Properties of Functions
Odd and Even Functions
Periodic Functions
Monotonic Functions
Bounded Functions
Maxima and Minima of Functions
Inverse Function
Sequence and Series
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This document contains lecture notes on evaluating definite integrals. It introduces the definition of the definite integral as a limit of Riemann sums, and properties of integrals such as additivity and comparison properties. It also states the Second Fundamental Theorem of Calculus, which relates definite integrals to indefinite integrals via the derivative of the integrand function. Examples are provided to illustrate how to use these properties and theorems to evaluate definite integrals.
This document discusses Taylor series and their applications. It begins by defining Taylor series and providing common examples. It then presents the general form of a Taylor series and explains how it can be used to approximate a function around a point by knowing its value and derivatives at that point. The document also discusses the error in Taylor series approximations and provides an example of using a Taylor series to estimate e1 to within an error of 10-6. It directs readers to additional online resources for further information on Taylor series.
* Evaluate a polynomial using the Remainder Theorem.
* Use the Factor Theorem to solve a polynomial equation.
* Use the Rational Zero Theorem to find rational zeros.
* Find zeros of a polynomial function.
* Use the Linear Factorization Theorem to find polynomials with given zeros.
* Use Descartes’ Rule of Signs.
This document defines antiderivatives (also called indefinite integrals) and provides examples of finding antiderivatives of various functions. The key points are:
- An antiderivative of a function f(x) is any function F(x) whose derivative is f(x).
- The general form of an antiderivative is F(x) + c, where c is an arbitrary constant.
- Rules are provided for finding antiderivatives of basic functions using integration formulas and techniques like substitution.
- Applications of antiderivatives include graphing functions from their derivatives, and solving boundary value problems in fields like physics.
This document provides a table of 100 functions with their derivatives worked out. It begins with introductory rules for deriving various types of functions like constants, polynomials, exponentials, logarithmic, trigonometric, and combined functions. Then it lists 100 specific functions and their derivatives to serve as worked examples applying the basic rules.
Properties of Functions
Odd and Even Functions
Periodic Functions
Monotonic Functions
Bounded Functions
Maxima and Minima of Functions
Inverse Function
Sequence and Series
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This document contains lecture notes on evaluating definite integrals. It introduces the definition of the definite integral as a limit of Riemann sums, and properties of integrals such as additivity and comparison properties. It also states the Second Fundamental Theorem of Calculus, which relates definite integrals to indefinite integrals via the derivative of the integrand function. Examples are provided to illustrate how to use these properties and theorems to evaluate definite integrals.
This document discusses Taylor series and their applications. It begins by defining Taylor series and providing common examples. It then presents the general form of a Taylor series and explains how it can be used to approximate a function around a point by knowing its value and derivatives at that point. The document also discusses the error in Taylor series approximations and provides an example of using a Taylor series to estimate e1 to within an error of 10-6. It directs readers to additional online resources for further information on Taylor series.
* Evaluate a polynomial using the Remainder Theorem.
* Use the Factor Theorem to solve a polynomial equation.
* Use the Rational Zero Theorem to find rational zeros.
* Find zeros of a polynomial function.
* Use the Linear Factorization Theorem to find polynomials with given zeros.
* Use Descartes’ Rule of Signs.
This document defines antiderivatives (also called indefinite integrals) and provides examples of finding antiderivatives of various functions. The key points are:
- An antiderivative of a function f(x) is any function F(x) whose derivative is f(x).
- The general form of an antiderivative is F(x) + c, where c is an arbitrary constant.
- Rules are provided for finding antiderivatives of basic functions using integration formulas and techniques like substitution.
- Applications of antiderivatives include graphing functions from their derivatives, and solving boundary value problems in fields like physics.
This document provides a table of 100 functions with their derivatives worked out. It begins with introductory rules for deriving various types of functions like constants, polynomials, exponentials, logarithmic, trigonometric, and combined functions. Then it lists 100 specific functions and their derivatives to serve as worked examples applying the basic rules.
This document provides a table of 100 functions with their derivatives worked out. It begins with introductory rules for deriving various types of functions like constants, polynomials, exponentials, logarithmic, trigonometric, and combined functions. Then it lists 100 specific functions and their derivatives, ranging from simple polynomials and exponentials to more complex expressions combining multiple functions. The purpose is to offer examples of applying the fundamental derivative rules to evaluate the derivatives of diverse mathematical expressions.
Here are the steps to solve this problem:
(a) Let t = time and y = height. Then the differential equation is:
dy/dt = -32 ft/sec^2
Integrate both sides:
∫dy = ∫-32 dt
y = -32t + C
Initial conditions: at t = 0, y = 0
0 = -32(0) + C
C = 0
Therefore, the equation is: y = -32t
When y = 0 (the maximum height), t = 0.625 sec
(b) Put t = 0.625 sec into the equation:
y = -32(0.625) = -20 ft
This document discusses evaluating functions by substituting values for variables. It provides examples of evaluating various functions, including price functions. Students should be able to evaluate functions and solve problems involving function evaluation. Functions can be even, odd, or neither based on whether f(-x) = f(x) or f(-x) = -f(x). Examples are provided to identify functions as even, odd, or neither. Exercises include evaluating functions for given values and determining function parity.
Application of partial derivatives with two variablesSagar Patel
Application of Partial Derivatives with Two Variables
Maxima And Minima Values.
Maximum And Minimum Values.
Tangent and Normal.
Error And Approximation.
Image sciences, image processing, image restoration, photo manipulation. Image and videos representation. Digital versus analog imagery. Quantization and sampling. Sources and models of noises in digital CCD imagery: photon, thermal and readout noises. Sources and models of blurs. Convolutions and point spread functions. Overview of other standard models, problems and tasks: salt-and-pepper and impulse noises, half toning, inpainting, super-resolution, compressed sensing, high dynamic range imagery, demosaicing. Short introduction to other types of imagery: SAR, Sonar, ultrasound, CT and MRI. Linear and ill-posed restoration problems.
This document discusses the continuity of functions. It defines a continuous function as one where the limit of the function as x approaches c exists and is equal to the value of the function at c. It also describes three types of discontinuity: removable, essential, and infinite. Examples are provided to demonstrate determining if a function is continuous at a given point by checking if the three conditions for continuity are met.
1. Partial derivatives describe how a function changes with respect to one variable while holding other variables constant. The partial derivative of Z with respect to x is denoted as ∂Z/∂x or fχ.
2. Optimization problems in calculus involve finding the maximum or minimum values of functions, which can be used to determine the best way to do something.
3. A function has a global/absolute maximum at c if it is greater than or equal to the function values at all other points, and a global/absolute minimum if it is less than or equal to all other points.
* Recognize characteristics of parabolas.
* Understand how the graph of a parabola is related to its quadratic function.
