Calculus Interlude - A Final Review

Weierstrass (Extreme Value) Theorem

A function continuous over an interval admits extrema and must take on all values between the endpoints over that interval. The same applies to the image.

Lagrange (Mean Value) Theorem

Rolle's Theorem

A function continuous over , and differentiable at at least that has the same value at two different points admits at least one point c such that its derivative at that point is zero.

"For them to be the same, what goes up must come down eventually, and what goes down must come up eventually."

Bolzano's Theorem of Existence of Zeroes

A function continuous over an interval over which it changes sign, must cross zero. In other words, if a continuous function over an interval switches signs it must have some point where it's equal to zero.

Summation Notation

Capital sigma symbol, with starting and ending conditions on bottom and top, and the dictating term.

Taylor Expansions

Tool used to approximate function around a point a based on its derivatives.

Maclaurin Expansions

Taylor expansions around the point a = 0.

Taylor and Maclaurin Polynomials in Function Analysis

Approximation error

Peano's Remainder and the Taylor Expansion

Maclaurin Expansions and function symmetry

If a function is even, its odd-order derivatives are odd, and its even-order derivatives are even.
Similarly, if a function is odd, its odd-order derivatives are even, and its even-order derivatives are odd.

Local Behavior of Functions

Given an n-th order Taylor Expansion of an n-times differentiable function, if the function at that point is equal to zero, and all its derivatives are zero up until a derivative of order , then

General theorem for analysis of functions based on their expansions

If you have zeroes for all parts of the expansion until a non-zero m-th order derivative (so, if everything is zero until some derivative of order m), for m at least 2:

  • If this m is even, this point a is a local extremum point. Maximum if , and minimum if

  • If this m is odd, the point a is an inflection point. Descending inflection point if , and ascending if

Welcome to the grownups table. Good luck. You'll need it.

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