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 021 slides)Matthew Leingang
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.
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 (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 14: Derivatives of Exponential and Logarithmic Functions (Section 041 ...Matthew Leingang
This document is a lecture on derivatives of exponential and logarithmic functions from a Calculus I class at New York University. It covers the objectives and outline, which include finding derivatives of exponential functions with any base, logarithmic functions with any base, and using logarithmic differentiation. It provides proofs and examples of finding derivatives, such as the derivative of the natural exponential function being itself and the derivative of the natural logarithm function.
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.
The document is a lecture note from an NYU Calculus I class covering the definite integral. It provides announcements about upcoming quizzes and exams. The content discusses computing definite integrals using Riemann sums and the limit of Riemann sums, as well as properties of the definite integral. Examples are given of calculating Riemann sums using different representative points in each interval.
Lesson 15: Exponential Growth and Decay (handout)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
The document introduces deterministic and stochastic observers. Deterministic observers estimate states using a model and measurements, like the Luenberger observer. Stochastic observers, like the Kalman filter, also account for noise. The document discusses open-loop and closed-loop observer designs, how to select observer eigenvalues, and approaches for partial state estimation.
Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)Matthew Leingang
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.
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 (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 14: Derivatives of Exponential and Logarithmic Functions (Section 041 ...Matthew Leingang
This document is a lecture on derivatives of exponential and logarithmic functions from a Calculus I class at New York University. It covers the objectives and outline, which include finding derivatives of exponential functions with any base, logarithmic functions with any base, and using logarithmic differentiation. It provides proofs and examples of finding derivatives, such as the derivative of the natural exponential function being itself and the derivative of the natural logarithm function.
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.
The document is a lecture note from an NYU Calculus I class covering the definite integral. It provides announcements about upcoming quizzes and exams. The content discusses computing definite integrals using Riemann sums and the limit of Riemann sums, as well as properties of the definite integral. Examples are given of calculating Riemann sums using different representative points in each interval.
Lesson 15: Exponential Growth and Decay (handout)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
The document introduces deterministic and stochastic observers. Deterministic observers estimate states using a model and measurements, like the Luenberger observer. Stochastic observers, like the Kalman filter, also account for noise. The document discusses open-loop and closed-loop observer designs, how to select observer eigenvalues, and approaches for partial state estimation.
This document discusses the application of vector spaces and subspaces in biotechnology. It begins by introducing the importance of linear algebra in scientific and technological development. It then defines the objectives as understanding vector spaces, subspaces, and dimensionality. Examples of vector spaces and subspaces are provided. Applications include using these concepts to create classification methods for diseases, animals and plants. In conclusions, it is stated that these concepts facilitate study and development in biotechnology.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (handout)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.
This document provides an introduction to blind source separation and non-negative matrix factorization. It describes blind source separation as a method to estimate original signals from observed mixed signals. Non-negative matrix factorization is introduced as a constraint-based approach to solving blind source separation using non-negativity. The alternating least squares algorithm is described for solving the non-negative matrix factorization problem. Experiments applying these methods to artificial and real image data are presented and discussed.
The document provides information about an upcoming calculus exam, quiz, and lecture on curve sketching. It outlines the procedure for sketching curves, including using the increasing/decreasing test and concavity test. It then provides an example of sketching a cubic function, showing the steps of finding critical points, inflection points, and putting together a sign chart to determine the function's monotonicity and concavity over intervals.
4. Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagona...Ceni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the fourth part which is discussing eigenvalues, eigenvectors and diagonalization.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
Here are the slides of the third part which is discussing factorization and linear transformations.
https://www.slideshare.net/CeniBabaogluPhDinMat/3-linear-algebra-for-machine-learning-factorization-and-linear-transformations-130813437
Lesson 2: A Catalog of Essential Functions (handout)Matthew Leingang
The document outlines topics to be covered in a Calculus I class, including essential functions such as linear, polynomial, rational, trigonometric, exponential, and logarithmic functions. It provides examples and definitions of these functions and notes how their graphs can be transformed through vertical and horizontal shifts. Assignments include WebAssign problems due January 31st and a written assignment due February 2nd.
This document outlines key concepts in linear models and estimation that will be covered in the STA721 Linear Models course, including:
1) Linear regression models decompose observed data into fixed and random components.