* Determine a quadratic function’s minimum or maximum value.
* Solve problems involving a quadratic function’s minimum or maximum value.
This document provides an overview of key calculus concepts and formulas taught in a Calculus I course at Miami Dade College - Hialeah Campus. The topics covered include limits and derivatives, integration, optimization techniques, and applications of calculus to economics, business, physics, and other fields. The document is intended as a study guide for students in the Calculus I class taught by Professor Mohammad Shakil.
Its states Periodic function, Fourier series for disontinous function, Fourier series, Intervals, Odd and even functions, Half range fourier series etc. Mostly used as active learning assignment in Degree 3rd sem students.
This document discusses antiderivatives and indefinite integrals. It begins by introducing the concept of an antiderivative, which is a function whose derivative is a known function. It then defines the indefinite integral as representing the set of all antiderivatives. Several properties of antiderivatives and indefinite integrals are presented, including: the constant of integration; basic integration rules like power, exponential, and logarithmic rules; and notation used to represent indefinite integrals. Examples are provided to illustrate key concepts and properties.
The document discusses techniques for finding the slope of a tangent line to a function at a given point using derivatives. It provides examples of finding the slope for various functions, including polynomials, trigonometric functions, and implicitly defined functions, using the definition of derivative and rules like the product rule. Approximations of slope using finite differences are also covered. Guidelines for performing implicit differentiation are outlined.
Regret Minimization in Multi-objective Submodular Function MaximizationTasuku Soma
This document presents algorithms for minimizing regret ratio in multi-objective submodular function maximization. It introduces the concept of regret ratio for evaluating the quality of a solution set for multiple objectives. It then proposes two algorithms, Coordinate and Polytope, that provide upper bounds on regret ratio by leveraging approximation algorithms for single objective problems. Experimental results on a movie recommendation dataset show the proposed algorithms achieve significantly lower regret ratios than a random baseline.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
The document discusses the inverse tangent function, tan-1(x). It begins by reviewing one-to-one functions and their inverses. Examples are provided to illustrate one-to-one functions. The lesson aim is to teach students the inverse tangent function. The domain of tan-1(x) is all real numbers, and the range is (-π/2, π/2). The graph of tan-1(x) is the reflection of the tangent function graph over the line y=x. Students are asked to find the values of various inverse tangent functions. Homework problems are assigned involving evaluating inverse tangent functions without a calculator.
Lesson 4- Math 10 - W4Q1_General Form of a Polynomial Function.pptxErlenaMirador1
The document discusses polynomial functions. It defines polynomials as algebraic expressions consisting of terms in the form axn where a is a real number and n is a non-negative integer. The degree of a polynomial refers to the highest exponent of its terms. A polynomial function P(x) is written as P(x) = anxn + an-1xn-1 + ... + a2x2 + a1x + a0 where n is the degree. The document provides examples of finding the degree of polynomials and performing operations like addition and multiplication on polynomial functions. It also demonstrates using synthetic division to divide one polynomial by another.
The document discusses convex functions and related concepts. It defines convex functions and provides examples of convex and concave functions on R and Rn, including norms, logarithms, and powers. It describes properties that preserve convexity, such as positive weighted sums and composition with affine functions. The conjugate function and quasiconvex functions are also introduced. Key concepts are illustrated with examples throughout.
This document discusses key concepts related to derivatives and their applications:
- It defines increasing and decreasing functions and explains how to determine if a function is increasing or decreasing based on the sign of the derivative.
- It introduces the use of derivatives to find maximum and minimum values, extreme points, and critical points of functions.
- Theorems 1 and 2 provide rules for determining if a critical point represents a maximum or minimum.
- Examples are provided to demonstrate finding the intervals where a function is increasing/decreasing, identifying extrema, and determining the greatest and least function values over an interval.
This document provides guidance for teachers on applications of differentiation for Years 11 and 12. It covers key topics like graph sketching, maxima and minima problems, and related rates. For graph sketching, it discusses increasing and decreasing functions, stationary points, local maxima and minima, and uses the first derivative test to determine the nature of stationary points. Examples are provided to illustrate these concepts.
Fortran is a general-purpose programming language, mainly intended for mathematical computations in
science applications
this chapter is the third chapter
1) The document discusses water flow in pipes, including descriptions of laminar and turbulent flow, the Reynolds number, energy in pipe flow, and head loss from pipe friction.
2) Key concepts covered include the Darcy-Weisbach equation for calculating head loss from pipe friction and empirical equations like the Hazen-Williams equation.
3) Friction factors are discussed for both laminar and turbulent flow and their relationship to parameters like the Reynolds number and pipe roughness.
Mais conteúdo relacionado
Semelhante a CAL1-CH0-NEW-CAL1-CH0-NEWCAL1-CH0-NEW.pdf
This document provides a table of 100 functions with their derivatives worked out. It begins with introductory rules for deriving various types of functions like constants, polynomials, exponentials, logarithmic, trigonometric, and combined functions. Then it lists 100 specific functions and their derivatives, ranging from simple polynomials and exponentials to more complex expressions combining multiple functions. The purpose is to offer examples of applying the fundamental derivative rules to evaluate the derivatives of diverse mathematical expressions.
Here are the steps to solve this problem:
(a) Let t = time and y = height. Then the differential equation is:
dy/dt = -32 ft/sec^2
Integrate both sides:
∫dy = ∫-32 dt
y = -32t + C
Initial conditions: at t = 0, y = 0
0 = -32(0) + C
C = 0
Therefore, the equation is: y = -32t
When y = 0 (the maximum height), t = 0.625 sec
(b) Put t = 0.625 sec into the equation:
y = -32(0.625) = -20 ft
This document discusses evaluating functions by substituting values for variables. It provides examples of evaluating various functions, including price functions. Students should be able to evaluate functions and solve problems involving function evaluation. Functions can be even, odd, or neither based on whether f(-x) = f(x) or f(-x) = -f(x). Examples are provided to identify functions as even, odd, or neither. Exercises include evaluating functions for given values and determining function parity.
Application of partial derivatives with two variablesSagar Patel
Application of Partial Derivatives with Two Variables
Maxima And Minima Values.
Maximum And Minimum Values.
Tangent and Normal.
Error And Approximation.
Image sciences, image processing, image restoration, photo manipulation. Image and videos representation. Digital versus analog imagery. Quantization and sampling. Sources and models of noises in digital CCD imagery: photon, thermal and readout noises. Sources and models of blurs. Convolutions and point spread functions. Overview of other standard models, problems and tasks: salt-and-pepper and impulse noises, half toning, inpainting, super-resolution, compressed sensing, high dynamic range imagery, demosaicing. Short introduction to other types of imagery: SAR, Sonar, ultrasound, CT and MRI. Linear and ill-posed restoration problems.
