Stewart6e4_3 - MATH 191 Sections 1 and 3 Calculus I Fall...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 4.3: Derivatives and the shape of things We’ve already seen ways in which we can tell stuff about f by examining its derivative f . For instance, if f ( x ) = 0 everywhere on an interval, then f is constant on that interval, as a consequence of the . More generally, if f ( x ) > 0 for all x on an interval, we expect that the function should be increasing, since the tangent lines “guide” it upwards; an analogous statement holds in case f ( x ) < 0 on an interval: The Increasing/Decreasing (I/D) Test. 1. If f ( x ) > 0 on an interval, then f is increasing on that interval. 2. If f ( x ) < 0 on an interval, then f is decreasing on that interval. Proof. This is easy! Let’s suppose that f ( x ) > 0 everywhere on the interval [ a, b ], and let’s take x 1 < x 2 inside of that interval. What can you say, by the MVT? Example. If f ( x ) = 3 x 4 - 4 x 3 - 12 x 2 + 5, where if f increasing? Where is it decreasing? Use the I/D test to find out!
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
The I/D Test gives us a way to check whether numbers give us local extrema: The First Derivative Test.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern