This document contains lecture notes on exponential and logarithmic functions from a Calculus I class at New York University. It begins with announcements about a graded midterm exam and an upcoming homework assignment. It then provides objectives and an outline for sections 3.1 and 3.2 on exponential functions. The bulk of the document derives definitions and properties of exponential functions for various exponents through examples and conventions to preserve desired properties.
Dar a conocer la importancia de los espacios y sub espacios vectoriales en la rama de la electrónica y automatización, también plantearemos ejercicios aplicando el teorema de wronksiano
1. The document discusses methods for solving systems of linear equations and calculating eigen values and eigen vectors of matrices. It describes direct and iterative methods for solving linear systems, including Gauss-Jacobi and Gauss-Seidel iterative methods.
2. It also covers the concepts of diagonal dominance and consistency conditions for linear systems. Rayleigh's power method is introduced for finding the dominant eigen value and vector of a matrix.
3. Examples are provided to illustrate solving linear systems by Jacobi's method and checking for diagonal dominance and consistency of systems. The convergence criteria for Gauss-Jacobi and Gauss-Seidel methods are also outlined.
This document provides an introduction to systems of linear equations and matrix operations. It defines key concepts such as matrices, matrix addition and multiplication, and transitions between different bases. It presents an example of multiplying two matrices using NumPy. The document outlines how systems of linear equations can be represented using matrices and discusses solving systems using techniques like Gauss-Jordan elimination and elementary row operations. It also introduces the concepts of homogeneous and inhomogeneous systems.
Here are the key steps to find the eigenvalues of the given matrix:
1) Write the characteristic equation: det(A - λI) = 0
2) Expand the determinant: (1-λ)(-2-λ) - 4 = 0
3) Simplify and factor: λ(λ + 1)(λ + 2) = 0
4) Find the roots: λ1 = 0, λ2 = -1, λ3 = -2
Therefore, the eigenvalues of the given matrix are -1 and -2.
The document provides examples to illustrate how to find the eigenvalues and eigenvectors of a matrix.
1) For a 2x2 matrix, the characteristic polynomial is computed by taking the determinant of the matrix minus the identity matrix. The roots of the characteristic polynomial are the eigenvalues. The corresponding eigenvectors are found by solving the original eigenvalue equation.
2) For a triangular matrix, the eigenvalues are the diagonal elements. The eigenvectors are found by setting rows corresponding to non-diagonal elements to zero.
3) The document provides a numerical example to demonstrate finding the eigenvalues (3, 1, -2) and eigenvectors of a 3x3 matrix.
A set of notes prepared for an introductory machine learning course, assuming very limited linear algebra background, because all linear algebra operations are fully written out. These notes go into thorough derivations of the generalized linear regression formulation, demonstrating how to write it out in matrix form.
The document discusses square matrices and determinants. It begins by noting that square matrices are the only matrices that can have inverses. It then presents an algorithm for calculating the inverse of a square matrix A by forming the partitioned matrix (A|I) and applying Gauss-Jordan reduction. The document also discusses determinants, defining them recursively as the sum of products of diagonal entries with signs depending on row/column position, for matrices larger than 1x1. Complexity increases exponentially with matrix size.
The document discusses eigenvectors and eigenvalues. It begins by defining diagonal matrices and provides examples. It then states that the goal is to understand diagonalization using eigenvectors and eigenvalues. Diagonalization involves finding a matrix such that when it transforms another matrix, the result is a diagonal matrix. This requires the eigenvectors of the original matrix to be linearly independent. The document provides examples of calculating eigenvectors and eigenvalues from matrices and shows how this relates to diagonalization. It also gives a brief introduction to linear independence and its implications for diagonalization.
Dar a conocer la importancia de los espacios y sub espacios vectoriales en la rama de la electrónica y automatización, también plantearemos ejercicios aplicando el teorema de wronksiano
1. The document discusses methods for solving systems of linear equations and calculating eigen values and eigen vectors of matrices. It describes direct and iterative methods for solving linear systems, including Gauss-Jacobi and Gauss-Seidel iterative methods.
2. It also covers the concepts of diagonal dominance and consistency conditions for linear systems. Rayleigh's power method is introduced for finding the dominant eigen value and vector of a matrix.
3. Examples are provided to illustrate solving linear systems by Jacobi's method and checking for diagonal dominance and consistency of systems. The convergence criteria for Gauss-Jacobi and Gauss-Seidel methods are also outlined.
This document provides an introduction to systems of linear equations and matrix operations. It defines key concepts such as matrices, matrix addition and multiplication, and transitions between different bases. It presents an example of multiplying two matrices using NumPy. The document outlines how systems of linear equations can be represented using matrices and discusses solving systems using techniques like Gauss-Jordan elimination and elementary row operations. It also introduces the concepts of homogeneous and inhomogeneous systems.
Here are the key steps to find the eigenvalues of the given matrix:
1) Write the characteristic equation: det(A - λI) = 0
2) Expand the determinant: (1-λ)(-2-λ) - 4 = 0
3) Simplify and factor: λ(λ + 1)(λ + 2) = 0
4) Find the roots: λ1 = 0, λ2 = -1, λ3 = -2
Therefore, the eigenvalues of the given matrix are -1 and -2.
The document provides examples to illustrate how to find the eigenvalues and eigenvectors of a matrix.
1) For a 2x2 matrix, the characteristic polynomial is computed by taking the determinant of the matrix minus the identity matrix. The roots of the characteristic polynomial are the eigenvalues. The corresponding eigenvectors are found by solving the original eigenvalue equation.
2) For a triangular matrix, the eigenvalues are the diagonal elements. The eigenvectors are found by setting rows corresponding to non-diagonal elements to zero.
3) The document provides a numerical example to demonstrate finding the eigenvalues (3, 1, -2) and eigenvectors of a 3x3 matrix.
A set of notes prepared for an introductory machine learning course, assuming very limited linear algebra background, because all linear algebra operations are fully written out. These notes go into thorough derivations of the generalized linear regression formulation, demonstrating how to write it out in matrix form.
The document discusses square matrices and determinants. It begins by noting that square matrices are the only matrices that can have inverses. It then presents an algorithm for calculating the inverse of a square matrix A by forming the partitioned matrix (A|I) and applying Gauss-Jordan reduction. The document also discusses determinants, defining them recursively as the sum of products of diagonal entries with signs depending on row/column position, for matrices larger than 1x1. Complexity increases exponentially with matrix size.
The document discusses eigenvectors and eigenvalues. It begins by defining diagonal matrices and provides examples. It then states that the goal is to understand diagonalization using eigenvectors and eigenvalues. Diagonalization involves finding a matrix such that when it transforms another matrix, the result is a diagonal matrix. This requires the eigenvectors of the original matrix to be linearly independent. The document provides examples of calculating eigenvectors and eigenvalues from matrices and shows how this relates to diagonalization. It also gives a brief introduction to linear independence and its implications for diagonalization.
This document discusses eigenvalues and eigenvectors. It introduces eigenvalues and eigenvectors and some of their applications in areas like engineering, science, control theory and physics. It defines diagonal matrices and explains how eigenvalues and eigenvectors are used to transform a given matrix into a diagonal matrix. It also discusses how this process can be used to solve coupled differential equations. It provides background on linear independence and explains that the eigenvectors of a matrix must be linearly independent for diagonalization.
Eigen values and eigen vectors engineeringshubham211
mathematics...for engineering mathematics.....learn maths...............................The individual items in a matrix are called its elements or entries.[4] Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.
Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function
...
The document discusses eigenvalues and eigenvectors. It defines an eigenvalue problem as finding scale constants (λ) and nonzero vectors (X) such that when a square matrix (A) multiplies a vector (X), it produces a vector in the same direction but scaled by λ. The characteristic polynomial is used to find the eigenvalues by setting its determinant equal to 0. Once the eigenvalues are obtained, the corresponding eigenvectors can be found by solving the homogeneous system (A - λI)X = 0. Examples are provided to demonstrate finding the eigenvalues and eigenvectors of different matrices.
A comparative analysis of predictve data mining techniques4Mintu246
Partial least squares (PLS) regression is a technique that predicts a set of dependent variables from independent variables. It addresses issues with having many independent variables by extracting orthogonal factors called latent variables that have maximum predictive power for the dependent variables. PLS regression decomposes both the predictor and dependent variable matrices to predict the dependent variables from the predictors. It finds latent vectors that maximize the covariance between the decomposed predictor and dependent variable matrices. PLS regression is useful when there are more predictor variables than observations.
The proof theoretic strength of the Steinitz exchange theorem - EACA 2006Michael Soltys
The document discusses the proof theoretic strength of the Steinitz Exchange Theorem from the perspectives of propositional proof systems and complexity theory. It presents the Steinitz Exchange Theorem and shows that it can be proved using polynomial time concepts and NC2 concepts. The theorem is then used to prove properties of matrices like the existence of powers of a matrix and the Cayley-Hamilton theorem, showing that these principles have proof strength beyond NC2.
K-Notes are concise study materials intended for quick revision near the end of preparation for exams like GATE. Each K-Note covers the concepts from a subject in 40 pages or less. They are useful for final preparation and travel. Students should use K-Notes in the last 2 months before the exam, practicing questions after reviewing each note. The document then provides a summary of key concepts in linear algebra and matrices, including matrix properties, operations, inverses, and systems of linear equations.
This document provides an overview of topics covered in a differential calculus course, including:
1. Limits and differential calculus concepts such as derivatives
2. Special functions and numbers used in calculus
3. A brief history of calculus and its founders Newton and Leibniz
4. Explanations and examples of key calculus concepts such as variables, constants, functions, and limits
The document discusses two iterative methods for solving systems of linear equations:
1. The Jacobi method, which solves for each diagonal element using the previous iteration's values for other elements. It converges to the solution by iterating this process.
2. The Gauss-Seidel method, which sequentially updates elements using values from the current iteration, making it converge faster than the Jacobi method. Both methods decompose the matrix and iteratively solve for the unknowns until the solution converges.
This document discusses iterative methods for solving systems of equations, including the Jacobi and Gauss-Seidel methods. The Jacobi method solves systems of equations by iteratively updating the estimates of the unknown variables. The Gauss-Seidel method similarly iteratively solves systems but updates the estimates sequentially from left to right. Examples applying both methods to solve systems are provided.
Numerical solution of eigenvalues and applications 2SamsonAjibola
This document provides an overview of eigenvalues and their applications. It discusses:
1) Eigenvalues arise in applications across science and engineering, including mechanics, control theory, and quantum mechanics. Numerical methods are used to solve increasingly large eigenvalue problems.
2) Common methods for small problems include the QR and power methods. For large, sparse problems, techniques like the Krylov subspace and Arnoldi methods are used to compute a few desired eigenvalues/eigenvectors.
3) The document outlines the structure of the thesis, which will investigate methods for finding eigenvalues like Krylov subspace, power, and QR. It will also explore applications in areas like biology, statistics, and engineering.
Aplicaciones y subespacios y subespacios vectoriales en laemojose107
se enfoca en la enseñanza del Álgebra Lineal en carreras de ingeniería. Los conceptos vinculados a esta rama de las matemáticas se estudian en los cursos básicos de los primeros años de los planes de estudio en esas carreras. Se estudian conceptos tales como vectores, matrices, sistemas de ecuaciones lineales, espacios vectoriales, transformaciones lineales, valores y. vectores propios, y diagonalización de matrices.
The document discusses numerical methods for solving differential equations that arise in game physics simulations. It begins with an overview of numerical methods and how they can be used to approximate solutions for difficult problems that cannot be solved exactly. It then reviews some common differential equations in physics, such as those for mass-spring systems and projectile motion. The document introduces the concept of solving differential equations numerically using finite difference methods and the explicit Euler method in particular. It demonstrates how explicit Euler can be used to simulate systems like mass-spring oscillations but notes that the method is unstable due to errors from extrapolation. The presentation aims to help understand numerical methods and their application to game physics.
The document provides an introduction to linear algebra concepts for machine learning. It defines vectors as ordered tuples of numbers that express magnitude and direction. Vector spaces are sets that contain all linear combinations of vectors. Linear independence and basis of vector spaces are discussed. Norms measure the magnitude of a vector, with examples given of the 1-norm and 2-norm. Inner products measure the correlation between vectors. Matrices can represent linear operators between vector spaces. Key linear algebra concepts such as trace, determinant, and matrix decompositions are outlined for machine learning applications.
This document contains notes on diagonalization, eigenvalues, and eigenvectors. It discusses how to solve recurrence relations using matrix multiplication and raises matrices to arbitrary powers by diagonalizing them. Diagonalization involves finding an invertible matrix P such that P-1MP is a diagonal matrix D. The columns of P are the eigenvectors of M, and the entries of D are the corresponding eigenvalues. This allows raising M to a power to be reduced to raising the simpler diagonal matrix D to the same power.
The document defines matrices and provides examples of different types of matrices. It discusses key concepts such as rows, columns, dimensions, entries, addition, subtraction, and multiplication of matrices. It also covers special matrices like identity matrices, inverse matrices, transpose of matrices, and using matrices to solve systems of linear equations. The document is a comprehensive overview of matrices that defines fundamental terms and concepts.
The document discusses iterative methods for solving systems of linear equations, including the Jacobi, Gauss-Seidel, and Gauss-Seidel relaxation methods. The Jacobi method works by rewriting the system in a form where the diagonal entries are isolated and computing successive approximations. The Gauss-Seidel method similarly computes approximations but uses the most recent values available at each step. Relaxation improves the Gauss-Seidel method's convergence by taking a weighted average of the current and previous iterations' results. Examples demonstrate applying the different methods to compute solutions.
The document discusses the eigenvalue-eigenvector problem, which has applications in solving differential equations, modeling population growth, and calculating matrix powers. It provides mathematical background on homogeneous systems of equations where the eigenvalues are the roots of the characteristic polynomial. Iterative methods like the power method are presented for finding the dominant or lowest eigenvalue of a matrix. Physical examples of mass-spring systems are given where the eigenvalues correspond to vibration frequencies and the eigenvectors to mode shapes.
Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)Mel Anthony Pepito
This document contains lecture notes from a Calculus I class at New York University on October 21, 2010. It discusses the objectives and outline for sections 3.1-3.2 on exponential and logarithmic functions. Key points include definitions of exponential functions, properties like ax+y = axay, and conventions for extending exponents to non-positive integers and irrational numbers.
Lesson 13: Exponential and Logarithmic Functions (Section 021 handout)Matthew Leingang
This document contains lecture notes from a Calculus I class at New York University on October 21, 2010 covering sections 3.1-3.2 on exponential functions. The notes include announcements about an upcoming midterm exam and homework assignment. Statistics on the recent midterm exam are provided, showing the average, median, standard deviation, and what constitutes a "good" or "great" score. The objectives of sections 3.1-3.2 are outlined as understanding exponential functions, their properties, and laws of logarithms. The notes provide definitions and derivations of exponential functions for various exponent values.
This document discusses eigenvalues and eigenvectors. It introduces eigenvalues and eigenvectors and some of their applications in areas like engineering, science, control theory and physics. It defines diagonal matrices and explains how eigenvalues and eigenvectors are used to transform a given matrix into a diagonal matrix. It also discusses how this process can be used to solve coupled differential equations. It provides background on linear independence and explains that the eigenvectors of a matrix must be linearly independent for diagonalization.
Eigen values and eigen vectors engineeringshubham211
mathematics...for engineering mathematics.....learn maths...............................The individual items in a matrix are called its elements or entries.[4] Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.
Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function
...
The document discusses eigenvalues and eigenvectors. It defines an eigenvalue problem as finding scale constants (λ) and nonzero vectors (X) such that when a square matrix (A) multiplies a vector (X), it produces a vector in the same direction but scaled by λ. The characteristic polynomial is used to find the eigenvalues by setting its determinant equal to 0. Once the eigenvalues are obtained, the corresponding eigenvectors can be found by solving the homogeneous system (A - λI)X = 0. Examples are provided to demonstrate finding the eigenvalues and eigenvectors of different matrices.
A comparative analysis of predictve data mining techniques4Mintu246
Partial least squares (PLS) regression is a technique that predicts a set of dependent variables from independent variables. It addresses issues with having many independent variables by extracting orthogonal factors called latent variables that have maximum predictive power for the dependent variables. PLS regression decomposes both the predictor and dependent variable matrices to predict the dependent variables from the predictors. It finds latent vectors that maximize the covariance between the decomposed predictor and dependent variable matrices. PLS regression is useful when there are more predictor variables than observations.
The proof theoretic strength of the Steinitz exchange theorem - EACA 2006Michael Soltys
The document discusses the proof theoretic strength of the Steinitz Exchange Theorem from the perspectives of propositional proof systems and complexity theory. It presents the Steinitz Exchange Theorem and shows that it can be proved using polynomial time concepts and NC2 concepts. The theorem is then used to prove properties of matrices like the existence of powers of a matrix and the Cayley-Hamilton theorem, showing that these principles have proof strength beyond NC2.
K-Notes are concise study materials intended for quick revision near the end of preparation for exams like GATE. Each K-Note covers the concepts from a subject in 40 pages or less. They are useful for final preparation and travel. Students should use K-Notes in the last 2 months before the exam, practicing questions after reviewing each note. The document then provides a summary of key concepts in linear algebra and matrices, including matrix properties, operations, inverses, and systems of linear equations.
This document provides an overview of topics covered in a differential calculus course, including:
1. Limits and differential calculus concepts such as derivatives
2. Special functions and numbers used in calculus
3. A brief history of calculus and its founders Newton and Leibniz
4. Explanations and examples of key calculus concepts such as variables, constants, functions, and limits
The document discusses two iterative methods for solving systems of linear equations:
1. The Jacobi method, which solves for each diagonal element using the previous iteration's values for other elements. It converges to the solution by iterating this process.
2. The Gauss-Seidel method, which sequentially updates elements using values from the current iteration, making it converge faster than the Jacobi method. Both methods decompose the matrix and iteratively solve for the unknowns until the solution converges.
This document discusses iterative methods for solving systems of equations, including the Jacobi and Gauss-Seidel methods. The Jacobi method solves systems of equations by iteratively updating the estimates of the unknown variables. The Gauss-Seidel method similarly iteratively solves systems but updates the estimates sequentially from left to right. Examples applying both methods to solve systems are provided.
Numerical solution of eigenvalues and applications 2SamsonAjibola
This document provides an overview of eigenvalues and their applications. It discusses:
1) Eigenvalues arise in applications across science and engineering, including mechanics, control theory, and quantum mechanics. Numerical methods are used to solve increasingly large eigenvalue problems.
2) Common methods for small problems include the QR and power methods. For large, sparse problems, techniques like the Krylov subspace and Arnoldi methods are used to compute a few desired eigenvalues/eigenvectors.
3) The document outlines the structure of the thesis, which will investigate methods for finding eigenvalues like Krylov subspace, power, and QR. It will also explore applications in areas like biology, statistics, and engineering.
Aplicaciones y subespacios y subespacios vectoriales en laemojose107
se enfoca en la enseñanza del Álgebra Lineal en carreras de ingeniería. Los conceptos vinculados a esta rama de las matemáticas se estudian en los cursos básicos de los primeros años de los planes de estudio en esas carreras. Se estudian conceptos tales como vectores, matrices, sistemas de ecuaciones lineales, espacios vectoriales, transformaciones lineales, valores y. vectores propios, y diagonalización de matrices.
The document discusses numerical methods for solving differential equations that arise in game physics simulations. It begins with an overview of numerical methods and how they can be used to approximate solutions for difficult problems that cannot be solved exactly. It then reviews some common differential equations in physics, such as those for mass-spring systems and projectile motion. The document introduces the concept of solving differential equations numerically using finite difference methods and the explicit Euler method in particular. It demonstrates how explicit Euler can be used to simulate systems like mass-spring oscillations but notes that the method is unstable due to errors from extrapolation. The presentation aims to help understand numerical methods and their application to game physics.
The document provides an introduction to linear algebra concepts for machine learning. It defines vectors as ordered tuples of numbers that express magnitude and direction. Vector spaces are sets that contain all linear combinations of vectors. Linear independence and basis of vector spaces are discussed. Norms measure the magnitude of a vector, with examples given of the 1-norm and 2-norm. Inner products measure the correlation between vectors. Matrices can represent linear operators between vector spaces. Key linear algebra concepts such as trace, determinant, and matrix decompositions are outlined for machine learning applications.
This document contains notes on diagonalization, eigenvalues, and eigenvectors. It discusses how to solve recurrence relations using matrix multiplication and raises matrices to arbitrary powers by diagonalizing them. Diagonalization involves finding an invertible matrix P such that P-1MP is a diagonal matrix D. The columns of P are the eigenvectors of M, and the entries of D are the corresponding eigenvalues. This allows raising M to a power to be reduced to raising the simpler diagonal matrix D to the same power.
The document defines matrices and provides examples of different types of matrices. It discusses key concepts such as rows, columns, dimensions, entries, addition, subtraction, and multiplication of matrices. It also covers special matrices like identity matrices, inverse matrices, transpose of matrices, and using matrices to solve systems of linear equations. The document is a comprehensive overview of matrices that defines fundamental terms and concepts.
The document discusses iterative methods for solving systems of linear equations, including the Jacobi, Gauss-Seidel, and Gauss-Seidel relaxation methods. The Jacobi method works by rewriting the system in a form where the diagonal entries are isolated and computing successive approximations. The Gauss-Seidel method similarly computes approximations but uses the most recent values available at each step. Relaxation improves the Gauss-Seidel method's convergence by taking a weighted average of the current and previous iterations' results. Examples demonstrate applying the different methods to compute solutions.