This document discusses the continuity of functions. It defines a continuous function as one where the limit of the function as x approaches c exists and is equal to the value of the function at c. It also describes three types of discontinuity: removable, essential, and infinite. Examples are provided to demonstrate determining if a function is continuous at a given point by checking if the three conditions for continuity are met.
1. Partial derivatives describe how a function changes with respect to one variable while holding other variables constant. The partial derivative of Z with respect to x is denoted as ∂Z/∂x or fχ.
2. Optimization problems in calculus involve finding the maximum or minimum values of functions, which can be used to determine the best way to do something.
3. A function has a global/absolute maximum at c if it is greater than or equal to the function values at all other points, and a global/absolute minimum if it is less than or equal to all other points.
* Recognize characteristics of parabolas.
* Understand how the graph of a parabola is related to its quadratic function.
* Determine a quadratic function’s minimum or maximum value.
* Solve problems involving a quadratic function’s minimum or maximum value.
This document provides an overview of key calculus concepts and formulas taught in a Calculus I course at Miami Dade College - Hialeah Campus. The topics covered include limits and derivatives, integration, optimization techniques, and applications of calculus to economics, business, physics, and other fields. The document is intended as a study guide for students in the Calculus I class taught by Professor Mohammad Shakil.
Its states Periodic function, Fourier series for disontinous function, Fourier series, Intervals, Odd and even functions, Half range fourier series etc. Mostly used as active learning assignment in Degree 3rd sem students.
This document discusses antiderivatives and indefinite integrals. It begins by introducing the concept of an antiderivative, which is a function whose derivative is a known function. It then defines the indefinite integral as representing the set of all antiderivatives. Several properties of antiderivatives and indefinite integrals are presented, including: the constant of integration; basic integration rules like power, exponential, and logarithmic rules; and notation used to represent indefinite integrals. Examples are provided to illustrate key concepts and properties.
The document discusses techniques for finding the slope of a tangent line to a function at a given point using derivatives. It provides examples of finding the slope for various functions, including polynomials, trigonometric functions, and implicitly defined functions, using the definition of derivative and rules like the product rule. Approximations of slope using finite differences are also covered. Guidelines for performing implicit differentiation are outlined.
Regret Minimization in Multi-objective Submodular Function MaximizationTasuku Soma
This document presents algorithms for minimizing regret ratio in multi-objective submodular function maximization. It introduces the concept of regret ratio for evaluating the quality of a solution set for multiple objectives. It then proposes two algorithms, Coordinate and Polytope, that provide upper bounds on regret ratio by leveraging approximation algorithms for single objective problems. Experimental results on a movie recommendation dataset show the proposed algorithms achieve significantly lower regret ratios than a random baseline.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
The document discusses the inverse tangent function, tan-1(x). It begins by reviewing one-to-one functions and their inverses. Examples are provided to illustrate one-to-one functions. The lesson aim is to teach students the inverse tangent function. The domain of tan-1(x) is all real numbers, and the range is (-π/2, π/2). The graph of tan-1(x) is the reflection of the tangent function graph over the line y=x. Students are asked to find the values of various inverse tangent functions. Homework problems are assigned involving evaluating inverse tangent functions without a calculator.
Lesson 4- Math 10 - W4Q1_General Form of a Polynomial Function.pptxErlenaMirador1
The document discusses polynomial functions. It defines polynomials as algebraic expressions consisting of terms in the form axn where a is a real number and n is a non-negative integer. The degree of a polynomial refers to the highest exponent of its terms. A polynomial function P(x) is written as P(x) = anxn + an-1xn-1 + ... + a2x2 + a1x + a0 where n is the degree. The document provides examples of finding the degree of polynomials and performing operations like addition and multiplication on polynomial functions. It also demonstrates using synthetic division to divide one polynomial by another.
The document discusses convex functions and related concepts. It defines convex functions and provides examples of convex and concave functions on R and Rn, including norms, logarithms, and powers. It describes properties that preserve convexity, such as positive weighted sums and composition with affine functions. The conjugate function and quasiconvex functions are also introduced. Key concepts are illustrated with examples throughout.
This document discusses key concepts related to derivatives and their applications:
- It defines increasing and decreasing functions and explains how to determine if a function is increasing or decreasing based on the sign of the derivative.
- It introduces the use of derivatives to find maximum and minimum values, extreme points, and critical points of functions.
- Theorems 1 and 2 provide rules for determining if a critical point represents a maximum or minimum.
- Examples are provided to demonstrate finding the intervals where a function is increasing/decreasing, identifying extrema, and determining the greatest and least function values over an interval.
This document provides guidance for teachers on applications of differentiation for Years 11 and 12. It covers key topics like graph sketching, maxima and minima problems, and related rates. For graph sketching, it discusses increasing and decreasing functions, stationary points, local maxima and minima, and uses the first derivative test to determine the nature of stationary points. Examples are provided to illustrate these concepts.
Fortran is a general-purpose programming language, mainly intended for mathematical computations in
science applications
this chapter is the third chapter
Semelhante a CAL1-CH0-NEW-CAL1-CH0-NEWCAL1-CH0-NEW.pdf (20)
1) The document discusses water flow in pipes, including descriptions of laminar and turbulent flow, the Reynolds number, energy in pipe flow, and head loss from pipe friction.
2) Key concepts covered include the Darcy-Weisbach equation for calculating head loss from pipe friction and empirical equations like the Hazen-Williams equation.
3) Friction factors are discussed for both laminar and turbulent flow and their relationship to parameters like the Reynolds number and pipe roughness.
This document discusses water pressure and pressure forces in three sections. The first section describes how water seeks a horizontal free surface where pressure is constant. The second section defines absolute and gauge pressures, noting that submerged objects experience hydrostatic pressure greater than atmospheric pressure. The third section introduces surfaces of equal pressure in water.
This document provides an overview of fundamental properties of water, including:
- Water can exist in three phases (solid, liquid, gas) depending on temperature and pressure, and requires latent heat to change phases.
- Water has a maximum density of 1 g/cm3 at 4°C and its density and viscosity depend on temperature and pressure.
- Atmospheric pressure at sea level is approximately 1 bar and exerts pressure on any surface in contact with air, including water surfaces.
- The document defines key terms including density, specific weight, viscosity, vapor pressure, and cavitation.
This document provides an overview of statics from the Department of Civil Engineering at Philadelphia University. It introduces mechanics, the fundamental concepts of length, time, mass and force. It describes idealizations like particles and rigid bodies. Newton's three laws of motion and the law of gravitational attraction are summarized. The document also discusses units of measurement and prefixes in the International System of Units. For homework, students are assigned problems from Chapter 1.