2) Maximum likelihood estimation finds parameter values that maximize the likelihood function.
3) Linear restrictions on the mean vector μ define a subspace and equivalent parameterizations represent the same subspace.
4) Inference should be independent of the parameterization or coordinate system used to represent μ.
This document discusses independent component analysis (ICA) for blind source separation. ICA is a method to estimate original signals from observed signals consisting of mixed original signals and noise. It introduces the ICA model and approach, including whitening, maximizing non-Gaussianity using kurtosis and negentropy, and fast ICA algorithms. The document provides examples applying ICA to separate images and discusses approaches to improve ICA, including using differential filtering. ICA is an important technique for blind source separation and independent component estimation from observed signals.
Lesson 15: Exponential Growth and Decay (Section 041 slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
Here is the solution again step-by-step:
We are given:
- At 3 hours, there are 10,000 bacteria
- At 5 hours, there are 40,000 bacteria
We know the model for bacterial growth is exponential: y' = ky
The general solution is: y = y0ekt
Setting up the equations:
- At 3 hours: 10,000 = y0ek(3)
- At 5 hours: 40,000 = y0ek(5)
Dividing the equations:
40,000/10,000 = ek(5)-k(3)
4 = e2k
Taking the ln of both sides:
ln
This document is from a Calculus I class at New York University. It provides an overview of functions, including the definition of a function, different ways functions can be represented (formulas, tables, graphs, verbal descriptions), properties of functions like monotonicity and symmetry, and examples of determining domains and ranges of functions. It aims to help students understand functions and their representations as a foundation for calculus.
Conditional trees use permutation tests and conditional inference at each step of recursive partitioning to overcome problems with traditional CART trees, such as selection bias and overfitting. The algorithm selects the variable with the strongest association using permutation tests, then searches for the best split point using a test statistic and permutation tests. It repeats this process recursively on the partitions until a stopping criterion is met where permutation tests show no variable has significant influence on the response. This approach aims to provide an unbiased and interpretable tree structure.
The document discusses methods for solving dynamic stochastic general equilibrium (DSGE) models. It outlines perturbation and projection methods for approximating the solution to DSGE models. Perturbation methods use Taylor series approximations around a steady state to derive linear approximations of the model. Projection methods find parametric functions that best satisfy the model equations. The document also provides an example of applying the implicit function theorem to derive a Taylor series approximation of a policy rule for a neoclassical growth model.
The document summarizes camera calibration techniques. It discusses:
1) Projecting 3D world points to 2D image points using a projection matrix with intrinsic and extrinsic parameters.
2) Computing camera parameters by estimating the projection matrix from known 3D points and corresponding 2D image points using linear and non-linear optimization methods.
3) Modeling lens distortion and different distortion types that must be accounted for during calibration.
The document discusses matrix factorization methods for solving systems of linear equations. It covers direct methods like LU, Cholesky, and QR factorizations that decompose a matrix into products of lower and upper triangular matrices. It also explains how to iteratively factorize a matrix A into A = LU by repeatedly subtracting outer products of rows/columns from submatrices. Examples are provided to demonstrate the factorization process.
Recently, there has been a surge in activity at the interface of optimal transport and statistics (with special emphasis on machine learning applications). The talk will summarize new results and challenges in this active area. For example, we will show how many of the most popular estimators in machine learning (such as Lasso and svm's) can be interpreted as games. This interpretation opens the door for new and potentially better estimators and algorithms, as well as questions about the underlying complexity of these new class of estimators.
(This talk is based on joint work with F. He, Y. Kang, K. Murthy, and F. Zhang)
CALCULUS:Ordinary Differential Equations,its history
Notation and Definitions,Solution methods,Order
Degree,Linearity,Homogeneity
Initial Value/Boundary value problems
Newton’s law of cooling, Heat balance equation (maths)
The document proposes a new fast algorithm for smooth non-negative matrix factorization (NMF) using function approximation. The algorithm uses function approximation to smooth the basis vectors, allowing for faster computation compared to existing methods. The method is extended to tensor decomposition models. Experimental results on image datasets show the proposed methods achieve better denoising and source separation performance compared to ordinary NMF and tensor decomposition methods, while being up to 300 times faster computationally. Future work includes extending the model to incorporate both common smoothness across factors and individual sparseness.