The document discusses the eigenvalue-eigenvector problem, which has applications in solving differential equations, modeling population growth, and calculating matrix powers. It provides mathematical background on homogeneous systems of equations where the eigenvalues are the roots of the characteristic polynomial. Iterative methods like the power method are presented for finding the dominant or lowest eigenvalue of a matrix. Physical examples of mass-spring systems are given where the eigenvalues correspond to vibration frequencies and the eigenvectors to mode shapes.
Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)Mel Anthony Pepito
This document contains lecture notes from a Calculus I class at New York University on October 21, 2010. It discusses the objectives and outline for sections 3.1-3.2 on exponential and logarithmic functions. Key points include definitions of exponential functions, properties like ax+y = axay, and conventions for extending exponents to non-positive integers and irrational numbers.
Lesson 13: Exponential and Logarithmic Functions (Section 021 handout)Matthew Leingang
This document contains lecture notes from a Calculus I class at New York University on October 21, 2010 covering sections 3.1-3.2 on exponential functions. The notes include announcements about an upcoming midterm exam and homework assignment. Statistics on the recent midterm exam are provided, showing the average, median, standard deviation, and what constitutes a "good" or "great" score. The objectives of sections 3.1-3.2 are outlined as understanding exponential functions, their properties, and laws of logarithms. The notes provide definitions and derivations of exponential functions for various exponent values.
Lesson 13: Exponential and Logarithmic Functions (Section 041 slides)Mel Anthony Pepito
This document contains lecture notes on exponential and logarithmic functions from a Calculus I class at New York University. The notes cover definitions of exponential functions, properties of exponential functions including laws of exponents, the natural exponential function e, and logarithmic functions. The objectives are to understand exponential functions, their properties, and laws of logarithms including the change of base formula.
Lesson 13: Exponential and Logarithmic Functions (Section 041 slides)Matthew Leingang
This document contains lecture notes on exponential and logarithmic functions from a Calculus I class at New York University. The notes cover definitions of exponential functions, properties of exponential functions including laws of exponents, the natural exponential function e, and logarithmic functions. The objectives are to understand exponential functions, their properties, and laws of logarithms including the change of base formula.
Lesson 13: Exponential and Logarithmic Functions (Section 041 handout)Matthew Leingang
This document summarizes sections 3.1-3.2 of a Calculus I course at New York University on exponential and logarithmic functions taught on October 20, 2010. It outlines definitions and properties of exponential functions, introduces the special number e and natural exponential function, and defines logarithmic functions. Announcements are made that the midterm exam is nearly graded and a WebAssign assignment is due the following week.
Lesson 13: Exponential and Logarithmic Functions (handout)Matthew Leingang
This document contains lecture notes from a Calculus I class covering exponential and logarithmic functions. The notes discuss definitions and properties of exponential functions, including the number e and the natural exponential function. Conventions for rational, irrational and negative exponents are also defined. Examples are provided to illustrate approximating exponential expressions with irrational exponents using rational exponents. The objectives are to understand exponential functions, their properties, and apply laws of logarithms including the change of base formula.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 021 ...Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 021 ...Mel Anthony Pepito
This document is from a Calculus I class at New York University and covers derivatives of exponential and logarithmic functions. It includes objectives, an outline, explanations of properties and graphs of exponential and logarithmic functions, and derivations of derivatives. Key points covered are the derivatives of exponential functions with any base equal the function times a constant, the derivative of the natural logarithm function, and using logarithmic differentiation to find derivatives of more complex expressions.
The document introduces concepts of vector algebra and geometry in two dimensions. It defines vectors, vector operations like addition and scalar multiplication, and properties including equality of vectors, parallelism, orthogonality, and the Cauchy-Schwarz inequality. It also covers vector representations geometrically as arrows, vector length, unit vectors, and linear combinations of vectors. Key concepts are illustrated with diagrams of vectors in the Cartesian plane.
This document provides an overview of unit 2 on the algebra of vectors from the course EMA 310: Vectors and Mechanics. It introduces the learning objectives which are to find the resultant of given vectors, add vectors using the parallelogram and triangle laws of addition, and establish and use properties of vector addition. Examples are given of applying the triangle and parallelogram laws of vector addition, along with activities for students to practice finding vector sums and multiplying vectors by scalars.
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
The document is a lecture on derivatives of exponential and logarithmic functions. It begins with announcements about homework and an upcoming midterm. It then provides objectives and an outline for sections on exponential and logarithmic functions. The body of the document defines exponential functions, establishes conventions for exponents of all types, discusses properties of exponential functions, and graphs various exponential functions. It focuses on setting up the necessary foundations before discussing derivatives of these functions.
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsMel Anthony Pepito
The document discusses conventions for defining exponential functions with exponents other than positive integers, such as negative exponents, fractional exponents, and exponents of zero. It defines exponential functions with these exponent types in a way that maintains important properties like ax+y = ax * ay. The goal is to extend the definition of exponential functions beyond positive integer exponents in a principled way.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 041 ...Mel Anthony Pepito
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
This document discusses vector algebra concepts including:
1. Vectors can represent quantities that have both magnitude and direction, unlike scalars which only have magnitude.
2. Common vector operations include addition, subtraction, and determining the resultant or sum of multiple vectors.
3. The dot product of two vectors produces a scalar value that can indicate whether vectors are parallel or perpendicular and define physical quantities like work and electric fields.
4. The cross product of two vectors produces a new vector that is perpendicular to the original vectors and can define quantities like angular velocity and motion in electromagnetic fields.
This document defines vectors and scalar quantities, and describes their key properties and relationships. It begins by defining physical quantities that can be measured, and distinguishes between scalar and vector quantities. Scalars have only magnitude, while vectors have both magnitude and direction. The document then provides a more rigorous definition of vectors as quantities that remain invariant under coordinate system rotations or translations. It describes how to represent and transform vectors between different coordinate systems. Vector addition, subtraction, and multiplication operations like the scalar and vector products are defined. Derivatives of vectors are also discussed. Examples of velocity and acceleration vectors in uniform circular motion are provided.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
This document discusses vectors and their properties. It provides examples of vector addition and multiplication. Some key points:
- Vectors have both magnitude and direction, while scalars only have magnitude. Vector addition follows the triangle and parallelogram laws.
- There are two types of vector multiplication: the dot product, which results in a scalar, and the cross product, which results in another vector.
- The dot product of two vectors is equal to their magnitudes multiplied by the cosine of the angle between them. It is used to calculate quantities like work and power.
- Vectors can be resolved into rectangular components using a set of base vectors like the i, j, k unit vectors. The magnitude
Similar to Lesson 13: Exponential and Logarithmic Functions (Section 021 slides) (20)
This document provides guidance on developing effective lesson plans for calculus instructors. It recommends starting by defining specific learning objectives and assessments. Examples should be chosen carefully to illustrate concepts and engage students at a variety of levels. The lesson plan should include an introductory problem, definitions, theorems, examples, and group work. Timing for each section should be estimated. After teaching, the lesson can be improved by analyzing what was effective and what needs adjustment for the next time. Advanced preparation is key to looking prepared and ensuring students learn.
Streamlining assessment, feedback, and archival with auto-multiple-choiceMatthew Leingang
Auto-multiple-choice (AMC) is an open-source optical mark recognition software package built with Perl, LaTeX, XML, and sqlite. I use it for all my in-class quizzes and exams. Unique papers are created for each student, fixed-response items are scored automatically, and free-response problems, after manual scoring, have marks recorded in the same process. In the first part of the talk I will discuss AMC’s many features and why I feel it’s ideal for a mathematics course. My contributions to the AMC workflow include some scripts designed to automate the process of returning scored papers
back to students electronically. AMC provides an email gateway, but I have written programs to return graded papers via the DAV protocol to student’s dropboxes on our (Sakai) learning management systems. I will also show how graded papers can be archived, with appropriate metadata tags, into an Evernote notebook.
This document discusses electronic grading of paper assessments using PDF forms. Key points include:
- Various tools for creating fillable PDF forms using LaTeX packages or desktop software.
- Methods for stamping completed forms onto scanned documents including using pdftk or overlaying in TikZ.
- Options for grading on tablets or desktops including GoodReader, PDFExpert, Adobe Acrobat.
- Extracting data from completed forms can be done in Adobe Acrobat or via command line with pdftk.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
g(x) represents the area under the curve of f(t) between 0 and x.
.
x
What can you say about g? 2 4 6 8 10f
The First Fundamental Theorem of Calculus
Theorem (First Fundamental Theorem of Calculus)
Let f be a con nuous func on on [a, b]. Define the func on F on [a, b] by
∫ x
F(x) = f(t) dt
a
Then F is con nuous on [a, b] and differentiable on (a, b) and for all x in (a, b),
F′(x
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
Lesson 27: Integration by Substitution (handout)Matthew Leingang
This document contains lecture notes on integration by substitution from a Calculus I class. It introduces the technique of substitution for both indefinite and definite integrals. For indefinite integrals, the substitution rule is presented, along with examples of using substitutions to evaluate integrals involving polynomials, trigonometric, exponential, and other functions. For definite integrals, the substitution rule is extended and examples are worked through both with and without first finding the indefinite integral. The document emphasizes that substitution often simplifies integrals and makes them easier to evaluate.
Lesson 26: The Fundamental Theorem of Calculus (handout)Matthew Leingang
1) The document discusses lecture notes on Section 5.4: The Fundamental Theorem of Calculus from a Calculus I course. 2) It covers stating and explaining the Fundamental Theorems of Calculus and using the first fundamental theorem to find derivatives of functions defined by integrals. 3) The lecture outlines the first fundamental theorem, which relates differentiation and integration, and gives examples of applying it.
This document contains notes from a calculus class lecture on evaluating definite integrals. It discusses using the evaluation theorem to evaluate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. The document also contains examples of evaluating definite integrals, properties of integrals, and an outline of the key topics covered.
This document contains lecture notes from a Calculus I class covering Section 5.3 on evaluating definite integrals. The notes discuss using the Evaluation Theorem to calculate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. Examples are provided to demonstrate evaluating definite integrals using the midpoint rule approximation. Properties of integrals such as additivity and the relationship between definite and indefinite integrals are also outlined.
Lesson 24: Areas and Distances, The Definite Integral (handout)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
Lesson 24: Areas and Distances, The Definite Integral (slides)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
This document contains lecture notes from a Calculus I class discussing optimization problems. It begins with announcements about upcoming exams and courses the professor is teaching. It then presents an example problem about finding the rectangle of a fixed perimeter with the maximum area. The solution uses calculus techniques like taking the derivative to find the critical points and determine that the optimal rectangle is a square. The notes discuss strategies for solving optimization problems and summarize the key steps to take.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
The document discusses curve sketching of functions by analyzing their derivatives. It provides:
1) A checklist for graphing a function which involves finding where the function is positive/negative/zero, its monotonicity from the first derivative, and concavity from the second derivative.
2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives.
3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.
The document contains lecture notes on curve sketching from a Calculus I class. It discusses using the first and second derivative tests to determine properties of a function like monotonicity, concavity, maxima, minima, and points of inflection in order to sketch the graph of the function. It then provides an example of using these tests to sketch the graph of the cubic function f(x) = 2x^3 - 3x^2 - 12x.
Lesson 20: Derivatives and the Shapes of Curves (slides)Matthew Leingang
This document contains lecture notes on derivatives and the shapes of curves from a Calculus I class taught by Professor Matthew Leingang at New York University. The notes cover using derivatives to determine the intervals where a function is increasing or decreasing, classifying critical points as maxima or minima, using the second derivative to determine concavity, and applying the first and second derivative tests. Examples are provided to illustrate finding intervals of monotonicity for various functions.
Lesson 20: Derivatives and the Shapes of Curves (handout)Matthew Leingang
This document contains lecture notes on calculus from a Calculus I course. It covers determining the monotonicity of functions using the first derivative test. Key points include using the sign of the derivative to determine if a function is increasing or decreasing over an interval, and using the first derivative test to classify critical points as local maxima, minima, or neither. Examples are provided to demonstrate finding intervals of monotonicity for various functions and applying the first derivative test.
Lesson 20: Derivatives and the Shapes of Curves (handout)
Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)
1. Sections 3.1–3.2
Exponential and Logarithmic Functions
V63.0121.021, Calculus I
New York University
October 21, 2010
Announcements
Midterm is graded and scores are on blackboard. Should get it
back in recitation.
There is WebAssign due Monday/Tuesday next week.
. . . . . .
2. . . . . . .
Announcements
Midterm is graded and
scores are on blackboard.
Should get it back in
recitation.
There is WebAssign due
Monday/Tuesday next
week.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 2 / 38
3. . . . . . .
Midterm Statistics
Average: 78.77%
Median: 80%
Standard Deviation: 12.39%
“good” is anything above average and “great” is anything more
than one standard deviation above average.
More than one SD below the mean is cause for concern.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 3 / 38
4. . . . . . .
Objectives for Sections 3.1 and 3.2
Know the definition of an
exponential function
Know the properties of
exponential functions
Understand and apply the
laws of logarithms,
including the change of
base formula.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 4 / 38
5. . . . . . .
Outline
Definition of exponential functions
Properties of exponential Functions
The number e and the natural exponential function
Compound Interest
The number e
A limit
Logarithmic Functions
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 5 / 38
6. . . . . . .
Derivation of exponential functions
Definition
If a is a real number and n is a positive whole number, then
an
= a · a · · · · · a
n factors
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 6 / 38
7. . . . . . .
Derivation of exponential functions
Definition
If a is a real number and n is a positive whole number, then
an
= a · a · · · · · a
n factors
Examples
23
= 2 · 2 · 2 = 8
34
= 3 · 3 · 3 · 3 = 81
(−1)5
= (−1)(−1)(−1)(−1)(−1) = −1
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 6 / 38
8. . . . . . .
Anatomy of a power
Definition
A power is an expression of the form ab
.
The number a is called the base.
The number b is called the exponent.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 7 / 38
9. . . . . . .
Fact
If a is a real number, then
ax+y
= ax
ay
(sums to products)
ax−y
=
ax
ay
(ax
)y
= axy
(ab)x
= ax
bx
whenever all exponents are positive whole numbers.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 8 / 38
10. . . . . . .