This document discusses reinforced concrete columns. It defines different types of columns including tied, spiral, composite, and steel pipe columns. It describes the behavior and analysis of axially loaded columns, including elastic behavior, creep effects, and nominal capacity. Design provisions from the ACI code are presented for reinforcement requirements of tied and spiral columns. The behavior of columns under combined bending and axial loads is discussed, including interaction diagrams. Examples are provided to demonstrate the design of columns for various load cases.
This document discusses shear stresses and design of shear reinforcement in reinforced concrete beams. It covers key topics such as:
1) The components that resist shear in an RC beam including aggregate interlock, dowel action, and shear reinforcement when used.
2) Design procedures for shear as outlined in ACI 318, including calculating the nominal shear provided by concrete (Vc) and shear reinforcement (Vs), and ensuring the shear capacity exceeds the factored shear demand (Vu).
3) Types and placement of common shear reinforcement such as stirrups. Minimum requirements for stirrups are also covered.
4) An example problem is worked through demonstrating shear design of a simply supported beam section using stirrup
This document discusses the flexural analysis and design of reinforced concrete T-beams. It addresses key aspects of T-beam analysis including effective flange width, the location of the neutral axis, and different analysis methods depending on if the neutral axis is within the flange or web. Example problems are provided for both analyzing and designing T-beams to resist a given bending moment. Geometries of common T-beam configurations are also shown.
This document discusses the flexural analysis and design of reinforced concrete beams. It introduces doubly reinforced beams, which have reinforcement in both the tension and compression zones. This allows beams to be designed when the maximum reinforcement ratio would otherwise be exceeded. The analysis procedure for doubly reinforced beams is described. An example problem is worked through, showing how to calculate the nominal moment capacity based on whether the compression steel yields or not. Finally, a design example is presented where the area of tension and compression reinforcement is determined for a beam carrying an ultimate moment of 287.2 kN.m.
1) The document discusses the flexural analysis and design of reinforced concrete beams. It covers typical beam behavior, stress calculations, flexural equations, and examples of determining nominal moment capacity.
2) Key aspects reviewed include the assumptions in flexural analysis, cracking moment calculations, strain distributions, balanced sections, and code limits on minimum and maximum steel ratios.
3) Practical considerations for concrete dimensions and reinforcement spacing are also addressed. Examples show how to calculate nominal moment strength and design flexural strength for given beam cross-sections.
This document provides information about reinforced concrete components and properties. It discusses:
1. The different types of cement used in concrete, including their compositions and applications. Common types are Type I, II, III, IV, and V cement.
2. The tests required to ensure the quality of aggregates used in concrete mixes, which include sieve analysis, impact value, absorption, and others.
3. The properties of concrete including how properties like strength develop over time and are affected by factors like water-cement ratio and curing method.
4. The properties of reinforcement steel used in reinforced concrete, including different standard bar sizes in both imperial and metric units, steel grades defined by minimum yield strength,
This document provides an introduction to reinforced concrete structures. It defines common structural elements like beams, columns, slabs, and frames. It discusses loads on structures like dead, live, and environmental loads. It explains concepts of serviceability, strength, and structural safety. Safety provisions in the ACI code are outlined. Two examples are provided to demonstrate designing a column and beam to withstand factored ultimate loads according to ACI code specifications. Design is defined as determining the shape and dimensions of a structure so it can safely perform its intended function over its useful life. The homework is to determine the nominal strength required for the members in the two examples.
ESPP presentation to EU Waste Water Network, 4th June 2024 “EU policies driving nutrient removal and recycling
and the revised UWWTD (Urban Waste Water Treatment Directive)”
PPT on Direct Seeded Rice presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
The technology uses reclaimed CO₂ as the dyeing medium in a closed loop process. When pressurized, CO₂ becomes supercritical (SC-CO₂). In this state CO₂ has a very high solvent power, allowing the dye to dissolve easily.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sérgio Sacani
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
Unlocking the mysteries of reproduction: Exploring fecundity and gonadosomati...AbdullaAlAsif1
The pygmy halfbeak Dermogenys colletei, is known for its viviparous nature, this presents an intriguing case of relatively low fecundity, raising questions about potential compensatory reproductive strategies employed by this species. Our study delves into the examination of fecundity and the Gonadosomatic Index (GSI) in the Pygmy Halfbeak, D. colletei (Meisner, 2001), an intriguing viviparous fish indigenous to Sarawak, Borneo. We hypothesize that the Pygmy halfbeak, D. colletei, may exhibit unique reproductive adaptations to offset its low fecundity, thus enhancing its survival and fitness. To address this, we conducted a comprehensive study utilizing 28 mature female specimens of D. colletei, carefully measuring fecundity and GSI to shed light on the reproductive adaptations of this species. Our findings reveal that D. colletei indeed exhibits low fecundity, with a mean of 16.76 ± 2.01, and a mean GSI of 12.83 ± 1.27, providing crucial insights into the reproductive mechanisms at play in this species. These results underscore the existence of unique reproductive strategies in D. colletei, enabling its adaptation and persistence in Borneo's diverse aquatic ecosystems, and call for further ecological research to elucidate these mechanisms. This study lends to a better understanding of viviparous fish in Borneo and contributes to the broader field of aquatic ecology, enhancing our knowledge of species adaptations to unique ecological challenges.
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
ESA/ACT Science Coffee: Diego Blas - Gravitational wave detection with orbita...Advanced-Concepts-Team
Presentation in the Science Coffee of the Advanced Concepts Team of the European Space Agency on the 07.06.2024.
Speaker: Diego Blas (IFAE/ICREA)
Title: Gravitational wave detection with orbital motion of Moon and artificial
Abstract:
In this talk I will describe some recent ideas to find gravitational waves from supermassive black holes or of primordial origin by studying their secular effect on the orbital motion of the Moon or satellites that are laser ranged.
When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
Authoring a personal GPT for your research and practice: How we created the Q...Leonel Morgado
Thematic analysis in qualitative research is a time-consuming and systematic task, typically done using teams. Team members must ground their activities on common understandings of the major concepts underlying the thematic analysis, and define criteria for its development. However, conceptual misunderstandings, equivocations, and lack of adherence to criteria are challenges to the quality and speed of this process. Given the distributed and uncertain nature of this process, we wondered if the tasks in thematic analysis could be supported by readily available artificial intelligence chatbots. Our early efforts point to potential benefits: not just saving time in the coding process but better adherence to criteria and grounding, by increasing triangulation between humans and artificial intelligence. This tutorial will provide a description and demonstration of the process we followed, as two academic researchers, to develop a custom ChatGPT to assist with qualitative coding in the thematic data analysis process of immersive learning accounts in a survey of the academic literature: QUAL-E Immersive Learning Thematic Analysis Helper. In the hands-on time, participants will try out QUAL-E and develop their ideas for their own qualitative coding ChatGPT. Participants that have the paid ChatGPT Plus subscription can create a draft of their assistants. The organizers will provide course materials and slide deck that participants will be able to utilize to continue development of their custom GPT. The paid subscription to ChatGPT Plus is not required to participate in this workshop, just for trying out personal GPTs during it.
