Functional Notations
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This document provides an introduction to calculus and functions. It discusses that calculus originated in the 17th century through the work of Newton and Leibniz. A function is defined as a relationship between variables where each value of one variable corresponds to a unique value of the other. Functions can relate multiple variables as well. Examples of functions include the area of a circle as a function of its radius and the perimeter of a rectangle as a function of its length. The document provides exercises on evaluating functions and relating composite functions.
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The document explains Gauss's divergence theorem and provides an example of its application. The theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the vector field over the region enclosed by the surface. It proves the theorem by imagining the volume divided into parallelepipeds and taking the limit. As an example, it applies the theorem to calculate the outward flux of the vector field F=x^2x+y^2y+z^2z over a unit cube, showing the surface integral equals the volume integral of the divergence, in agreement with the theorem.
A study on connectivity in graph theory june 18 pdf
This document provides an introduction and two chapters on connectivity in graphs. The introduction discusses the history and applications of graph theory. Chapter 1 defines key concepts related to connectivity such as bridges, cut vertices, and k-connectivity. It presents theorems characterizing when an edge is a bridge and when a graph is a tree. Chapter 2 discusses applications of connectivity in graphs.
A Unifying theory for blockchain and AI
This document proposes a unifying theory connecting blockchain and artificial intelligence technologies. It introduces the Lam-Visser theory and how it fits within the Damhof Quadrants framework. The document provides definitions related to the main result, which states that there exists a minimal, ultra-connected, almost everywhere linear and generic solvable, semi-countable polytope if a certain condition is met. It then discusses applications of this theory to questions of associativity and the computation of analytically independent subalgebras.
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Functional Notations
- 1. LIMITS Chapter 1 CALCULUS 1 Samar College Galina V. Panela
- 2. Brief History on Calculus: • It is a branch of mathematics which started to develop in the 17th century. • Sir Isaac Newton and Wilhelm Leibniz created calculus independently of each other and about the same time. CALCULUS 1 Samar College Galina V. Panela
- 3. Brief History on Calculus: • The word calculus is derived from the Latin word for stone or pebble. • Calculus roughly means a “method of calculation” CALCULUS 1 Samar College Galina V. Panela
- 4. FUNCTIONAL NOTATION Lesson 1.1 CALCULUS 1 Samar College Galina V. Panela
- 5. Concept of Function: • A function is a certain law of correspondence, • It is generally associated with a formula. • Like the formula of the area of the circle 𝐴 = 𝜋𝑟2 . We say that A is a function of r and in symbol, we write 𝐴 = 𝑓(𝑟). CALCULUS 1 Samar College Galina V. Panela
- 6. Concept of Function: • In general, if there is a relation between two variables x and y such that each value of x corresponds to a value of y, then we say, y is a function of x. 𝑦 = 𝑓(𝑥) CALCULUS 1 Samar College Galina V. Panela
- 7. Concept of Function: This concept may be extended to relations between more than two variables such that if 𝑧 = 𝑓 𝑥, 𝑦 then the value of z is determined when x and y are given. CALCULUS 1 Samar College Galina V. Panela
- 8. Example: Express the perimeter P of the rectangle as a function of the length x of one side if the area of the rectangle is 10 square meters. CALCULUS 1 Samar College Galina V. Panela
- 9. Exercises: Evaluate the function, 𝑓 𝑥 = 2𝑥2 − 4𝑥 + 2, if • 𝑓 −10 • 𝑓 𝑦2 + 8 • 𝑓 𝑥 + ∆𝑥 • 𝑓 𝑥 + 1 − 𝑓(𝑥 − 1) CALCULUS 1 Samar College Galina V. Panela
- 10. Exercises: Express the area of an equilateral triangle as a function of its side x. If 𝑓 𝑥 = 𝑥2 + 1, find 𝑓 𝑥+ℎ −𝑓(𝑥) ℎ , where h ≠ 0. If f(𝑥) 4 𝑥+5 and 𝑔 𝑥 = 𝑥2 − 3, find f(g(x)) and g(f(x)). CALCULUS 1 Samar College Galina V. Panela