This document provides an overview of engineering risk analysis and event logic. It discusses how to define events, represent them using Venn diagrams and indicator variables, and perform logic operations like unions and intersections on events. Simple reliability models are presented, including series, parallel and 2-out-of-3 systems. Methods for analyzing events are covered, such as event trees, fault trees, cut sets, and minimal cut sets. Examples of initiating events in risk analysis are also listed.
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 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 the application of vector spaces and subspaces in biotechnology. It begins by introducing the importance of linear algebra in scientific and technological development. It then defines the objectives as understanding vector spaces, subspaces, and dimensionality. Examples of vector spaces and subspaces are provided. Applications include using these concepts to create classification methods for diseases, animals and plants. In conclusions, it is stated that these concepts facilitate study and development in biotechnology.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (handout)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.
This document provides an introduction to blind source separation and non-negative matrix factorization. It describes blind source separation as a method to estimate original signals from observed mixed signals. Non-negative matrix factorization is introduced as a constraint-based approach to solving blind source separation using non-negativity. The alternating least squares algorithm is described for solving the non-negative matrix factorization problem. Experiments applying these methods to artificial and real image data are presented and discussed.
The document provides information about an upcoming calculus exam, quiz, and lecture on curve sketching. It outlines the procedure for sketching curves, including using the increasing/decreasing test and concavity test. It then provides an example of sketching a cubic function, showing the steps of finding critical points, inflection points, and putting together a sign chart to determine the function's monotonicity and concavity over intervals.
4. Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagona...Ceni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the fourth part which is discussing eigenvalues, eigenvectors and diagonalization.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
Here are the slides of the third part which is discussing factorization and linear transformations.
https://www.slideshare.net/CeniBabaogluPhDinMat/3-linear-algebra-for-machine-learning-factorization-and-linear-transformations-130813437
Lesson 2: A Catalog of Essential Functions (handout)Matthew Leingang
The document outlines topics to be covered in a Calculus I class, including essential functions such as linear, polynomial, rational, trigonometric, exponential, and logarithmic functions. It provides examples and definitions of these functions and notes how their graphs can be transformed through vertical and horizontal shifts. Assignments include WebAssign problems due January 31st and a written assignment due February 2nd.
This document outlines key concepts in linear models and estimation that will be covered in the STA721 Linear Models course, including:
1) Linear regression models decompose observed data into fixed and random components.
2) Maximum likelihood estimation finds parameter values that maximize the likelihood function.
3) Linear restrictions on the mean vector μ define a subspace and equivalent parameterizations represent the same subspace.
4) Inference should be independent of the parameterization or coordinate system used to represent μ.
This document discusses independent component analysis (ICA) for blind source separation. ICA is a method to estimate original signals from observed signals consisting of mixed original signals and noise. It introduces the ICA model and approach, including whitening, maximizing non-Gaussianity using kurtosis and negentropy, and fast ICA algorithms. The document provides examples applying ICA to separate images and discusses approaches to improve ICA, including using differential filtering. ICA is an important technique for blind source separation and independent component estimation from observed signals.
Lesson 15: Exponential Growth and Decay (Section 041 slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
Here is the solution again step-by-step:
We are given:
- At 3 hours, there are 10,000 bacteria
- At 5 hours, there are 40,000 bacteria
We know the model for bacterial growth is exponential: y' = ky
The general solution is: y = y0ekt
Setting up the equations:
- At 3 hours: 10,000 = y0ek(3)
- At 5 hours: 40,000 = y0ek(5)
Dividing the equations:
40,000/10,000 = ek(5)-k(3)
4 = e2k
Taking the ln of both sides:
ln
This document is from a Calculus I class at New York University. It provides an overview of functions, including the definition of a function, different ways functions can be represented (formulas, tables, graphs, verbal descriptions), properties of functions like monotonicity and symmetry, and examples of determining domains and ranges of functions. It aims to help students understand functions and their representations as a foundation for calculus.
Conditional trees use permutation tests and conditional inference at each step of recursive partitioning to overcome problems with traditional CART trees, such as selection bias and overfitting. The algorithm selects the variable with the strongest association using permutation tests, then searches for the best split point using a test statistic and permutation tests. It repeats this process recursively on the partitions until a stopping criterion is met where permutation tests show no variable has significant influence on the response. This approach aims to provide an unbiased and interpretable tree structure.
