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Calculus: The First Derivative Test

by Thinkwell

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About this Lesson

  • Type: Video Tutorial
  • Length: 2:46
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 29 MB
  • Posted: 11/18/2008

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus Review (48 lessons, $95.04)
Calculus: Final Exam Test Prep and Review (45 lessons, $64.35)
Calculus: Curve Sketching (20 lessons, $25.74)
Calculus: Critical Points (4 lessons, $8.91)

To find the maxima and minima of a graph, you first have to find all of the critical points. Thus, you will always find the max or min at one of these points where slope is 0 or is undefined (though the function is defined). The first derivative test is a way to determine if a critical point is a max, a min, or neither. To do the test, you look at how the derivative is changing around the point. If slope is positive before and negative after, it is a max. If it is negative before and positive after, it is a min. Note that these maximums and minimums are relative maximums and relative minimums. If the derivative of the function does not change sign around the critical point, it is neither a maximum nor a minimum.

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/calculus. The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hôpital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

Thinkwell
Thinkwell
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Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.

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Curve Sketching
Critical Points
The First Derivative Test Page [1 of 1]
So, I’d like to reflect just a second to take a look back at what we just figured out, because it was really neat and not that difficult. First of all, if you want to find maxima and minima for graphs, what do you do? Well, what you do is you first find all the critical points. Because it turns out that’s the pool of candidates where you will have either max or mins. And that’s the only place you have to look for max or mins are critical points. What are the critical points? Places where the derivative equals zero or places where the derivative is undefined, but the function is defined there.
Okay and we looked at a variety of examples, and then, using that little sign chart I showed you, you are able to figure out if the function was increasing or decreasing. And I want to take a look at one of the examples we looked at, in fact, the second example—example b. Here was the slide that we had. And we figured out, by using this little sign chart, that, in fact, the function was increasing here, decreasing here, and increasing here.
Well, that gave us for free, on this little sign chart picture, the fact that this point has to be a max, and this point has to be a min. And we already concluded that in those examples. That’s pretty easy. Turns out, this has a name. This is called the first derivative test.
The first derivative test is just a test that we can use on critical points to determine if those critical points are max, mins or neither. And the test is quite simple. What you do is, you find the critical point, and then you look a little bit to the right and a little bit to the left, and you see what the derivative is doing in terms of its sign.
If you have a positive first and then a negative afterwards, then you know, by the first derivative test, you must have a max. If you have a negative before and a positive after, then you must be decreasing then increasing, so you must have a min. If you have a positive before and then a positive after, that means you’re increasing and then increasing more, so you have neither a max nor a min. It’s not a peak. It’s not a valley. And similarly, if you’re decreasing and then decreasing, it’s neither a max nor a min.
And that’s all the first derivative test says. It’s a fancy name for this sign chart that we just saw. So, to recap, if you see a change in sign in the derivative as you cross the critical point, it will either be a max or a min, depending if you go from positive to negative or negative to positive. But if you go positive, positive, that’s not a max and it’s not a min. If you go negative, and then afterwards, negative, it’s not a max—not a min.
We call it the first derivative test. I think about it as just this little sign chart, but now you know the fancy name. Next up, we’ll take up the issue of very, very subtle curvature. See you in a bit.

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