The cost of acquiring information by natural selectionCarl Bergstrom
This is a short talk that I gave at the Banff International Research Station workshop on Modeling and Theory in Population Biology. The idea is to try to understand how the burden of natural selection relates to the amount of information that selection puts into the genome.
It's based on the first part of this research paper:
The cost of information acquisition by natural selection
Ryan Seamus McGee, Olivia Kosterlitz, Artem Kaznatcheev, Benjamin Kerr, Carl T. Bergstrom
bioRxiv 2022.07.02.498577; doi: https://doi.org/10.1101/2022.07.02.498577
1. Lecture Notes for Calculus 101
Chapter 0 : Before Calculus
Feras Awad
Philadelphia University
2. Functions
Idea and Definition
Definition 1
A function f is a rule that associates with each input, a unique
(exactly ONE) output.
✓ ✗
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 2 / 81
3. Functions
Idea and Definition
Example 1
Let f (x) = 3x2
− 4x + 2. Then
f (−1) = 3(−1)2
− 4(−1) + 2
= 3 + 4 + 2 = 9
NOTE: There are 4-ways to represent functions:
Table of values
Words
Graph
Formula
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 3 / 81
4. Functions
Some Graphs
y = x
y = x2
y = x3
y =
√
x
y = 3
√
x y =
1
x
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 4 / 81
5. Functions
Vertical Line Test
A curve in the xy−plane is the graph of some function f if and only
if no vertical line intersects the curve more than once
Example 2
Which of the following graphs is a function?
✓ ✗ ✗
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 5 / 81
6. Functions
Domain (and Range) of Functions
The set of all allowable inputs
is the Domain of f .
The set of all resulting
outputs is the Range of f .
NOTE: If no domain is mentioned, we always assume the largest
possible domain, called the Natural Domain.
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 6 / 81
7. Functions
Examples on Function Domain
Example 3
Find the natural domain of the function f (x) = x3
− 2x2
+ 1.
dom(f ) = all real numbers
= (−∞, ∞)
= R
−∞ ∞
-2 -1 0 1 2
NOTE: Any polynomial
P(x) = anxn
+ an−1xn−1
+ · · · + a1x + a0
has domain R.
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 7 / 81
8. Functions
Examples on Function Domain
Example 4
Find the natural domain of the function f (x) =
1
(x + 1)(x − 3)
.
dom(f ) = all real numbers except −1 and 3
= R − {−1, 3}
= (−∞, −1) ∪ (−1, 3) ∪ (3, ∞)
−∞ ∞
-1 3
NOTE: For rational functions, avoid division by 0.
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 8 / 81
9. Functions
Examples on Function Domain
Example 5
Find the natural domain of the function f (x) =
√
x − 1.
dom(f ) = all real numbers such that x − 1 ≥ 0
= all x ∈ R such that x ≥ 1
= [1, ∞)
−∞ ∞
1
NOTE: Avoid even roots of negative numbers.
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 9 / 81
10. Functions
Examples on Function Domain
Example 6
Find the natural domain of the function f (x) =
√
x2 − 5x + 6.
dom(f ) = all real numbers such that x2
− 5x + 6 ≥ 0
(x − 2)(x − 3) ≥ 0
= (−∞, 2] ∪ [3, ∞)
−∞ ∞
2 3
+ − +
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 10 / 81
11. Functions
Examples on Function Domain
Example 7
Find the natural domain of the function f (x) = 3
√
4 − 3x2.
dom(f ) = all real numbers
= R
NOTE: Odd root accepts +, −, 0 but depends on the function inside
it.
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 11 / 81
12. Functions
Examples on Function Domain
Example 8
Find the natural domain of the function f (x) = 3
s
1
x
.
dom(f ) = all real numbers except 0
= (−∞, 0) ∪ (0, ∞)
−∞ ∞
0
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 12 / 81
13. Functions
Examples on Function Domain
Example 9
Find the natural domain of the function f (x) =
√
x
x2 − 4
.
dom(f ) = all real numbers such that x ≥ 0 except ± 2
= [0, 2) ∪ (2, ∞)
−∞ ∞
-2 0 2
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 13 / 81
14. Functions
Examples on Function Domain
Example 10
Find the natural domain of the function f (x) =
√
x + 1 −
√
2 − x.
dom
√
x + 1
= x ∈ R; x + 1 ≥ 0
x ≥ −1
= [−1, ∞)
dom
√
2 − x
= x ∈ R; 2 − x ≥ 0
− x ≥ −2
x ≤ 2
= (−∞, 2]
−∞ ∞
-1
−∞ ∞
2
−∞ ∞
-1 2
∴ dom(f ) = [−1, ∞) ∩ (−∞, 2] = [−1, 2]
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 14 / 81
15. Functions
Examples on Function Domain
Exercise 1
(1) Compare the domain of g(x) = x and f (x) =
x2
+ x
1 + x
.
(2) Find the domain of each of the following.
a) f (x) =
p
x2 + 2x + 4
b) g(x) =
x
x2 − x − 2
c) f (x) =
√
9 − x2
1 − x2
d) g(x) =
x − 3
1 −
√
x
e) f (x) =
x − 3
1 +
√
x
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 15 / 81
16. Functions
Piecewise Functions
Functions can be defined in pieces (cases).
Example 11
Let f (x) =
0 : x ≤ −1
√
1 − x2 : −1 x 1
x : x ≥ 1
f (0) =
√
1 − 02 = 1
f (−3) = 0
f (5) = 5
f (1) = 1
-1 1
1
NOTE: x = −1 and x = 1 are called piecewise points.