The document discusses methods for solving dynamic stochastic general equilibrium (DSGE) models. It outlines perturbation and projection methods for approximating the solution to DSGE models. Perturbation methods use Taylor series approximations around a steady state to derive linear approximations of the model. Projection methods find parametric functions that best satisfy the model equations. The document also provides an example of applying the implicit function theorem to derive a Taylor series approximation of a policy rule for a neoclassical growth model.
The document summarizes camera calibration techniques. It discusses:
1) Projecting 3D world points to 2D image points using a projection matrix with intrinsic and extrinsic parameters.
2) Computing camera parameters by estimating the projection matrix from known 3D points and corresponding 2D image points using linear and non-linear optimization methods.
3) Modeling lens distortion and different distortion types that must be accounted for during calibration.
The document discusses matrix factorization methods for solving systems of linear equations. It covers direct methods like LU, Cholesky, and QR factorizations that decompose a matrix into products of lower and upper triangular matrices. It also explains how to iteratively factorize a matrix A into A = LU by repeatedly subtracting outer products of rows/columns from submatrices. Examples are provided to demonstrate the factorization process.
Recently, there has been a surge in activity at the interface of optimal transport and statistics (with special emphasis on machine learning applications). The talk will summarize new results and challenges in this active area. For example, we will show how many of the most popular estimators in machine learning (such as Lasso and svm's) can be interpreted as games. This interpretation opens the door for new and potentially better estimators and algorithms, as well as questions about the underlying complexity of these new class of estimators.
(This talk is based on joint work with F. He, Y. Kang, K. Murthy, and F. Zhang)
CALCULUS:Ordinary Differential Equations,its history
Notation and Definitions,Solution methods,Order
Degree,Linearity,Homogeneity
Initial Value/Boundary value problems
Newton’s law of cooling, Heat balance equation (maths)
The document proposes a new fast algorithm for smooth non-negative matrix factorization (NMF) using function approximation. The algorithm uses function approximation to smooth the basis vectors, allowing for faster computation compared to existing methods. The method is extended to tensor decomposition models. Experimental results on image datasets show the proposed methods achieve better denoising and source separation performance compared to ordinary NMF and tensor decomposition methods, while being up to 300 times faster computationally. Future work includes extending the model to incorporate both common smoothness across factors and individual sparseness.
This document provides an overview of engineering risk analysis and event logic. It discusses how to define events, represent them using Venn diagrams and indicator variables, and perform logic operations like unions and intersections on events. Simple reliability models are presented, including series, parallel and 2-out-of-3 systems. Methods for analyzing events are covered, such as event trees, fault trees, cut sets, and minimal cut sets. Examples of initiating events in risk analysis are also listed.
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 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.
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.
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 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 15: Exponential Growth and Decay (Section 041 slides)Mel Anthony Pepito
The document is notes from a Calculus I class covering exponential growth and decay. It discusses solving differential equations of the form y' = ky, with applications to population growth, radioactive decay, cooling, and interest. It provides examples of solving equations for various growth rates k, and uses an example of bacterial population growth over time to find the initial population from given later populations.
This document is from an NYU Calculus I class and outlines the topics of exponential growth and decay that will be covered in Section 3.4. It includes announcements about an upcoming quiz, objectives to solve differential equations involving exponential functions, and an outline of topics like modeling population growth, radioactive decay, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving simple differential equations and finding general solutions that involve exponential and logarithmic functions.
Lesson 15: Exponential Growth and Decay (Section 021 slides)Mel Anthony Pepito
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
No, this is not a function because the same input of 2 is mapped to two different outputs of 4 and 5. For a relation to be a function, each input must map to a unique output.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 17 / 33
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 041 ...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 ...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.
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.
International Journal of Computational Engineering Research(IJCER)ijceronline
This document discusses solving matrix games when the payoff values are intuitionistic fuzzy numbers rather than precise values. It begins by introducing intuitionistic fuzzy sets and intuitionistic fuzzy numbers. It then defines various score functions that can be used to defuzzify or obtain a crisp value from an intuitionistic fuzzy number, including the linearizing score function. The document goes on to define an intuitionistic matrix game where the payoff values are intuitionistic fuzzy numbers rather than precise values. It proposes using a score function approach to solve the game by obtaining defuzzified payoff values.
