Tuesday, February 06, 2007

Chapter 4-1

Applications of the First Derivative





Vocabulary:




relative maximum- for a function f, the relative maximum occurs at x=c if there exists an open interval (a,b) containing c such that f(x)≤f(c) for all x in (a,b)




relative minimum- for a function f, the relative minimum occurs at x=c if there exists an open interval (a,b) containing c such that f(x) ≥f(c) for all x in (a,b)



Notes:


If two points exist, and y2>y1 then the function is increasing


If two points exist, and y2

For the derivative:



If f'(x) >0 then the function is increasing... If the tangent line has a positive slope, then the function is increasing.





If f'(x)<0>


A relative maximum or minimum is the maximum or minimum point in a "neighborhood."




At a critical point, the derivative of the function is either equal to zero or is undefined at point c. Maximum and Minimum points will only occur at a critical point.
At the relative maximum or minimum, the derivative is equal to ZERO or when the derivative is UNDEFINED.


If a relative maximum or minimum does exist, the one of the above not necessarily both MUST be true.



The First Derivative Test



The first derivative test is used to see if a point f(c) is a relative maximum or minimum.


If f'(x)>0 for xc then f(c) is a relative maximum.


If f'(x)<0>0 for x>c then f(c) is a relative minimum.





For the first derivative test, it is always helpful to make a sign chart and find where the graph is increasing and decreasing around the point c. This chart allows you to see if the point is a relative maximum or minimum.



Several Questions can be phrased different ways about the first derivative:


At what x-values do relative maximum and minimum points occur?


For the above, you simply have to list the values of x.




What are the values at the relative maximum and minimum points?


For the above, the question is asking for the value, which refers to the value of y at x. In order to find the value, simply plug in the x-values into your original equation (not the derivative equation) and find the y-value. Only give the y-value.... not the x and y value!




What are the coordinates at the relative maximum and minimum points?


For the above, you are looking for the points, so you plug the x-values back into your original equation, and pair that answer with the x-value to make a point. This point should reflect where the maximum or minimum point occurs.




When is the graph increasing?


For the above, you should make a sign chart! Where the sign chart shows positive numbers, the graph is increasing. Where the sign chart shows negative numbers that decrease, the graph is decreasing.



SOME EXCEPTIONS DO OCCUR:


The horizontal tangent line does fit the rule of having the derivative equal zero or be undefined, but there is no maximum or minimum point.



The derivative does not exist, but a relative minimum does exist. It is not differentiable at x=0, but the relative minimum does occur there.




Example: What are x-values for the relative maximum and minimum points in the following function:



Take the derivative to find that the derivative equation is:



You can factor out the 2 to get 2(x+7)(x-5). Set that equal to zero and find that x=-7 and x=5. Make a sign chart to find that the graph is increasing from negative infinite to -7 and decreases from -7 to 5. So we can establish that the point at x = -7 is a maximum. And by making a sign chart we find that that the graph is decreasing from -7 to 5 and that it increases from 5 to infinite. So we can establish that x = 5 is a minimum.




X= -7 is a maximum and x=5 is a minimum.




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Next for the Blog is Kjirsten for Chapter 4.2 Applications of the Second Derivative on February, Wednesday 7 2007.



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