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Topic: First derivative test

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	Derivative - Wikipedia, the free encyclopedia
		In this case, the Second Derivative Test can still be used to characterize critical points, by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point.
		The common thread is that the derivative at a point serves as a linear approximation of the function at that point.
		Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles.
	en.wikipedia.org /wiki/Derivative (2146 words)

	First derivative test - Wikipedia, the free encyclopedia
		In calculus, a branch of mathematics, the first derivative test determines whether a given critical point of a function is a maximum, a minimum, or neither.
		If f is differentiable in a neighbourhood of x, we can rephrase the conditions of being increasing or decreasing in terms of the derivative of f.
		When the derivative of f is positive, then f is increasing, and when the derivative of f is negative, then f is decreasing.
	en.wikipedia.org /wiki/First_derivative_test (368 words)

	Concavity and the Second Derivative Test - HMC Calculus Tutorial
		The Second Derivative Test provides a means of classifying relative extreme values by using the sign of the second derivative at the critical number.
		To appreciate this test, it is first necessary to understand the concept of concavity.
		The Second Derivative Test relates the concepts of critical points, extreme values, and concavity to give a very useful tool for determining whether a critical point on the graph of a function is a relative minimum or maximum.
	www.math.hmc.edu /calculus/tutorials/secondderiv (849 words)

	MAT 270 Exam #3 Review (Site not responding. Last check: 2007-10-13)
		Critical numbers also correspond to cases when the first derivative fails to exist (such as at a sharp corner or a vertical asymptote).
		If the derivative is positive, then the graph of f is increasing on that interval; if the derivative is negative, then the graph of f is decreasing on that interval.
		Be able to use the first and second derivatives of a function to obtain precise information about the graph of the function and be able to fill out the derivative table for the function you are trying to analyze and graph.
	math.la.asu.edu /~kolossa/mat270/review3.html (702 words)

	Maple Handout on the First Derivative Test (Site not responding. Last check: 2007-10-13)
		Set the derivative equal to zero and solve for x to find the local maxima and local minima of the original function.
		Then test the x values from step 2 to see where the original function is increasing or decreasing.
		Now use the First Derivative Test by plugging in x values to the left and right of the x values from step 2.
	ellerbruch.nmu.edu /Maple/MapleHandouts/FirstDerivTestHandout.html (208 words)

	The First Derivative Test (Site not responding. Last check: 2007-10-13)
		Similarly, for a relative minimum, on the right the function is decreasing and on the left the function is increasing.
		First set the first derivative equal to zero to locate the critical points.
		Notice also that there is a critical point at x = 1 since the first derivative is undefined there (notice the negative exponent -3/5).
	www.ltcconline.net /greenl/courses/115/applications/frsttst.htm (307 words)

	Calculus I (Math 2413) - Applications of Derivatives - The Shape of a Graph, Part II (Site not responding. Last check: 2007-10-13)
		In the previous section we saw how we could use the first derivative of a function to get some information about the graph of a function. In this section we are going to look at the information that the second derivative of a function can give us a about the graph of a function.
		Note that from the first derivative test we can also say that x=-1 is a relative maximum and that x=1 is a relative minimum. Also x=0 is neither a relative minimum or maximum.
		The first thing that we need to do is identify the possible inflection points. These will be where the second derivative is zero or doesn’t exist. The second derivative in this case is a polynomial and so will exist everywhere. It will be zero at the following points.
	tutorial.math.lamar.edu /AllBrowsers/2413/ShapeofGraphPtII.asp (1148 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		1 Apr. The focus of the test is sections 3.4, 3.5, 3.6, 3.7, 3.9 and 4.1.
		Use the first derivative test to identify a critical point of a function as a local maximum, a local minimum or neither.
		Use the second derivative test to identify a critical point of a function as a local maximum, a local minimum or neither.
	www.cbu.edu /~yanushka/k1/r.3 (571 words)