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 is from a Calculus I class at New York University and covers inverse trigonometric functions. It begins with announcements about midterm grades and an upcoming quiz. The objectives are to learn the definitions, domains, ranges and derivatives of inverse trig functions such as arcsin, arccos, arctan, arcsec and arccsc. Examples are provided to demonstrate calculating values of these inverse functions.
Similar to Lesson 13: Exponential and Logarithmic Functions (Section 041 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.
GraphSummit Singapore | The Art of the Possible with Graph - Q2 2024Neo4j
Neha Bajwa, Vice President of Product Marketing, Neo4j
Join us as we explore breakthrough innovations enabled by interconnected data and AI. Discover firsthand how organizations use relationships in data to uncover contextual insights and solve our most pressing challenges – from optimizing supply chains, detecting fraud, and improving customer experiences to accelerating drug discoveries.
“An Outlook of the Ongoing and Future Relationship between Blockchain Technologies and Process-aware Information Systems.” Invited talk at the joint workshop on Blockchain for Information Systems (BC4IS) and Blockchain for Trusted Data Sharing (B4TDS), co-located with with the 36th International Conference on Advanced Information Systems Engineering (CAiSE), 3 June 2024, Limassol, Cyprus.
Removing Uninteresting Bytes in Software FuzzingAftab Hussain
Imagine a world where software fuzzing, the process of mutating bytes in test seeds to uncover hidden and erroneous program behaviors, becomes faster and more effective. A lot depends on the initial seeds, which can significantly dictate the trajectory of a fuzzing campaign, particularly in terms of how long it takes to uncover interesting behaviour in your code. We introduce DIAR, a technique designed to speedup fuzzing campaigns by pinpointing and eliminating those uninteresting bytes in the seeds. Picture this: instead of wasting valuable resources on meaningless mutations in large, bloated seeds, DIAR removes the unnecessary bytes, streamlining the entire process.
In this work, we equipped AFL, a popular fuzzer, with DIAR and examined two critical Linux libraries -- Libxml's xmllint, a tool for parsing xml documents, and Binutil's readelf, an essential debugging and security analysis command-line tool used to display detailed information about ELF (Executable and Linkable Format). Our preliminary results show that AFL+DIAR does not only discover new paths more quickly but also achieves higher coverage overall. This work thus showcases how starting with lean and optimized seeds can lead to faster, more comprehensive fuzzing campaigns -- and DIAR helps you find such seeds.
- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
DevOps and Testing slides at DASA ConnectKari Kakkonen
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zkStudyClub - Reef: Fast Succinct Non-Interactive Zero-Knowledge Regex ProofsAlex Pruden
This paper presents Reef, a system for generating publicly verifiable succinct non-interactive zero-knowledge proofs that a committed document matches or does not match a regular expression. We describe applications such as proving the strength of passwords, the provenance of email despite redactions, the validity of oblivious DNS queries, and the existence of mutations in DNA. Reef supports the Perl Compatible Regular Expression syntax, including wildcards, alternation, ranges, capture groups, Kleene star, negations, and lookarounds. Reef introduces a new type of automata, Skipping Alternating Finite Automata (SAFA), that skips irrelevant parts of a document when producing proofs without undermining soundness, and instantiates SAFA with a lookup argument. Our experimental evaluation confirms that Reef can generate proofs for documents with 32M characters; the proofs are small and cheap to verify (under a second).
Paper: https://eprint.iacr.org/2023/1886
UiPath Test Automation using UiPath Test Suite series, part 6DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 6. In this session, we will cover Test Automation with generative AI and Open AI.
UiPath Test Automation with generative AI and Open AI webinar offers an in-depth exploration of leveraging cutting-edge technologies for test automation within the UiPath platform. Attendees will delve into the integration of generative AI, a test automation solution, with Open AI advanced natural language processing capabilities.
Throughout the session, participants will discover how this synergy empowers testers to automate repetitive tasks, enhance testing accuracy, and expedite the software testing life cycle. Topics covered include the seamless integration process, practical use cases, and the benefits of harnessing AI-driven automation for UiPath testing initiatives. By attending this webinar, testers, and automation professionals can gain valuable insights into harnessing the power of AI to optimize their test automation workflows within the UiPath ecosystem, ultimately driving efficiency and quality in software development processes.