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 16 / 81
17. Functions
Absolute Value
The absolute value of x ∈ R is defined by
|x| =
(
x : x ≥ 0
−x : x 0
For example, |4| = 4, |0| = 0, −1
2
= 1
2
. -1 1
1
NOTE:
(1) |x| = a ⇔ x = ±a. Ex. |x| = 4 ⇔ x = ±4
(2) |x| ≤ a ⇔ −a ≤ x ≤ a. Ex. |x| ≤ 4 ⇔ −4 ≤ x ≤ 4
(3) |x| ≥ a ⇔ x ≥ a or x ≤ −a. Ex. |x| ≥ 4 ⇔ x ≤ −4 or x ≥ 4
(4)
√
x2 = |x|
(5) dom|g(x)| = dom(g(x))
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 17 / 81
19. Functions
Trigonometric Ratios (Right Triangle)
a
c b
θ
a : adjacent
b : opposite
c : hypotenuse
sin θ =
b
c
, cos θ =
a
c
, tan θ =
b
a
sec θ =
c
a
, csc θ =
c
b
, cot θ =
a
b
NOTE:
sec θ =
1
cos θ
, csc θ =
1
sin θ
, cot θ =
1
tan θ
tan θ =
sin θ
cos θ
, cot θ =
cos θ
sin θ
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 19 / 81
20. Functions
Angle Measures and Standard Position
We measure angles either by
degrees or by radians, where
π rad = 180◦
Example 13
90◦
= 90 ×
π
180
=
π
2
2π
3
=
2π
3
×
180
π
= 120◦
Initial Ray
Vertex
Terminal Ray
θ
Counterclockwise
θ is positive
Initial Ray
Vertex
Terminal Ray
θ
Clockwise
θ is negative
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 20 / 81
21. Functions
Angle Measures and Standard Position
Example 14
Right Acute Obtuse Straight
90◦
= π/2 30◦
= π/6 150◦
= 5π/6 180◦
= π
270◦
= 3π/2 −90◦
= −π/2 −45◦
= −π/4 −540◦
= −3π
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 21 / 81
31. Functions
Reference Angle
Example 15
Find all angles θ such that
cos θ = −1/2.
θ1 = π −
π
3
=
2π
3
± 2nπ
θ2 = π +
π
3
=
4π
3
± 2nπ
where n = 0, 1, 2, · · ·
0◦
90◦
180◦
270◦
α
π − α
π + α 2π − α
= π/3
✓
✓
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 31 / 81
32. Functions
Reference Angle
Example 16
Find all angles θ such that
sin θ = −1/
√
2.
θ1 = π +
π
4
=
5π
4
± 2nπ
θ2 = 2π −
π
4
=
7π
4
± 2nπ
where n = 0, 1, 2, · · ·
0◦
90◦
180◦
270◦
α
π − α
π + α 2π − α
= π/4
✓ ✓
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 32 / 81
33. Functions
Reference Angle
Example 17
Find all angles θ such that
tan θ =
√
3.
θ1 =
π
3
± 2nπ
θ2 = π +
π
3
=
4π
3
± 2nπ
∴ θ =
π
3
± nπ
where n = 0, 1, 2, · · ·
0◦
90◦
180◦
270◦
α
π − α
π + α 2π − α
= π/3
✓
✓
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 33 / 81
34. Functions
Reference Angle
Example 18
Find all angles θ such that sin θ = 0.
θ1 = 0 ± 2nπ
θ2 = π ± 2nπ
∴ θ = 0 ± nπ = ±nπ
where n = 0, 1, 2, · · ·
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 34 / 81
35. Functions
Reference Angle
Example 19
Find all angles θ such that cos θ = 0.
θ1 =
π
2
± 2nπ
θ2 =
3π
2
± 2nπ
∴ θ =
π
2
± nπ
where n = 0, 1, 2, · · ·
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 35 / 81
36. Functions
Reference Angle
Example 20
Find all angles θ such that cos θ = −1.
θ = π ± 2nπ =
1 ± 2n
π where n = 0, 1, 2, · · ·
Example 21
Find all angles θ such that sin θ = 1.
θ =
π
2
± 2nπ where n = 0, 1, 2, · · ·
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 36 / 81
37. Functions
Reference Angle
Example 22
Find the value of sin
5π
3
.
2π − α =
5π
3
α = 2π −
5π
3
=
π
3
∴ sin
5π
3
= − sin
π
3
= −
√
3
2
0◦
90◦
180◦
270◦
α
π − α
π + α 2π − α
= π/3
✓
5π/3 = 300◦
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 37 / 81
38. Functions
Reference Angle
Example 23
Find the value of
tan
−3π
4
= − tan
3π
4
π − α =
3π
4
α = π −
3π
4
=
π
4
∴ tan
−3π
4
= − tan
3π
4
= −(−) tan
π
4
= 1
0◦
90◦
180◦
270◦
α
π − α
π + α 2π − α
= π/4
✓
3π/4 = 135◦
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 38 / 81
39. Functions
Reference Angle
Example 24
Find the domain of f (x) = tan x =
sin x
cos x
dom(tan x) = R −
n
cos x = 0
o
= R −
n
x = π/2 ± nπ
o
= dom(sec x)
Example 25
Find the domain of f (x) = csc x =
1
sin x
dom(csc x) = R −
n
sin x = 0
o
= R −
n
x = ±nπ
o
= dom(cot x)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 39 / 81
40. New Functions from Old
Operations on Functions
Let f and g be functions, then
f
×
+
−
g
(x) = f (x)
×
+
−
g(x) ; x ∈ dom(f ) ∩ dom(g)
(f ÷ g)(x) = f (x) ÷ g(x) ; x ∈ dom(f ) ∩ dom(g) −
n
g(x) = 0
o
Example 26
Let f (x) =
√
x → dom(f ) = [0, ∞).
g(x) =
√
x → dom(g) = [0, ∞)
Find (f · g)(x) and its domain.
(f · g)(x) =
√
x ·
√
x =
√
x
2
= x
dom(f · g) = [0, ∞) ∩ [0, ∞) = [0, ∞)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 40 / 81
41. New Functions from Old
Composition of Functions
x
g
g(x)
f
f (g(x))
f ◦ g
f circle g of x
f after g of x
Definition 2
(f ◦ g)(x) = f (g(x))
dom(f ◦ g) = set of all x ∈ dom(g) such that g(x) ∈ dom(f )
NOTE: In general, f ◦ g ̸= g ◦ f .
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 41 / 81
42. New Functions from Old
Composition of Functions
Example 27
Given that f (1) = 1, f (−1) = 0, g(−1) = 1 and g(0) = 0. Find
(f ◦ g)(−1).
(f ◦ g)(−1) = f (g(−1)) = f (1) = 1
Example 28
Let f (x) = x2
+ 3 and g(x) =
√
x. then
(1) (f ◦ g)(x) = f (g(x)) = f (
√
x) =
√
x
2
+ 3 = x + 3
dom(f ◦ g) = [0, ∞) ∩ R = [0, ∞)
(2) (g ◦ f )(x) = g(f (x)) = g(x2
+ 3) =
√
x2 + 3
dom(g ◦ f ) = R ∩ R = R
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 42 / 81
43. New Functions from Old
How Operations Affect Functions Graphs
(Translation)
Example f (x) + c f (x) − c f (x + c) f (x − c)
f (x) = x2
up down left right
x2
+ c x2
− c (x + c)2
(x − c)2
(0, 0)
(0, c)
(0, −c) (−c, 0) (c, 0)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 43 / 81
44. New Functions from Old
How Operations Affect Functions Graphs
(Translation)
Example 29
Draw the graph of f (x) = (x + 1)2
− 1.
