College Algebra 5.2
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The document discusses logarithms and log functions. It defines what a logarithm is using the example "Log 10 100 = 2". It explains how to convert from logarithmic form to exponential form. It discusses the common log and natural log, and properties of logarithmic functions. It describes the graph of a basic log function, including that it has an asymptote at the y-axis and an x-intercept of (1,0). It asks the reader to graph and find features of shifted log functions, like asymptotes, domains and x-intercepts.
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This document discusses simplifying radical expressions using the product, quotient, and power rules for radicals. It also covers adding, subtracting, multiplying, and dividing radicals. Rationalizing denominators is explained as well as solving radical equations. Key steps include isolating the radical term, squaring both sides to remove the radical, and checking solutions in the original equation.
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Simplifying radical expressions, rational exponents, radical equations
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Logarithmic functions (2)
Logarithmic functions relate the exponent of a number to the logarithm of its base. They are inverse functions of exponentials, where the domain and range are flipped. Key properties include:
1) Logarithmic graphs have a domain of positive real numbers and range of all real numbers, while being always increasing with a vertical asymptote at y=0.
2) Logarithms can have various bases, with common bases being base 10 (common log), base e (natural log), and base 2 or 10.
3) Logarithmic equations can be solved by converting them to exponential form and then solving the exponential equation.
Logarithmic Functions
The document defines logarithmic functions and provides examples of converting between logarithmic and exponential forms. It discusses the properties of logarithmic equality and solving logarithmic equations by rewriting them in exponential form. Key points include: logarithmic and exponential forms are inverses; the exponent becomes the logarithm; logarithmic functions have a domain of positive real numbers; and solving logarithms involves setting arguments equal when bases are the same.
Real World Haskell: Lecture 6
This document summarizes a lecture about models of evaluation in Haskell. It discusses lazy evaluation and call-by-need evaluation in Haskell. It provides examples showing that expressions are not evaluated unless their values are needed. Common functions like map and fold can be written in terms of each other due to their shared structural transformation of lists. Both left and right folds are discussed, with right folds constructing nested applications and left folds avoiding this issue.
Common natural logarithms
The document discusses common logarithms and natural logarithms. It defines common logarithms as logarithms with base 10, which are often used in real-world problems. It shows how to use properties of logarithms to solve exponential equations and by graphing. The document then defines natural logarithms as logarithms with base e, which are the inverse of exponential functions with base e. It demonstrates using natural logarithms and exponential base e to solve equations and expressions.
Math130 ch09
This document provides an overview of exponential and logarithmic functions. It covers composite and inverse functions, exponential functions, logarithmic functions, properties of logarithms, common logarithms, exponential and logarithmic equations, and natural exponential and logarithmic functions. Example problems are provided to illustrate each concept.
8.4 logarithmic functions
Logarithms were invented to solve exponential equations like 2x = 6, where x is between 2 and 3. A logarithm with base b is defined as logby = x if and only if bx = y. Common properties of logarithms include logb1 = 0 and logbb = 1. Logarithmic functions have inverse relationships with exponential functions and can be graphed by shifting and reflecting the standard logarithmic curve.
Chapter 5 algebra
This document provides definitions and examples of exponential and logarithmic functions:
- It defines logarithms as the inverse of exponential functions, where evaluating bx is equivalent to taking the log base b of y equals x.
- It introduces the important number e, called the natural exponential, which arises from calculations of continuous compound interest as the number of compounding periods increases infinitely.
- It provides the continuous growth formula A=Pe^rt and examples of using it to model bacterial growth over time.
Programming in Scala - Lecture Two
This document provides an overview of a lecture on functional programming in Scala. It covers the following topics:
1. A recap of functional programming principles like functions as first-class values and no side effects.
2. An introduction to the Haskell programming language including its syntax for defining functions.
3. How functions are defined in Scala and how they are objects at runtime.
4. Examples of defining the factorial function recursively in Haskell and Scala, and making it tail recursive.
5. Concepts of first-class functions, currying, partial application, and an example of implementing looping in Scala using these techniques.
8.4 logarithms1
This document discusses evaluating and graphing logarithmic functions. It defines logarithms and explains how to evaluate logarithmic expressions without a calculator by rewriting them as exponential expressions. Examples are provided such as log 3 81 = 4 and log 2 (1/32) = -5. The definitions of common (base 10), natural (base e), and general logarithmic functions are given. Graphs of logarithmic functions are asymptotic to the horizontal axis and shift right or left depending on the base.
Log properties
Logs have many useful properties that allow them to be simplified. Logs with the same base can have their expressions combined by multiplying or dividing the expressions, depending on if they are added or subtracted. Additionally, if the expression inside a log is raised to a power, the exponent can be brought down as a coefficient to the outside of the log. However, exponents only apply to the expression inside the log, not the log as a whole.
Log properties
Logs have many useful properties that allow them to be simplified. Logs with the same base can have their expressions combined by multiplying or dividing the expressions, depending on if they are added or subtracted. Additionally, if the expression inside a log is raised to a power, the exponent can be brought down as a coefficient to the outside of the log. However, exponents only apply to the expression inside the log, not the log as a whole.
Exponents
The document discusses exponents and exponential functions. It begins by defining key terms like base, exponent, and exponential form. It then provides examples of how to write numbers in exponential form and evaluate expressions with positive, negative, and zero exponents. The document also discusses exponential growth, logarithmic functions, and their inverse relationships. Exponential functions have a constant rate of growth, while logarithmic functions are the inverse of exponential functions. Graphs of exponential functions increase rapidly, while logarithmic functions slowly increase.
