chase the dream not the competition

Over 1000+ articles, updated everyday...for Free SMS Alerts click here , Engineering Q&A forum here

Invite Friends
Search:     Advanced Search
Browse by category:
 

Basic Definitions of Asymptotes, Concavity, Critical Points Etc

Vist the new KnowledgeBin forum to ask all your questions!

KnowledgBin.org SMS Registration

Click Here to Register Online

Creative Commons License
KnowledgeBin.org is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.5 India License.

Asymptotes
Definition of a horizontal asymptote: The line y = y0 is a "horizontal asymptote" of f(x) if and only if f(x) approaches y0 as x approaches + or - inf.
Definition of a vertical asymptote: The line x = x0 is a "vertical asymptote" of f(x) if and only if f(x) approaches + or - inf as x approaches x0 from the left or from the right.
Definition of a slant asymptote: the line y = ax + b is a "slant asymptote" of f(x) if and only if lim (x-->+/-inf) f(x) = ax + b.
Concavity
Definition of a concave up curve: f(x) is "concave up" at x0 if and only if f '(x) is increasing at x0
Definition of a concave down curve: f(x) is "concave down" at x0 if and only if f '(x) is decreasing at x0
The second derivative test: If f ''(x) exists at x0 and is positive, then f ''(x) is concave up at x0. If f ''(x0) exists and is negative, then f(x) is concave down at x0. If f ''(x) does not exist or is zero, then the test fails.
Critical Points
Definition of a critical point: a critical point on f(x) occurs at x0 if and only if either f '(x0) is zero or the derivative doesn't exist.
Extrema (Maxima and Minima)
Local (Relative) Extrema
Definition of a local maxima: A function f(x) has a local maximum at x0 if and only if there exists some interval I containing x0 such that f(x0) >= f(x) for all x in I.
Definition of a local minima: A function f(x) has a local minimum at x0 if and only if there exists some interval I containing x0 such that f(x0) <= f(x) for all x in I.
Occurrence of local extrema: All local extrema occur at critical points, but not all critical points occur at local extrema.
The first derivative test for local extrema: If f(x) is increasing (f '(x) > 0) for all x in some interval (a, x0] and f(x) is decreasing (f '(x) < 0) for all x in some interval [x0, b), then f(x) has a local maximum at x0. If f(x) is decreasing (f '(x) < 0) for all x in some interval (a, x0] and f(x) is increasing (f '(x) > 0) for all x in some interval [x0, b), then f(x) has a local minimum at x0.
The second derivative test for local extrema: If f '(x0) = 0 and f ''(x0) > 0, then f(x) has a local minimum at x0. If f '(x0) = 0 and f ''(x0) < 0, then f(x) has a local maximum at x0.
Absolute Extrema
Definition of absolute maxima: y0 is the "absolute maximum" of f(x) on I if and only if y0 >= f(x) for all x on I.
Definition of absolute minima: y0 is the "absolute minimum" of f(x) on I if and only if y0 <= f(x) for all x on I.
The extreme value theorem: If f(x) is continuous in a closed interval I, then f(x) has at least one absolute maximum and one absolute minimum in I.
Occurrence of absolute maxima: If f(x) is continuous in a closed interval I, then the absolute maximum of f(x) in I is the maximum value of f(x) on all local maxima and endpoints on I.
Occurrence of absolute minima: If f(x) is continuous in a closed interval I, then the absolute minimum of f(x) in I is the minimum value of f(x) on all local minima and endpoints on I.
Alternate method of finding extrema: If f(x) is continuous in a closed interval I, then the absolute extrema of f(x) in I occur at the critical points and/or at the endpoints of I.
(This is a less specific form of the above.)
Increasing/Decreasing Functions
Definition of an increasing function: A function f(x) is "increasing" at a point x0 if and only if there exists some interval I containing x0 such that f(x0) > f(x) for all x in I to the left of x0 and f(x0) < f(x) for all x in I to the right of x0.
Definition of a decreasing function: A function f(x) is "decreasing" at a point x0 if and only if there exists some interval I containing x0 such that f(x0) < f(x) for all x in I to the left of x0 and f(x0) > f(x) for all x in I to the right of x0.
The first derivative test: If f '(x0) exists and is positive, then f '(x) is increasing at x0. If f '(x) exists and is negative, then f(x) is decreasing at x0. If f '(x0) does not exist or is zero, then the test tells fails.
Inflection Points
Definition of an inflection point: An inflection point occurs on f(x) at x0 if and only if f(x) has a tangent line at x0 and there exists and interval I containing x0 such that f(x) is concave up on one side of x0 and concave down on the other side
 

SMS Alerts

SocialTwist Tell-a-Friend

Admission Updates

KnowledgeBin Forum Invite Friends

KnowledgeBin Forum Discuss In Forum

Views: 3769
Votes: 0

Others In This Category
document Maths Solving made easy!!!!
document Mathematics - Algebra Formula ShrotCut Booklet - 1
document Mathematics - Calculus Formula ShrotCut Booklet - 1
document Mathematics - Geometry Formula ShrotCut Booklet - 1
document Mathematics - Statistics Formula ShrotCut Booklet - 1
document Mathematics - Trigonometry Formula ShrotCut Booklet - 1
document Vedic Mathematics
document How To Find The Square Of Any Number Quickly!! [A must have tool for IIT-JEE and AIEEE]
document Maxima, Minima and Inflection Points
document IIT-JEE Mathematics Cheat Sheets and Tables - Must Have!
document IIT - JEE Mathematics: Permutations and Combination Basic Fundamentals
document Quick revision for IIT-JEE general formulas, Straight lines and Parabola
document Matrices and Determinants for IIT-JEE and AIEEE - Quick Revision
About Us | Contact Us | Feedback | Copyright © 2008 KnowledgeBin.org™ All rights reserved



RSS