Fact
If a is a real number, then
ax+y
= ax
ay
(sums to products)
ax−y
=
ax
ay
(differences to quotients)
(ax
)y
= axy
(ab)x
= ax
bx
whenever all exponents are positive whole numbers.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 8 / 38
11. . . . . . .
Fact
If a is a real number, then
ax+y
= ax
ay
(sums to products)
ax−y
=
ax
ay
(differences to quotients)
(ax
)y
= axy
(repeated exponentiation to multiplied powers)
(ab)x
= ax
bx
whenever all exponents are positive whole numbers.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 8 / 38
12. . . . . . .
Fact
If a is a real number, then
ax+y
= ax
ay
(sums to products)
ax−y
=
ax
ay
(differences to quotients)
(ax
)y
= axy
(repeated exponentiation to multiplied powers)
(ab)x
= ax
bx
(power of product is product of powers)
whenever all exponents are positive whole numbers.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 8 / 38
13. . . . . . .
Fact
If a is a real number, then
ax+y
= ax
ay
(sums to products)
ax−y
=
ax
ay
(differences to quotients)
(ax
)y
= axy
(repeated exponentiation to multiplied powers)
(ab)x
= ax
bx
(power of product is product of powers)
whenever all exponents are positive whole numbers.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 8 / 38
14. . . . . . .
Fact
If a is a real number, then
ax+y
= ax
ay
(sums to products)
ax−y
=
ax
ay
(differences to quotients)
(ax
)y
= axy
(repeated exponentiation to multiplied powers)
(ab)x
= ax
bx
(power of product is product of powers)
whenever all exponents are positive whole numbers.
Proof.
Check for yourself:
ax+y
= a · a · · · · · a
x + y factors
= a · a · · · · · a
x factors
· a · a · · · · · a
y factors
= ax
ay
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 8 / 38
15. . . . . . .
Let's be conventional
The desire that these properties remain true gives us conventions
for ax
when x is not a positive whole number.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 9 / 38
16. . . . . . .
Let's be conventional
The desire that these properties remain true gives us conventions
for ax
when x is not a positive whole number.
For example, what should a0
be? We cannot write down zero a’s
and multiply them together. But we would want this to be true:
an
= an+0 !
= an
· a0
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 9 / 38
17. . . . . . .
Let's be conventional
The desire that these properties remain true gives us conventions
for ax
when x is not a positive whole number.
For example, what should a0
be? We cannot write down zero a’s
and multiply them together. But we would want this to be true:
an
= an+0 !
= an
· a0
=⇒ a0 !
=
an
an
= 1
(The equality with the exclamation point is what we want.)
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 9 / 38
18. . . . . . .
Let's be conventional
The desire that these properties remain true gives us conventions
for ax
when x is not a positive whole number.
For example, what should a0
be? We cannot write down zero a’s
and multiply them together. But we would want this to be true:
an
= an+0 !
= an
· a0
=⇒ a0 !
=
an
an
= 1
(The equality with the exclamation point is what we want.)
Definition
If a ̸= 0, we define a0
= 1.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 9 / 38
19. . . . . . .
Let's be conventional
The desire that these properties remain true gives us conventions
for ax
when x is not a positive whole number.
For example, what should a0
be? We cannot write down zero a’s
and multiply them together. But we would want this to be true:
an
= an+0 !
= an
· a0
=⇒ a0 !
=
an
an
= 1
(The equality with the exclamation point is what we want.)
Definition
If a ̸= 0, we define a0
= 1.
Notice 00
remains undefined (as a limit form, it’s indeterminate).
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 9 / 38
20. . . . . . .
Conventions for negative exponents
If n ≥ 0, we want
an+(−n) !
= an
· a−n
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 10 / 38
21. . . . . . .
Conventions for negative exponents
If n ≥ 0, we want
an+(−n) !
= an
· a−n
=⇒ a−n !
=
a0
an
=
1
an
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 10 / 38
22. . . . . . .
Conventions for negative exponents
If n ≥ 0, we want
an+(−n) !
= an
· a−n
=⇒ a−n !
=
a0
an
=
1
an
Definition
If n is a positive integer, we define a−n
=
1
an
.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 10 / 38
23. . . . . . .
Conventions for negative exponents
If n ≥ 0, we want
an+(−n) !
= an
· a−n
=⇒ a−n !
=
a0
an
=
1
an
Definition
If n is a positive integer, we define a−n
=
1
an
.
Fact
The convention that a−n
=
1
an
“works” for negative n as well.
If m and n are any integers, then am−n
=
am
an
.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 10 / 38
24. . . . . . .
Conventions for fractional exponents
If q is a positive integer, we want
(a1/q
)q !
= a1
= a
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 11 / 38
25. . . . . . .
Conventions for fractional exponents
If q is a positive integer, we want
(a1/q
)q !
= a1
= a =⇒ a1/q !
= q
√
a
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 11 / 38
26. . . . . . .
Conventions for fractional exponents
If q is a positive integer, we want
(a1/q
)q !
= a1
= a =⇒ a1/q !
= q
√
a
Definition
If q is a positive integer, we define a1/q
= q
√
a. We must have a ≥ 0 if q
is even.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 11 / 38
27. . . . . . .
Conventions for fractional exponents
If q is a positive integer, we want
(a1/q
)q !
= a1
= a =⇒ a1/q !
= q
√
a
Definition
If q is a positive integer, we define a1/q
= q
√
a. We must have a ≥ 0 if q
is even.
Notice that
q
√
ap =
( q
√
a
)p
. So we can unambiguously say
ap/q
= (ap
)1/q
= (a1/q
)p
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 11 / 38
28. . . . . . .
Conventions for irrational exponents
So ax
is well-defined if a is positive and x is rational.
What about irrational powers?
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 12 / 38
29. . . . . . .
Conventions for irrational exponents
So ax
is well-defined if a is positive and x is rational.
What about irrational powers?
Definition
Let a > 0. Then
ax
= lim
r→x
r rational
ar
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 12 / 38
30. . . . . . .
Conventions for irrational exponents
So ax
is well-defined if a is positive and x is rational.
What about irrational powers?
Definition
Let a > 0. Then
ax
= lim
r→x
r rational
ar
In other words, to approximate ax
for irrational x, take r close to x but
rational and compute ar
.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 12 / 38
31. . . . . . .
Approximating a power with an irrational exponent
r 2r
3 23
= 8
3.1 231/10
=
10
√
231
≈ 8.57419
3.14 2314/100
=
100
√
2314
≈ 8.81524
3.141 23141/1000
=
1000
√
23141
≈ 8.82135
The limit (numerically approximated is)
2π
≈ 8.82498
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 13 / 38
32. . . . . . .
Graphs of various exponential functions
. .x
.y
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 14 / 38
33. . . . . . .
Graphs of various exponential functions
. .x
.y
.y = 1x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 14 / 38
34. . . . . . .
Graphs of various exponential functions
. .x
.y
.y = 1x
.y = 2x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 14 / 38
35. . . . . . .
Graphs of various exponential functions
. .x
.y
.y = 1x
.y = 2x.y = 3x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 14 / 38
36. . . . . . .
Graphs of various exponential functions
. .x
.y
.y = 1x
.y = 2x.y = 3x
.y = 10x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 14 / 38
42. . . . . . .
