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# 1314L10 - Lesson 10 Applications of the First Derivative...

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Lesson 10 – Applications of the First Derivative 1 Lesson 10 Applications of the First Derivative Determining the Intervals on Which a Function is Increasing or Decreasing From the Graph of f A function is increasing on an interval ( a, b ) if, for any two numbers 1 x and 2 x in ( a, b ), ) ( ) ( 2 1 x f x f < , whenever 2 1 x x < . A function is decreasing on an interval ( a, b ) if, for any two numbers 1 x and 2 x in ( a, b ), ) ( ) ( 2 1 x f x f , whenever 2 1 x x < . In other words, if the y values are getting bigger as we move from left to right across the graph of the function, the function is increasing. If they are getting smaller, then the function is decreasing. Example 1: You are given the graph of f ( x ). State the interval(s) on which f is increasing and the interval(s) on which f is decreasing. The rate of change of a function at a point is given by the derivative of the function at that point. So we can use the derivative to determine where a function is increasing and where a function is decreasing.

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Lesson 10 – Applications of the First Derivative 2 Let’s look at the slopes of some lines that are tangent to this graph at various points: Here are some generalizations: At a point where the derivative is positive, a function is increasing.
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