	Calculus I (Math 2413) - Applications of Derivatives - The Shape of a Graph, Part I
		In the previous section we saw how to use the derivative to determine the absolute minimum and maximum values of a function. However, there is a lot more information about a graph that can be determined from the first derivative of a function. We will start looking at that information in this section.
		Recall that we know that the derivative will be the same sign in each region. The only place that the derivative can change signs is at the critical points and we’ve marked the only critical points on the number line.
		Using the first derivative to give us information about a whether a function is increasing or decreasing is a very important application of derivatives and arises on a fairly regular basis in many areas.
	tutorial.math.lamar.edu /AllBrowsers/2413/ShapeofGraphPtI.asp (1436 words)

	Calculus World
		It is possible to find the behavior of a function using the derivative and second derivative.
		The first derivative test funds the intervals where the function is increasing or decreasing.
		Using the second derivative test, it is possible to find the concavity and minimums of a function
	home.earthlink.net /~sfmm84/calculus/derivativetests.html (404 words)

	June25.html
		is increasing," and the first derivative test for relative extrema.
		An immediate consequence of this is the famous first derivative test for local min/max.
		We could define "concave up" to mean that the second derivative is positive, but then the theorem that the graph is concave up if the second derivative is positive would be circular.
	www.uwec.edu /smithaj/Summer710/June25.html (439 words)

	unitles6 (Site not responding. Last check: 2007-10-13)
		Students must be able to identify maximum and minimum by using the first derivative test.
		Students will learn about the second derivative test and be able to perform it.
		Go through some of yesterdays homework problems using the second derivative test instead of the first derivative test at the board with the students.
	www.mste.uiuc.edu /courses/educ362sp04/folders/fischer/unitles6.htm (388 words)

	First Derivatives on Maple (Site not responding. Last check: 2007-10-13)
		The first derivative of a function can help us understand the whole graph by showing us the critical points of the function.
		We can find the local maxima and local minima by finding where the first derivative of a function equals zero and then use the First Derivative Test to determine whether these points are local maxima and/or local minima.
		Algebraically we could plug values into the derived function (Deriv) to see where the derivative is increasing or decreasing.
	ellerbruch.nmu.edu /Maple/FirstDeriv.html (237 words)

	Visual Calculus - Mean Value Theorem (Site not responding. Last check: 2007-10-13)
		We develop the First Derivative Test and look at some examples where the First Derivative Test is applied.
		to use the First Derivative Test to find intervals where a function is increasing or decreasing and to use this information to find local maxima and local minima.
		Let f be a differentiable function such that the derivative f ' is negative on the closed interval [a, b].
	archives.math.utk.edu /visual.calculus/3/mvt.3 (271 words)

	curvesketching.nb
		One needs to calculate the second derivative, f"(x), and solve for the values of x where either the numerator or denominator are 0.
		Since the curvature must be entirely in one direction or the other between adjacent inflection points, one can test the second derivative at any convenient point in that interval to identify whether the function is concave up or concave down there.
		This information can either be used to supplement the picture of the function one has formed using the First Derivative Test (using smooth curves rather than straight lines), or as the basis for the Second Derivative Test (using the critical values as the points of interest).
	www.people.vcu.edu /~ldwibber/mgmt212/curvesketching (817 words)

	[No title]
		It is clear from this definition (and Corollary 3.3) that to discern information about concavity we must take the second derivative and where the second derivative is positive is where the first derivative will be increasing, and hence where the function is concave up.
		Vice versa, when the second derivative is negative, the function is concave down.
		The discussion of how to graph a function based on its first and second derivatives is presented in a straightforward manner on p.213-216 with examples that are easy to follow.
	www.math.colostate.edu /~butler/m160/m160_lecture16.doc (401 words)

	Concavity and the Second Derivative Test: Sec. 3.4 (Site not responding. Last check: 2007-10-13)
		To find points of inflection, you set the second derivative of the particular graph equal to zero.
		Find the values of x for which the second derivative is equal to zero and where it is undefined.
		Let f be a function such that f' (c)=0 and the second derivative of f exists on an open interval containing c.
	www.kent.k12.wa.us /staff/DavidWright/calculus/book/34 (268 words)

	The First Derivative: Maxima and Minima - HMC Calculus Tutorial
		We cannot find regions on which f is increasing or decreasing, relative maxima or minima, or the absolute maximum or minimum value of f on [-2,3] by inspection.
		We can use the first derivative of f, however, to find all these things quickly and easily.
		By the First Derivative Test, f has a relative maximum at x = 0 and relative minima at x = -1 and x = 2.
	www.math.hmc.edu /calculus/tutorials/extrema (509 words)