Go Beyond Higher Order Functions: A Journey into Functional Programming
Here's an introduction to folks using FP in various programming languages (especially those with built-in support for first class functions).
Join Lex Shehan as he shares his journey into FP as a retrospective; it should be great introduction to folks to using FP in other languages (especially those with built-in support for first class functions).
Our journey will include:
* high order functions in javascript (lodash)
* functional programming in Ruby
* going deeper in FP with Scala
* learning Haskell to gain a broader understanding of FP
* the history of FP and what Category Theory has to do with FP
* challenges & opportunities of incorporating FP techniques in Go
02. haskell motivation
This document provides an overview of key concepts in Haskell including:
- Recursion is important in Haskell because computations are defined by declaring what something is, not how to compute it.
- Functions are defined through pattern matching and recursion rather than variables and loops.
- Types in Haskell include integers, floats, booleans, characters, strings, lists, and tuples. Typeclasses define properties like ordering.
- Functions can be made polymorphic through typeclasses to allow different data types rather than limiting them to specific types.
- Lazy evaluation only evaluates expressions as needed, avoiding unnecessary computations.
Clojure introduction by Jason Basanese
This document introduces several core Clojure concepts including iteration functions like reduce, map, and filter. It provides examples of using reduce to iterate and accumulate values, map to apply a function to each element in a collection, and filter to select elements. It also demonstrates state with atoms, destructuring for easier parameter passing, and formatting code with zprint. The document encourages asking questions and provides additional Clojure learning resources.
Lambda Calculus
Lambda calculus (also written as λ-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine and was first introduced by mathematician Alonzo Church in the 1930s as part of his research of the foundations of mathematics.
Lambda calculus consists of constructing lambda terms and performing reduction operations on them
Logarithm
Lots of students find logarithm complicated and tricky.For them this slide may prove useful.This is the most easy way to learn logarithm.Best of luck!! :)
Introduction to Functional Languages
The document provides information about functional programming languages and concepts including:
1) Haskell and ML are introduced as functional languages with features like strong typing, algebraic data types, and pattern matching.
2) Core functional programming concepts are explained like referential transparency, higher-order functions, and recursion instead of iteration.
3) Fold functions are summarized as a way to iterate over lists in functional languages in both a left and right oriented way.
A taste of Functional Programming
Scala, Haskell and LISP are examples of programming languages using the functional programming paradigm. Join us in this TechTalk to know why functional programming is so important, how to implement some of its core concepts in your existing programming languages, and how functional programming inspired Google's Map Reduce, Twitter's Algebird, and many other technologies.
By Mohammad Ghabboun - Senior Software Engineer, SOUQ.com
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Go Beyond Higher Order Functions: A Journey into Functional Programming
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This document contains 10 multiple choice questions about finding the equation of lines given characteristics like two points on the line, a point and slope, or a point and being parallel or perpendicular to another line. The questions ask the learner to determine the equation for lines matching each given description.
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This document contains homework problems involving algebraic expressions and functions. Problem 1 asks to evaluate expressions involving square roots. Problem 2 asks to identify the meaning of an equality involving a function. Problems 3-5 ask to evaluate and simplify expressions involving square roots and composite functions. Problem 6 asks to find two functions whose composition is a given function.
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This document contains 10 math and functions questions: questions 1-5 ask to find values of x for which certain equations are true, questions 6 asks to identify which sentence correctly describes functions, and questions 7-10 ask to find values of expressions given values of variables.
2.2 homework a
This document contains 10 multiple choice and short answer questions that assess understanding of functions and function notation. Questions 1 and 2 ask students to identify which charts and ordered pairs represent functions. Questions 3-7 require students to evaluate specific functions for given input values. Questions 8-10 deal with piecewise functions, asking students to evaluate expressions and find output values.
College algebra hw 1.4
This homework assignment contains 10 algebra problems to solve using different techniques: problems 1-4 involve factoring quadratic equations; problems 5-6 require extracting the square root of equations to find exact solutions; problems 7-8 involve finding the discriminant of equations to determine the number of real solutions; and problems 9-10 should be solved using the quadratic formula.
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This document contains 10 math homework problems from a College Algebra 1 class taught by Dr. Moseley. The problems include finding x-intercepts and y-intercepts of linear equations, writing expressions for distance and discounted prices, solving equations for unknown numbers, calculating original prices from sale prices, finding required test scores to get a grade average, calculating trip times with constant speed, and finding dimensions of a picture frame given its perimeter and a ratio of its width to height.
Math 111 College Algebra 1.2
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P.5 homework a
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P.3 homework
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P 2 homework b
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This document contains a math homework assignment with 38 problems involving exponents, radicals, and simplifying expressions. Students are asked to identify components of exponential and radical expressions, evaluate expressions, and simplify expressions using properties of exponents. They will rewrite expressions with positive exponents and find values of expressions for given variables.
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This document contains a list of 43 questions about concepts in college algebra including: 1) numbers with non-repeating decimal representations, 2) properties of real numbers and their representations on a number line, 3) classifying numbers as natural, whole, integer, rational, irrational or real, 4) evaluating algebraic expressions, and 5) finding distances between points on a number line. Students are instructed to submit their answers on Moodle by the specified due date.
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Magic Money
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We three sisters
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LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
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Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
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Chapter 4
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Chapter 5
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Chapter 6
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1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
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