What will you get from this session?
1. Insights into integrating generative AI.
2. Understanding how this integration enhances test automation within the UiPath platform
3. Practical demonstrations
4. Exploration of real-world use cases illustrating the benefits of AI-driven test automation for UiPath
Topics covered:
What is generative AI
Test Automation with generative AI and Open AI.
UiPath integration with generative AI
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
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Mike Del Balso, CEO & Co-Founder at Tecton, presents "Full RAG," a novel approach to AI recommendation systems, aiming to push beyond the limitations of traditional models through a deep integration of contextual insights and real-time data, leveraging the Retrieval-Augmented Generation architecture. This talk will outline Full RAG's potential to significantly enhance personalization, address engineering challenges such as data management and model training, and introduce data enrichment with reranking as a key solution. Attendees will gain crucial insights into the importance of hyperpersonalization in AI, the capabilities of Full RAG for advanced personalization, and strategies for managing complex data integrations for deploying cutting-edge AI solutions.
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In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
Threats to mobile devices are more prevalent and increasing in scope and complexity. Users of mobile devices desire to take full advantage of the features
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Rapid and secure feature delivery is a goal across every application team and every branch of the DoD. The Navy’s DevSecOps platform, Party Barge, has achieved:
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Development teams can ship efficiently and ensure applications are cyber ready for Navy Authorizing Officials (AOs). In this webinar, Sigma Defense and Anchore will give attendees a look behind the scenes and demo secure pipeline automation and security artifacts that speed up application ATO and time to production.
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- How to remove silos in DevSecOps
- How to build efficient development pipeline roles and component templates
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Pakdata Cf is a groundbreaking system designed to streamline and facilitate access to CNIC information. This innovative platform leverages advanced technology to provide users with efficient and secure access to their CNIC details.
In his public lecture, Christian Timmerer provides insights into the fascinating history of video streaming, starting from its humble beginnings before YouTube to the groundbreaking technologies that now dominate platforms like Netflix and ORF ON. Timmerer also presents provocative contributions of his own that have significantly influenced the industry. He concludes by looking at future challenges and invites the audience to join in a discussion.
UiPath Test Automation using UiPath Test Suite series, part 5DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 5. In this session, we will cover CI/CD with devops.
Topics covered:
CI/CD with in UiPath
End-to-end overview of CI/CD pipeline with Azure devops
Speaker:
Lyndsey Byblow, Test Suite Sales Engineer @ UiPath, Inc.
Elizabeth Buie - Older adults: Are we really designing for our future selves?
Lesson 13: Exponential and Logarithmic Functions (Section 041 slides)
1. Sections 3.1–3.2
Exponential and Logarithmic Functions
V63.0121.041, Calculus I
New York University
October 20, 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 2 / 37
3. 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 3 / 37
4. 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
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 4 / 37
5. 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
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 5 / 37
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
Examples
23 = 2 · 2 · 2 = 8
34 = 3 · 3 · 3 · 3 = 81
(−1)5 = (−1)(−1)(−1)(−1)(−1) = −1
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 5 / 37
7. 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 6 / 37
8. Fact
If a is a real number, then
ax+y = ax ay (sums to products)
ax
ax−y = y
a
(ax )y = axy
(ab)x = ax bx
whenever all exponents are positive whole numbers.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 7 / 37
9. Fact
If a is a real number, then
ax+y = ax ay (sums to products)
ax
ax−y = y (differences to quotients)
a
(ax )y = axy
(ab)x = ax bx
whenever all exponents are positive whole numbers.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 7 / 37
10. Fact
If a is a real number, then
ax+y = ax ay (sums to products)
ax
ax−y = y (differences to quotients)
a
(ax )y = axy (repeated exponentiation to multiplied powers)
(ab)x = ax bx
whenever all exponents are positive whole numbers.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 7 / 37
11. Fact
If a is a real number, then
ax+y = ax ay (sums to products)
ax
ax−y = y (differences to quotients)
a
(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.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 7 / 37
12. Fact
If a is a real number, then
ax+y = ax ay (sums to products)
ax
ax−y = y (differences to quotients)
a
(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.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 7 / 37
13. Fact
If a is a real number, then
ax+y = ax ay (sums to products)
ax
ax−y = y (differences to quotients)
a
(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 = a · a · · · · · a · a · a · · · · · a = ax ay
x + y factors x factors y factors
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 7 / 37
14. Let's be conventional
The desire that these properties remain true gives us conventions
for ax when x is not a positive whole number.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 8 / 37
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.
