Home The Pillars: A Road Map A picture is worth 1000 words The basic trig functions Basic 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 Exponentials with positive integer exponents Fractional and negative powers The function $f(x)=a^x$ and its graph Exponential growth and decay Inverse Functions How to find a formula for an inverse function Logarithms as Inverse Exponentials Inverse Trig Functions Close is good enough Definition One-sided Limits How can a limit fail to exist? Infinite Limits and Vertical Asymptotes Summary A summary of Limit Laws Why do these laws work? Two limit theorems How to algebraically manipulate a 0/0? Limits with fractions Limits with Absolute Values Limits involving Rationalization Limits of Piece-wise Functions The Squeeze Theorem Definition of continuity Continuity and piece-wise functions Continuity properties Types of discontinuities The Intermediate Value Theorem Examples of continuous functions Limits at infinity and horizontal asymptotes Limits at infinity of rational functions Which functions grow the fastest? Vertical asymptotes (Redux) Toolbox of graphs Tracking change Average and instantaneous velocity Instantaneous rate of change of any function Finding tangent line equations Definition of derivative The derivative function Sketching the graph of $f’$ Differentiability Notation and higher-order derivatives The Power Rule and other basic rules The derivative of $e^x$ The Product Rule The Quotient Rule Two important Limits Sine and Cosine Tangent, Cotangent, Secant, and Cosecant Summary Two forms of the chain rule Version 1 Version 2 Why does it work? A hybrid chain rule Introduction and Examples Derivatives of Inverse Trigs via Implicit Differentiation A Summary Formulas and Examples Logarithmic Differentiation In Physics In Economics In Biology Overview How to tackle the problems Example (ladder) Example (shadow) Overview Examples An example with negative $dx$ Basic Building Blocks Advanced Building Blocks Product and Quotient Rules The Chain Rule Combining Rules Implicit Differentiation Logarithmic Differentiation Conclusions and Tidbits Definitions The Extreme Value Theorem Fermat’s Theorem How-to Rolle’s Theorems The Mean Value Theorem Finding $c$ Increasing/Decreasing Test and Critical Numbers How-to The First Derivative Test Concavity, Points of Inflection, and the Second Derivative Test What does $\frac{0}{0}$ equal? Indeterminate Differences Indeterminate Powers Three Versions of L’Hospital’s Rule Proofs Strategies Another Example The Idea of Newton’s Method An Example Solving Transcendental Equations When NM doesn’t work Anti-derivatives and Physics Some formulas Anti-derivatives are not Integrals The Area Problem and Examples Riemann Sums Notation Summary Definition Properties What is integration good for? More Examples Three Different Quantities The Whole as Sum of Partial Changes The Indefinite Integral as Antiderivative The FTC and the Chain Rule |
The Extreme Value Theorem tells us that the minimum and maximum of a function have to be somewhere. But where should we look? The answer lies in the third of the Six Pillars of Calculus: Places where the derivative either This is the idea behind one of Fermat’s theorems: Equivalently, if $f'(c)$ exists and is not zero, then $f(c)$ is neither a maximum nor a minimum. |

# Absolute and Local Extrema

Fermat’s little theorem proof | Number Theory

Fermat’s little theorem proof | Number Theory