Home The Six Pillars of CalculusThe Pillars: A Road MapA picture is worth 1000 words Trigonometry ReviewThe basic trig functionsBasic trig identities The unit circle Addition of angles, double and half angle formulas The law of sines and the law of cosines Graphs of Trig Functions Exponential FunctionsExponentials with positive integer exponentsFractional and negative powers The function $f(x)=a^x$ and its graph Exponential growth and decay Logarithms and Inverse functionsInverse FunctionsHow to find a formula for an inverse function Logarithms as Inverse Exponentials Inverse Trig Functions Intro to LimitsOverviewDefinition Onesided Limits When limits don't exist Infinite Limits Summary Limit Laws and ComputationsLimit LawsIntuitive idea of why these laws work Two limit theorems How to algebraically manipulate a 0/0? Indeterminate forms involving fractions Limits with Absolute Values Limits involving indeterminate forms with square roots Limits of Piecewise Functions The Squeeze Theorem Continuity and the Intermediate Value TheoremDefinition of continuityContinuity and piecewise functions Continuity properties Types of discontinuities The Intermediate Value Theorem Summary of using continuity to evaluate limits Limits at InfinityLimits at infinity and horizontal asymptotesLimits at infinity of rational functions Which functions grow the fastest? Vertical asymptotes (Redux) Summary and selected graphs Rates of ChangeAverage velocityInstantaneous velocity Computing an instantaneous rate of change of any function The equation of a tangent line The Derivative of a Function at a Point The Derivative FunctionThe derivative functionSketching the graph of $f'$ Differentiability Notation and higherorder derivatives Basic Differentiation RulesThe Power Rule and other basic rulesThe derivative of $e^x$ Product and Quotient RulesThe Product RuleThe Quotient Rule Derivatives of Trig FunctionsNecessary LimitsDerivatives of Sine and Cosine Derivatives of Tangent, Cotangent, Secant, and Cosecant Summary The Chain RuleTwo Forms of the Chain RuleVersion 1 Version 2 Why does it work? A hybrid chain rule Implicit DifferentiationIntroductionExamples Derivatives of Inverse Trigs via Implicit Differentiation A Summary Derivatives of LogsFormulas and ExamplesLogarithmic Differentiation Derivatives in ScienceIn PhysicsIn Economics In Biology Related RatesOverviewHow to tackle the problems Example (ladder) Example (shadow) Linear Approximation and DifferentialsOverviewExamples An example with negative $dx$ Differentiation ReviewHow to take derivativesBasic Building Blocks Advanced Building Blocks Product and Quotient Rules The Chain Rule Combining Rules Implicit Differentiation Logarithmic Differentiation Conclusions and Tidbits Absolute and Local ExtremaDefinitionsThe Extreme Value Theorem Critical Numbers Steps to Find Absolute Extrema The Mean Value and other TheoremsRolle's TheoremsThe Mean Value Theorem Finding $c$ $f$ vs. $f'$Increasing/Decreasing Test and Critical NumbersProcess for finding intervals of increase/decrease The First Derivative Test Concavity Concavity, Points of Inflection, and the Second Derivative Test The Second Derivative Test Visual Wrapup Indeterminate Forms and L'Hospital's RuleWhat does $\frac{0}{0}$ equal?Examples Indeterminate Differences Indeterminate Powers Three Versions of L'Hospital's Rule Proofs OptimizationStrategiesAnother Example Newton's MethodThe Idea of Newton's MethodAn Example Solving Transcendental Equations When NM doesn't work AntiderivativesAntiderivativesCommon antiderivatives Initial value problems Antiderivatives are not Integrals The Area under a curveThe Area Problem and ExamplesRiemann Sum Notation Summary Definite IntegralsDefinition of the IntegralProperties of Definite Integrals What is integration good for? More Applications of Integrals The Fundamental Theorem of CalculusThree Different ConceptsThe Fundamental Theorem of Calculus (Part 2) The Fundamental Theorem of Calculus (Part 1) More FTC 1 The Indefinite Integral and the Net ChangeIndefinite Integrals and AntiderivativesA Table of Common Antiderivatives The Net Change Theorem The NCT and Public Policy SubstitutionSubstitution for Indefinite IntegralsExamples to Try Revised Table of Integrals Substitution for Definite Integrals Examples Area Between CurvesComputation Using IntegrationTo Compute a Bulk Quantity The Area Between Two Curves Horizontal Slicing Summary VolumesSlicing and Dicing SolidsSolids of Revolution 1: Disks Solids of Revolution 2: Washers More Practice 
DefinitionsIntuitively, the absolute maximum value
is the largest of the possible values of $f(x)$, and similarly for
minimum. Notice that a function may reach its maximum (or
minimum) at more than one $x$value.
A value of $f(x)$ may not be the largest (or smallest) of
all, but it might be the largest (or smallest) compared to nearby
values. We call these local
extrema, or local extreme
values.
Notice that when a function is defined on a closed interval, an absolute extreme value may occur at the endpoint of that interval of domain, since the endpoint of the interval may yield the largest value of $f$ on that closed interval. This situation is different with local extrema. When we say $x$ is near $c$, we mean on an open interval containing $c$, so on either side of $c$. When $f$ is defined on a closed interval, there is no open interval containing an endpoint of the closed interval on which $f$ is defined. Hence, a local extreme value cannot occur at the endpoint of an interval of domain. This is a definition, and it could be defined differently. This video will help clarify these concepts. Notice that the absolute and local maxima
and minima are $y$values. Graphically, this means
that the max/min value is the maximum/minimum height of the graph at some
$x=c$. Then $x=c$ is where
the max/min occurs. Suppose that, along the way, you go up, then at mile marker 324 (where your altitude is 6,200 feet) you start going down. We would say that you reached a local maximum altitude of 6,200 feet at $x=324$. You keep driving, some up, some down (attaining local maximum and maximum altitudes) until you reach Vail Pass (the Continental Divide) at mile marker 278, where the altitude is 10,662 feet. We say you reached an absolute maximum altitude of 10,662 feet at $x=278$. If you are traveling from Denver at 5280 feet high, to Grand Junction at 4560 feet high, what is your guess about the absolute minimum altitude you attain, and where it occurs?