(0, 0) (−1, 0)
(−1, −1)
x2
(x + 1)2
(x + 1)2
− 1
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 44 / 81
45. New Functions from Old
How Operations Affect Functions Graphs
(Reflection)
About y−axis About x−axis
f (−x) −f (x)
f (x)
f (−x)
f (x)
−f (x)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 45 / 81
46. New Functions from Old
How Operations Affect Functions Graphs
(Reflection)
Example 30
Draw the graph of f (x) = 4 − |x − 2|
(0, 0) (2, 0)
(2, 0)
(2, 4)
|x| |x − 2| −|x − 2| 4 − |x − 2|
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 46 / 81
47. New Functions from Old
How Operations Affect Functions Graphs
(Reflection)
NOTE: Follow this order of transformations:
(1) Horizontal shifts: f (x ± c)
(2) Reflection: f (−x) and/or −f (x)
(3) Vertical shifts: f (x) ± c
Example 31
Draw the graph of f (x) = 2 −
√
3 − x
(0, 0) (−3, 0) (3, 0)
(3, 0)
(3, 2)
√
x
√
x + 3
√
−x + 3 −
√
3 − x 2 −
√
3 − x
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 47 / 81
48. New Functions from Old
Functions with Symmetric Graphs
Definition 3
f (x) is an even function if f (−x) = f (x). In this case, f is
symmetric about the y−axis.
−1 1 −3/2 3/2 −π/3 π/3
x2
|x| cos x
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 48 / 81
49. New Functions from Old
Functions with Symmetric Graphs
Definition 4
f (x) is an odd function if f (−x) = −f (x). In this case, f is
symmetric about the origin.
−1 1 −π/2 π/2 −π/4 π/4
x3
sin x − tan x = tan(−x)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 49 / 81
50. New Functions from Old
Functions with Symmetric Graphs
NOTE:
(1) Some functions are Odd, some are Even, and some neither Odd
nor Even like f (x) = x + 2 cos x.
(2) ▶ Odd ± Odd = Odd
▶ Even ± Even = Even
▶ Odd × Odd = Even , Odd ÷ Odd = Even
▶ Even × Even = Even , Even ÷ Even = Even
▶ Odd × Even = Odd , Odd ÷ Even = Odd
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 50 / 81
51. Inverse Functions
Idea of Inverse Functions
When two functions UNDO
each other.
The inverse function is denoted by
f −1
and read f −inverse.
NOTE:
* f −1
̸=
1
f
* If y = x3
, then x = 3
√
y
* The inputs of f , are the
outputs of f −1
and vice-versa.
x y = x3
f
f −inverse
domain
f −1
= range (f )
range
f −1
= domain (f )
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 51 / 81
52. Inverse Functions
When Do the Inverse Exist?
Function but NOT 1 − 1 Function and 1 − 1
A function f has inverse ⇔ it is one-to-one (1 − 1)
⇔ x1 ̸= x2, then f (x1) ̸= f (x2) .
⇔ The graph intersects any horizontal
⇔ line at most once (Horizontal Line Test)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 52 / 81
53. Inverse Functions
When Do the Inverse Exist?
Example 32
Let f (x) = x3
.
* The domain of f is R.
* If we take x1, x2 ∈ R such
that x1 ̸= x2, then
x3
1 ̸= x3
2 ⇒ f (x1) ̸= f (x1)
* So, x3
is 1 − 1 on R.
−1 1
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 53 / 81
54. Inverse Functions
When Do the Inverse Exist?
Example 33
Let f (x) = x2
.
* The domain of f is R.
* Note that −1 ̸= 1, but
(−1)2
= (1)2
.
* So, x2
is NOT 1 − 1 on R.
* But, x2
is 1 − 1 on [0, ∞).
−1 1
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 54 / 81
55. Inverse Functions
Definition of Inverse Function
Definition 5
The functions f , f −1
are inverses provided both
f ◦ f −1
(x) = f
f −1
(x)
= x ; x ∈ dom
f −1
f −1
◦ f
(x) = f −1
(f (x)) = x ; x ∈ dom (f )
Example 34
Let f (x) = 2x3
+ 5x + 3. Find x if f −1
(x) = 1.
f −1
(x) = 1 ⇒ f
f −1
(x)
= f (1)
⇒ x = f (1) = 10
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 55 / 81
56. Inverse Functions
Finding the Inverse Function
To find the inverse function of f (x):
(1) Write y = f (x).
(2) Solve the equation for x as a function of y.
(3) Replace x by f −1
(x), and y by x.
Example 35
Find f −1
(x) given that f (x) = x3
.
y = x3
3
√
y =
3
√
x3
x = 3
√
y
f −1
(x) = 3
√
x
x3
3
√
x
y = x
(3/4, 27/64)
(27/64, 3/4)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 56 / 81
57. Inverse Functions
Finding the Inverse Function
Example 36
Find f −1
(x) given that
f (x) =
√
1 + 2x.
y =
√
1 + 2x ⇒ y2
=
√
1 + 2x
2
⇒ y2
= 1 + 2x
⇒ x =
y2
− 1
2
⇒ f −1
(x) =
1
2
x2
− 1
√
1 + 2x
x2
− 1
2
y = x
(1, 0)
(0, 1)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 57 / 81
58. Inverse Functions
Finding the Inverse Function
Example 37
Find f −1
(x) given that f (x) =
x + 1
x − 1
.
y =
x + 1
x − 1
⇒ (x − 1)y = x + 1
⇒ xy − y = x + 1
⇒ xy − x = y + 1
⇒ x(y − 1) = y + 1
⇒ x =
y + 1
y − 1
⇒ f −1
(x) =
x + 1
x − 1
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 58 / 81
59. Inverse Functions
Inverse Trigonometric Functions
f −1
Domain Range Graph Note
sin−1
x [−1, 1]
h
−π
2
, π
2
i
−1 1
π/2
−π/2
arcsin x
cos−1
x [−1, 1] [0, π]
−1 1
π
arccos x
tan−1
x R
−π
2
, π
2
π/2
−π/2
arctan x
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 59 / 81
62. Inverse Functions
Inverse Trigonometric Functions
Example 41
Find the exact value.
(1) sin
sin−1
(1/4)
= 1/4
Note that dom
sin−1
= [−1, 1] and 1/4 ∈ [−1, 1]
Also, sin and sin−1
are inverses and cancel each other
(2) tan (tan−1
(−17/9)) = −17/9
Note that dom(tan−1
) = R and −17/9 ∈ R
Also, tan and tan−1
are inverses and cancel each other
(3) cos (cos−1
(−2/3)) = −2/3
Note that dom(cos−1
) = [−1, 1] and −2/3 ∈ [−1, 1]
Also, cos and cos−1
are inverses and cancel each other
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 62 / 81
64. Inverse Functions
Inverse Trigonometric Functions
Example 43
Find the exact value of
tan−1
tan
5π
6
.