Outline
Definition of exponential functions
Properties of exponential Functions
The number e and the natural exponential function
Compound Interest
The number e
A limit
Logarithmic Functions
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 15 / 38
43. . . . . . .
Properties of exponential Functions
.
.
Theorem
If a > 0 and a ̸= 1, then f(x) = ax
is a continuous function with domain
(−∞, ∞) and range (0, ∞). In particular, ax
> 0 for all x. For any real numbers
x and y, and positive numbers a and b we have
ax+y
= ax
ay
ax−y
=
ax
ay
(ax
)y
= axy
(ab)x
= ax
bx
Proof.
This is true for positive integer exponents by natural definition
Our conventional definitions make these true for rational exponents
Our limit definition make these for irrational exponents, too
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 16 / 38
44. . . . . . .
Properties of exponential Functions
.
.
Theorem
If a > 0 and a ̸= 1, then f(x) = ax
is a continuous function with domain
(−∞, ∞) and range (0, ∞). In particular, ax
> 0 for all x. For any real numbers
x and y, and positive numbers a and b we have
ax+y
= ax
ay
ax−y
=
ax
ay
(negative exponents mean reciprocals)
(ax
)y
= axy
(ab)x
= ax
bx
Proof.
This is true for positive integer exponents by natural definition
Our conventional definitions make these true for rational exponents
Our limit definition make these for irrational exponents, too
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 16 / 38
45. . . . . . .
Properties of exponential Functions
.
.
Theorem
If a > 0 and a ̸= 1, then f(x) = ax
is a continuous function with domain
(−∞, ∞) and range (0, ∞). In particular, ax
> 0 for all x. For any real numbers
x and y, and positive numbers a and b we have
ax+y
= ax
ay
ax−y
=
ax
ay
(negative exponents mean reciprocals)
(ax
)y
= axy
(fractional exponents mean roots)
(ab)x
= ax
bx
Proof.
This is true for positive integer exponents by natural definition
Our conventional definitions make these true for rational exponents
Our limit definition make these for irrational exponents, too
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 16 / 38
46. . . . . . .
Simplifying exponential expressions
Example
Simplify: 82/3
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 17 / 38
51. . . . . . .
Limits of exponential functions
Fact (Limits of exponential
functions)
If a > 1, then lim
x→∞
ax
= ∞
and lim
x→−∞
ax
= 0
If 0 < a < 1, then
lim
x→∞
ax
= 0 and
lim
x→−∞
ax
= ∞ . .x
.y
.y =
.y = 2x
.y = 3x
.y = 10x
.y =.y = (1/2)x.y = (1/3)x
.y = (1/10)x
.y = (2/3)x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 18 / 38
52. . . . . . .
Outline
Definition of exponential functions
Properties of exponential Functions
The number e and the natural exponential function
Compound Interest
The number e
A limit
Logarithmic Functions
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 19 / 38
53. . . . . . .
Compounded Interest
Question
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have
After one year?
After two years?
after t years?
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 20 / 38
54. . . . . . .
Compounded Interest
Question
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100 + 10% = $110
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 20 / 38
55. . . . . . .
Compounded Interest
Question
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100 + 10% = $110
$110 + 10% = $110 + $11 = $121
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 20 / 38
56. . . . . . .
Compounded Interest
Question
Suppose you save $100 at 10% annual interest, with interest
compounded once a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100 + 10% = $110
$110 + 10% = $110 + $11 = $121
$100(1.1)t
.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 20 / 38
57. . . . . . .
Compounded Interest: quarterly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded four times a year. How much do you have
After one year?
After two years?
after t years?
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 21 / 38
58. . . . . . .
Compounded Interest: quarterly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded four times a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100(1.025)4
= $110.38,
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 21 / 38
59. . . . . . .
Compounded Interest: quarterly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded four times a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100(1.025)4
= $110.38, not $100(1.1)4
!
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 21 / 38
60. . . . . . .
Compounded Interest: quarterly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded four times a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100(1.025)4
= $110.38, not $100(1.1)4
!
$100(1.025)8
= $121.84
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 21 / 38
61. . . . . . .
Compounded Interest: quarterly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded four times a year. How much do you have
After one year?
After two years?
after t years?
Answer
$100(1.025)4
= $110.38, not $100(1.1)4
!
$100(1.025)8
= $121.84
$100(1.025)4t
.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 21 / 38
62. . . . . . .
Compounded Interest: monthly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded twelve times a year. How much do you have after t
years?
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 22 / 38
63. . . . . . .
Compounded Interest: monthly
Question
Suppose you save $100 at 10% annual interest, with interest
compounded twelve times a year. How much do you have after t
years?
Answer
$100(1 + 10%/12)12t
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 22 / 38
64. . . . . . .
Compounded Interest: general
Question
Suppose you save P at interest rate r, with interest compounded n
times a year. How much do you have after t years?
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 23 / 38
65. . . . . . .
Compounded Interest: general
Question
Suppose you save P at interest rate r, with interest compounded n
times a year. How much do you have after t years?
Answer
B(t) = P
(
1 +
r
n
)nt
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 23 / 38
66. . . . . . .
Compounded Interest: continuous
Question
Suppose you save P at interest rate r, with interest compounded every
instant. How much do you have after t years?
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 24 / 38
67. . . . . . .
Compounded Interest: continuous
Question
Suppose you save P at interest rate r, with interest compounded every
instant. How much do you have after t years?
Answer
B(t) = lim
n→∞
P
(
1 +
r
n
)nt
= lim
n→∞
P
(
1 +
1
n
)rnt
= P
[
lim
n→∞
(
1 +
1
n
)n
independent of P, r, or t
]rt
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 24 / 38
68. . . . . . .
The magic number
Definition
e = lim
n→∞
(
1 +
1
n
)n
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 25 / 38
69. . . . . . .
The magic number
Definition
e = lim
n→∞
(
1 +
1
n
)n
So now continuously-compounded interest can be expressed as
B(t) = Pert
.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 25 / 38
70. . . . . . .
Existence of e
See Appendix B
n
(
1 +
1
n
)n
1 2
2 2.25
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 26 / 38
71. . . . . . .
Existence of e
See Appendix B
n
(
1 +
1
n
)n
1 2
2 2.25
3 2.37037
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 26 / 38
72. . . . . . .
Existence of e
See Appendix B
n
(
1 +
1
n
)n
1 2
2 2.25
3 2.37037
10 2.59374
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 26 / 38
73. . . . . . .
Existence of e
See Appendix B
n
(
1 +
1
n
)n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 26 / 38
74. . . . . . .
Existence of e
See Appendix B
n
(
1 +
1
n
)n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
1000 2.71692
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 26 / 38
75. . . . . . .
Existence of e
See Appendix B
n
(
1 +
1
n
)n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
1000 2.71692
106
2.71828
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 26 / 38
76. . . . . . .
Existence of e
See Appendix B
We can experimentally
verify that this number
exists and is
e ≈ 2.718281828459045 . . .
n
(
1 +
1
n
)n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
1000 2.71692
106
2.71828
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 26 / 38
77. . . . . . .
Existence of e
See Appendix B
We can experimentally
verify that this number
exists and is
e ≈ 2.718281828459045 . . .
e is irrational
n
(
1 +
1
n
)n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
1000 2.71692
106
2.71828
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 26 / 38
78. . . . . . .