	[No title]
		This information is precisely what we need for the first derivative test for local extreme values.
		Theorem 3.6: First Derivative Test (for Local Extreme Values at Interior Points) Suppose that c was a critical point of f(x).
		Theorem 3.7: First Derivative Test (for Local Extreme Values at Endpoints) At a left endpoint a: If f’ is positive (respectively, negative) for x > a, then f has a local maximum (respectively, local minimum) value at a.
	www.math.colostate.edu /~butler/m160/m160_lecture15.doc (203 words)

	THE FIRST DERIVATIVE TEST!! (Site not responding. Last check: 2007-10-13)
		In order to find out where the derivative is positive or negative, first find the points where the derivative is equal to zero or undefined.
		So, when the derivative of a function changes from positive to negative, the function itself changes from increasing to decreasing.
		On the other hand, when the derivative changes from negative to positive at a critical point C, that point will represent the relative minimum.
	www.kent.k12.wa.us /staff/DavidWright/calculus/book/33 (370 words)

	3.2.htm
		Relative extrema occur at the critical points for the first derivative.
		This does not mean that every critical point will be a relative maximum or a relative minimum.
		Using the Frist-Derivative Test we conclude that x = 1 is a relative minimum.
	www.howardcc.edu /math/MA145/3.2/3.2.htm (570 words)

	4 (Site not responding. Last check: 2007-10-13)
		The First Derivative Test Suppose that the function
		The Second Derivative Test: (It uses concavity to test critical numbers to see if they are the
		= 0, then the test fails and the First Derivative Test must be used.
	www.gpc.edu /~jcraig/calc1_ch4/4s3_deriv_and_curves.htm (261 words)

	Untitled (Site not responding. Last check: 2007-10-13)
		The first-derivative test for relative optimums: Given a function, y = f(x), the necessary condition for a relative max (min) is f(x
		The derivative of a derivative: f(x), f'(x), f''(x), f'''(x), f''''(x), etc.
		Second derivative test: A sufficient condition for a local max (min) is f''(x) < (>) 0.
	core.ecu.edu /econ/whiteheadj/5360/ch9.htm (181 words)

	Increasing and Decreasing Functions AND Extrema and the First derivative test
		Increasing and Decreasing Functions AND Extrema and the First derivative test
		then we can use the derivative of f to test if x=c is a maximum or a minimum.
		is undefined, then we can use the derivative of f to test if x=c is a maximum or a minimum.
	www.runet.edu /~wyang/121/maxmin/node1.html (290 words)

	Study Guide Exam #3
		Remember, the suggested exercises will help you prepare for the types of things you’ll be asked about—you should not assume that every problem on the test will be “just like” these problems here, although it’s quite possible that several of the questions will resemble problems listed below.
		be able to find critical points of functions, be able to determine intervals on which a function is inc/dec, and be able to apply the first derivative test to find local max/min values
		be able to apply the second derivative test to find local max/min values
	spider.georgetowncollege.edu /mpc/harris/mat109/studyguide3.htm (303 words)

	Finding Relative Extrema -- TC3 MATH201 (Brown) (Site not responding. Last check: 2007-10-13)
		First Derivative Test, shows how to find relative max or min of a continuous and differentiable function:
		If the function is continuous at a point but not differentiable at that single point, you can still apply the First Derivative Test.
		If f′(x)=f″(x)=0, you must use the First Derivative Test: look at sign changes in f′.
	www.acad.sunytccc.edu /instruct/sbrown/calc/ln033.htm (355 words)

	PinkMonkey.com Calculus Study Guide - Section 5.9 First Derivative Test For Local Extrema
		If the derivative of a function changes its sign while passing through a critical point along a given curve (i.e.
		(II) If the derivative changes its sign from negative (decreasing function) to positive (increasing function), the function has a Local minima at that critical point.
		Note : There is no guarantee that the derivative will change signs, therefore it is essential to test each Interval around a critical point.
	www.pinkmonkey.com /studyguides/subjects/calc/chap5/c0505901.asp (255 words)

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