Thomas 4 - Apps of Derivatives

"This chapter studies some of the important applications of derivatives. We learn how derivatives are used to find extreme values of functions, to determine and analyze the shapes of graphs, to calculate limits of fractions whose numerators and denominators both approach zero or infinity, and to find numerically where a function equals zero. We also consider the process of recovering a function from its derivative. The key to many of these accomplishments is the Mean Value Theorem, a theorem whose corollaries provide the gateway to integral calculus."
(George Thomas)


In calculus, the mean value theorem states, roughly, that given an arc of a smooth differentiable (continuous) curve, there is at least one point on that arc at which the derivative (slope) of the curve is equal (parallel) to the "average" derivative of the arc. Briefly, a suitable infinitesimal element of the arc is parallel to the secant chord connecting the endpoints of the arc. The theorem is used to prove theorems that make global conclusions about a function on an interval starting from local hypotheses about derivatives at points of the interval.






* Extreme values of functions Absolute Maximum, Absolute Minimum

* Determine & analyze the shapes of graphs

* Calculate limits of fractions whose numerators & denominators both approach zero or infinity

* Find where a function equals zero

* Recovering a function from its derivative / Antiderivative

* Mean Value Theorem


Definitions & Theorems; Corallaries; Laws


D 4.1 - Absolute Maximum, Absolute Minimum

T 4.1 - Extreme value theorem

D 4.2 - Local Maximum, Local Minimum

T 4.2 - The First Derivative Theorem for Local Extreme Values

D 4.3 - Critical Point

T 4.3 - Rolle's theorem

T 4.4 - Mean Value theorem

C 4.1 - Functions with Zero Derivatives are Constant

C 4.2 - Functions with the same derivative differ by a constant

- Laws of Exponents for e^x

D 4.4 - Increasing, Decreasing Function

C 4.3 - First derivative test for monotonic functions

- First derivative test for local extrema

D 4.5 - Concave up, Concave down

- Second derivative test for concavity


Comments

Post a Comment

Popular posts from this blog

Squeeze Theorem - "2 policeman & a drunk"

In calculus, the squeeze theorem (known also as the pinching theorem, the sandwich theorem, the sandwich rule and sometimes the squeeze lemma) is a theorem regarding the limit of a function. The squeeze theorem is a technical result that is very important in proofs in calculus and mathematical analysis. It is typically used to confirm the limit of a function via comparison with two other functions whose limits are known or easily computed. It was first used geometrically by the mathematicians Archimedes and Eudoxus in an effort to compute π, and was formulated in modern terms by Gauss. In Italy, Russia and France, the squeeze theorem is also known as the two carabinieri theorem, two militsioner theorem, two gendarmes theorem, or two policemen and a drunk theorem. The story is that if two policemen are escorting a drunk prisoner between them, and both officers go to a cell, then (regardless of the path taken, and the fact that the prisoner may be wobbling about between the policemen) t...
Read more

Surreal Numbers

Image
In mathematics , the surreal number system is a totally ordered class containing the real numbers as well as infinite and infinitesimal numbers , respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field . [a] If formulated in Von Neumann–Bernays–Gödel set theory , the surreal numbers are the largest possible ordered field; all other ordered fields, such as the rationals, the reals, the rational functions , the Levi-Civita field , the superreal numbers , and the hyperreal numbers , can be realized as subfields of the surreals. [1] It has also been shown (in Von Neumann–Bernays–Gödel set theory ) that the maximal class hyperreal field is isomorphic to the maximal class surreal field; in theories without the axiom of global choice , this need not be the case, and in such the...
Read more