Existence of e
See Appendix B
We can experimentally
verify that this number
exists and is
e ≈ 2.718281828459045 . . .
e is irrational
e is transcendental
n
(
1 +
1
n
)n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
1000 2.71692
106
2.71828
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 26 / 38
79. . . . . . .
Meet the Mathematician: Leonhard Euler
Born in Switzerland, lived
in Prussia (Germany) and
Russia
Eyesight trouble all his life,
blind from 1766 onward
Hundreds of contributions
to calculus, number theory,
graph theory, fluid
mechanics, optics, and
astronomy
Leonhard Paul Euler
Swiss, 1707–1783
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 27 / 38
80. . . . . . .
A limit
.
.
Question
What is lim
h→0
eh − 1
h
?
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 28 / 38
81. . . . . . .
A limit
.
.
Question
What is lim
h→0
eh − 1
h
?
Answer
e = lim
n→∞
(1 + 1/n)n
= lim
h→0
(1 + h)1/h
. So for a small h, e ≈ (1 + h)1/h
. So
eh − 1
h
≈
[
(1 + h)1/h
]h
− 1
h
= 1
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 28 / 38
82. . . . . . .
A limit
.
.
Question
What is lim
h→0
eh − 1
h
?
Answer
e = lim
n→∞
(1 + 1/n)n
= lim
h→0
(1 + h)1/h
. So for a small h, e ≈ (1 + h)1/h
. So
eh − 1
h
≈
[
(1 + h)1/h
]h
− 1
h
= 1
It follows that lim
h→0
eh − 1
h
= 1.
This can be used to characterize e: lim
h→0
2h
− 1
h
= 0.693 · · · < 1 and
lim
h→0
3h
− 1
h
= 1.099 · · · > 1
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 28 / 38
83. . . . . . .
Outline
Definition of exponential functions
Properties of exponential Functions
The number e and the natural exponential function
Compound Interest
The number e
A limit
Logarithmic Functions
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 29 / 38
84. . . . . . .
Logarithms
Definition
The base a logarithm loga x is the inverse of the function ax
y = loga x ⇐⇒ x = ay
The natural logarithm ln x is the inverse of ex
. So
y = ln x ⇐⇒ x = ey
.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 30 / 38
85. . . . . . .
Logarithms
Definition
The base a logarithm loga x is the inverse of the function ax
y = loga x ⇐⇒ x = ay
The natural logarithm ln x is the inverse of ex
. So
y = ln x ⇐⇒ x = ey
.
Facts
(i) loga(x1 · x2) = loga x1 + loga x2
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 30 / 38
86. . . . . . .
Logarithms
Definition
The base a logarithm loga x is the inverse of the function ax
y = loga x ⇐⇒ x = ay
The natural logarithm ln x is the inverse of ex
. So
y = ln x ⇐⇒ x = ey
.
Facts
(i) loga(x1 · x2) = loga x1 + loga x2
(ii) loga
(
x1
x2
)
= loga x1 − loga x2
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 30 / 38
87. . . . . . .
Logarithms
Definition
The base a logarithm loga x is the inverse of the function ax
y = loga x ⇐⇒ x = ay
The natural logarithm ln x is the inverse of ex
. So
y = ln x ⇐⇒ x = ey
.
Facts
(i) loga(x1 · x2) = loga x1 + loga x2
(ii) loga
(
x1
x2
)
= loga x1 − loga x2
(iii) loga(xr
) = r loga x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 30 / 38
88. . . . . . .
Logarithms convert products to sums
Suppose y1 = loga x1 and y2 = loga x2
Then x1 = ay1 and x2 = ay2
So x1x2 = ay1 ay2 = ay1+y2
Therefore
loga(x1 · x2) = loga x1 + loga x2
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 31 / 38
89. . . . . . .
Example
Write as a single logarithm: 2 ln 4 − ln 3.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 32 / 38
90. . . . . . .
Example
Write as a single logarithm: 2 ln 4 − ln 3.
Solution
2 ln 4 − ln 3 = ln 42
− ln 3 = ln
42
3
not
ln 42
ln 3
!
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 32 / 38
91. . . . . . .
Example
Write as a single logarithm: 2 ln 4 − ln 3.
Solution
2 ln 4 − ln 3 = ln 42
− ln 3 = ln
42
3
not
ln 42
ln 3
!
Example
Write as a single logarithm: ln
3
4
+ 4 ln 2
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 32 / 38
92. . . . . . .
Example
Write as a single logarithm: 2 ln 4 − ln 3.
Solution
2 ln 4 − ln 3 = ln 42
− ln 3 = ln
42
3
not
ln 42
ln 3
!
Example
Write as a single logarithm: ln
3
4
+ 4 ln 2
Answer
ln 12
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 32 / 38
93. . . . . . .
Graphs of logarithmic functions
. .x
.y
.y = 2x
.y = log2 x
. .(0, 1)
..(1, 0)
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 33 / 38
94. . . . . . .
Graphs of logarithmic functions
. .x
.y
.y = 2x
.y = log2 x
. .(0, 1)
..(1, 0)
.y = 3x
.y = log3 x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 33 / 38
95. . . . . . .
Graphs of logarithmic functions
. .x
.y
.y = 2x
.y = log2 x
. .(0, 1)
..(1, 0)
.y = 3x
.y = log3 x
.y = 10x
.y = log10 x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 33 / 38
96. . . . . . .
Graphs of logarithmic functions
. .x
.y
.y = 2x
.y = log2 x
. .(0, 1)
..(1, 0)
.y = 3x
.y = log3 x
.y = 10x
.y = log10 x
.y = ex
.y = ln x
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 33 / 38
97. . . . . . .
Change of base formula for exponentials
Fact
If a > 0 and a ̸= 1, and the same for b, then
loga x =
logb x
logb a
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 34 / 38
98. . . . . . .
Change of base formula for exponentials
Fact
If a > 0 and a ̸= 1, and the same for b, then
loga x =
logb x
logb a
Proof.
If y = loga x, then x = ay
So logb x = logb(ay
) = y logb a
Therefore
y = loga x =
logb x
logb a
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 34 / 38
99. . . . . . .
Example of changing base
Example
Find log2 8 by using log10 only.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 35 / 38
100. . . . . . .
Example of changing base
Example
Find log2 8 by using log10 only.
Solution
log2 8 =
log10 8
log10 2
≈
0.90309
0.30103
= 3
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 35 / 38
101. . . . . . .
Example of changing base
Example
Find log2 8 by using log10 only.
Solution
log2 8 =
log10 8
log10 2
≈
0.90309
0.30103
= 3
Surprised?
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 35 / 38
102. . . . . . .
Example of changing base
Example
Find log2 8 by using log10 only.
Solution
log2 8 =
log10 8
log10 2
≈
0.90309
0.30103
= 3
Surprised? No, log2 8 = log2 23
= 3 directly.
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 35 / 38
103. . . . . . .
Upshot of changing base
The point of the change of base formula
loga x =
logb x
logb a
=
1
logb a
· logb x = constant · logb x
is that all the logarithmic functions are multiples of each other. So just
pick one and call it your favorite.
Engineers like the common logarithm log = log10
Computer scientists like the binary logarithm lg = log2
Mathematicians like natural logarithm ln = loge
Naturally, we will follow the mathematicians. Just don’t pronounce it
“lawn.”
V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 36 / 38