π − α =
5π
6
α = π −
5π
6
=
π
6
tan−1
tan
5π
6
= − tan−1
tan
π
6
=
−π
6
0◦
90◦
180◦
270◦
α
π − α
π + α 2π − α
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 64 / 81
65. Inverse Functions
Inverse Trigonometric Functions
Example 44
Simplify the expression tan
sin−1
x
.
Let θ = sin−1
x
sin θ = sin
sin−1
x
sin θ = x ; x ∈ [−1, 1]
∴ tan
sin−1
x
= tan θ =
x
√
1 − x2
where x ∈ (−1, 1)
θ
x
1
√
1 − x2
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 65 / 81
66. Inverse Functions
Inverse Trigonometric Functions
Example 45
Find the exact value of sin (2 sec−1
3).
Let
θ = sec−1
3
sec θ = sec
sec−1
3
sec θ = 3
∴ sin
2 sec−1
3
= sin(2θ) = 2 sin θ cos θ
= 2 ×
√
8
3
×
1
3
=
4
√
2
9
θ
√
8
3
1
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 66 / 81
67. Exponential and Logarithmic Functions
The Exponential Function
Definition 6
The exponential function to the
base b is f (x) = bx
where b 0
and b ̸= 1.
domain of bx
= R
range of bx
= (0, ∞)
b 1
2x
0 b 1
1
2
x
(0, 1)
Example 46
2x
, πx
, (1/3)x
, · · · are exponential
x2
, xπ
, x
1/3
, · · · are NOT exponential
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 67 / 81
68. Exponential and Logarithmic Functions
The Exponential Function
Example 47
Find the domain of the function f (x) = 4
x
x2−4 .
dom(f ) = R − {x2
− 4 = 0} = R − {±2}
NOTE:
The number e ≈ 2.7182818285 · · · is
called the Natural Number.
The exponential function f (x) = ex
is
called the Natural Exponential
Function.
ex
(0, 1)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 68 / 81
70. Exponential and Logarithmic Functions
The Exponential Function: Properties
Example 48
Find the exact value:
(1) (−8)
2
3 = 3
q
(−8)2 = 3
√
64 = 4
(2) 9−1/2
=
1
91/2
=
1
√
9
=
1
3
Example 49
Solve the equation 2x
= 64.
2x
= 64 ⇒ 2x
= 26
⇒ x = 6
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 70 / 81
71. Exponential and Logarithmic Functions
The Logarithmic Function
Definition 7
The logarithmic function to the
base b is f (x) = logb x where
b 0 and b ̸= 1.
domain of logb x = (0, ∞)
range of logb x = R
b 1
log2 x
0 b 1
log1/2 x
(1, 0)
NOTE: The function loge x is called the Natural Logarithmic
Function and denoted by ln x.
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 71 / 81
72. Exponential and Logarithmic Functions
The Logarithmic Function
NOTE: dom (logb g(x)) = All real numbers such that g(x) 0
and defined.
Example 50
Find the domain of f (x) = ln (9 − x2
).
dom ln
9 − x2
= All x ∈ R such that 9 − x2
0
⇒ x2
9
⇒
√
x2
√
9
⇒ |x| 3
⇒ −3 x 3
⇒ x ∈ (−3, 3)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 72 / 81
73. Exponential and Logarithmic Functions
The Logarithmic Function: Properties
(1) logb 1 = 0
Ex: log10 1 = 0
(2) logb b = 1
Ex: ln e = loge e = 1
(3) logb (xn
) = n logb x
Ex: log3 9 = log3
32
= 2
(4) logb a =
ln a
ln b
Ex: log4 3 = ln 3/ln 4
(5) logb(xy) = logb x + logb y
(6) logb (x/y) = logb x − logb y
(7) logb x = logb y ⇔ x = y
One-to-One Property
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 73 / 81
75. Exponential and Logarithmic Functions
Logarithmic Exponential are Inverses
Function Domain Range
bx
R (0, ∞)
logb x (0, ∞) R
logb (bx
) = x ; x ∈ R
blogb x
= x ; x ∈ (0, ∞)
y = x
bx
logb x
Example 53
Find the exact value of 52 log5 4
.
52 log5 4
= 5log5(42
) = 42
= 16
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 75 / 81
76. Exponential and Logarithmic Functions
Logarithmic Exponential are Inverses
Example 54
Solve the following equations.
(1) ex
= 7 ⇒ ln (ex
) = ln 7 ⇒ x = ln 7
(2) log10 x = −2 ⇒ 10log10 x
= 10−2
⇒ x =
1
102
= 0.01
(3) ln x = 3 ⇒ eln x
= e3
⇒ x = e3
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 76 / 81
77. Exponential and Logarithmic Functions
Logarithmic Exponential are Inverses
Example 55
Solve the equation e2x
+ ex
− 6 = 0.
e2x
+ ex
− 6 = 0 ⇒ (ex
)2
+ ex
− 6 = 0 (Let y = ex
)
⇒ y2
+ y − 6 = 0
⇒ (y + 3)(y − 2) = 0
Solution (1) ⇒ y = −3
⇒ ex
= −3 ✗
Solution (2) ⇒ y = 2
⇒ ex
= 2 ✓
⇒ x = ln 2
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 77 / 81
79. Exponential and Logarithmic Functions
Logarithmic Exponential are Inverses
Example 57
True or False?
“The functions ln (x2
) and 2 ln x have the same domain !!”
domain of ln
x2
= R − {0}
domain of 2 ln (x) = (0, ∞) ∴ FALSE
Example 58
Find the inverse function of f (x) = ln(x + 3).
y = ln(x + 3) ⇒ ey
= eln(x+3)
⇒ x + 3 = ey
⇒ x = ey
− 3 ∴ f −1
(x) = ex
− 3
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 79 / 81
80. Exponential and Logarithmic Functions
Logarithmic Exponential are Inverses
Example 59
Find the inverse function of g(x) = e2x−1
.
y = e2x−1
⇒ ln y = ln
e2x−1
⇒ 2x − 1 = ln y
⇒ x =
1 + ln y
2
∴ g−1
(x) =
1 + ln x
2
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 80 / 81
81. Exponential and Logarithmic Functions
One More Example
Example 60
Let f (x) = x2
− 2x − 3.
(1) Draw the graph of f by shifting the graph of x2
.
x2
− 2x
− 3 =
x2
− 2x + 1
− 1 − 3
= (x − 1)2
− 4
(2) Find f −1(x).
y = (x − 1)2
− 4
x =
p
y + 4 + 1
f −1
(x) =
√
x + 4 + 1
(3) What is the range of f .
range of f = [−4, ∞)
= domain of f −1
(1, −4)
Feras Awad (Philadelphia University) Lecture Notes for Calculus 101 81 / 81