing that f 0 ( - 4) = f 0 (2) = 0, so f might have a maximum or minimum at - 4 or 2. How can we find out this? Let us take a better look to the graph of f 0 . Note that for x < - 4 the derivative f 0 is negative and thus f is decreasing there. The following table collects some more information about f and f 0 . Intervals Sign of f 0 Info about f ( -∞ , - 4) - Decreasing ( - 4 , 2) + Increasing (2 , + ) - Decreasing Since f is decreasing to the left of x = - 4 and increasing to the right, then f has a minimum at x = - 4. Similarly, since f is increasing to the left of x = 2 and decreasing to the right, then f has a maximum at x = 2. With all this information in mind, we can conclude that the graph of f is the last option, namely: To practice, visit the following websites: http://www.flashandmath.com/advanced/derdraw/DerivativeDraw.html http://webspace.ship.edu/msrenault/GeoGebraCalculus/derivative_try_ to_graph.html http://webspace.ship.edu/msrenault/GeoGebraCalculus/derivative_matching_ antiderivative.html http://webspace.ship.edu/msrenault/GeoGebraCalculus/derivative_app_ 1_graph_AD.html
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36 J. S´ anchez-Ortega 3.10 Differentiation Rules In this section we will review how to differentiate constant functions, poly- nomial functions and power functions. We will learn how to derive the sum, difference, product, quotient and composition of two functions. Let us start with the simplest of all functions, the constant function f ( x ) = c . The graph of this function is the horizontal line y = c , which has slope 0, so it is not surprising that f 0 ( x ) = 0. In Leibniz notation, we have the following rule: Derivative of a Constant Function d dx ( c ) = 0 We next look at the functions f ( x ) = x n , where n is any real number . For example, f ( x ) = x , f ( x ) = 1 /x = x - 1 and f ( x ) = x = x 1 / 2 . If n = 1, the graph of f ( x ) = x is the line y = x , which has slope 1 and so f 0 ( x ) = 1. In general: The Power Rule. If n is any real number , then d dx ( x n ) = nx n - 1 Example 3.35. Using the Power Rule , compute the following deriva- tives. a. d dx h 3 x 2 i b. d dx 1 x 2 Solution: a. Notice that y = 3 x 2 = x 2 / 3 , i. e., n = 2 / 3 in the Power Rule.
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