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
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 8 / 37
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 = an+0 = an · a0 =⇒ a0 = =1
an
(The equality with the exclamation point is what we want.)
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 8 / 37
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 = an+0 = an · a0 =⇒ a0 = =1
an
(The equality with the exclamation point is what we want.)
Definition
If a ̸= 0, we define a0 = 1.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 8 / 37
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 = an+0 = an · a0 =⇒ a0 = =1
an
(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).
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 8 / 37
19. Conventions for negative exponents
If n ≥ 0, we want
an+(−n) = an · a−n
!
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 9 / 37
20. Conventions for negative exponents
If n ≥ 0, we want
a0 1
an+(−n) = an · a−n =⇒ a−n =
! !
n
= n
a a
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 9 / 37
21. Conventions for negative exponents
If n ≥ 0, we want
a0 1
an+(−n) = an · a−n =⇒ a−n =
! !
n
= n
a a
Definition
1
If n is a positive integer, we define a−n = .
an
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 9 / 37
22. Conventions for negative exponents
If n ≥ 0, we want
a0 1
an+(−n) = an · a−n =⇒ a−n =
! !
n
= n
a a
Definition
1
If n is a positive integer, we define a−n = .
an
Fact
1
The convention that a−n = “works” for negative n as well.
an
am
If m and n are any integers, then am−n = n .
a
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 9 / 37
23. Conventions for fractional exponents
If q is a positive integer, we want
!
(a1/q )q = a1 = a
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 10 / 37
24. Conventions for fractional exponents
If q is a positive integer, we want
! ! √
(a1/q )q = a1 = a =⇒ a1/q = q
a
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 10 / 37
25. 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 10 / 37
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.
√q
( √ )p
Notice that ap = q a . So we can unambiguously say
ap/q = (ap )1/q = (a1/q )p
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 10 / 37
27. Conventions for irrational exponents
So ax is well-defined if a is positive and x is rational.
What about irrational powers?
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 11 / 37
28. 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 ar
r→x
r rational
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 11 / 37
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 ar
r→x
r rational
In other words, to approximate ax for irrational x, take r close to x but
rational and compute ar .
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 11 / 37
30. Approximating a power with an irrational exponent
r 2r
3 23
√ =8
10
3.1 231/10 = √ 31 ≈ 8.57419
2
100
3.14 2314/100 = √2314 ≈ 8.81524
1000
3.141 23141/1000 = 23141 ≈ 8.82135
The limit (numerically approximated is)
2π ≈ 8.82498
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 12 / 37
31. Graphs of various exponential functions
y
.
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
32. Graphs of various exponential functions
y
.
. = 1x
y
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
33. Graphs of various exponential functions
y
.
. = 2x
y
. = 1x
y
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
34. Graphs of various exponential functions
y
.
. = 3x. = 2x
y y
. = 1x
y
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
35. Graphs of various exponential functions
y
.
. = 10x= 3x. = 2x
y y
. y
. = 1x
y
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
36. Graphs of various exponential functions
y
.
. = 10x= 3x. = 2x
y y
. y . = 1.5x
y
. = 1x
y
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
37. Graphs of various exponential functions
y
.
. = (1/2)x
y . = 10x= 3x. = 2x
y y
. y . = 1.5x
y
. = 1x
y
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
38. Graphs of various exponential functions
x
y
.
. = (1/2)x (1/3)
y y
. = . = 10x= 3x. = 2x
y y
. y . = 1.5x
y
. = 1x
y
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
39. Graphs of various exponential functions
y
.
. = (1/2)x (1/3)
y y
. = x
. = (1/10)x. = 10x= 3x. = 2x
y y y
. y . = 1.5x
y
. = 1x
y
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
40. Graphs of various exponential functions
y
.
y yx
.. = ((1/2)x (1/3)x
y = 2/. )=
3 . = (1/10)x. = 10x= 3x. = 2x
y y y
. y . = 1.5x
y
. = 1x
y
. x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
41. 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
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 14 / 37
42. 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
ax−y = y
a
(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
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 15 / 37
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
ax−y = y (negative exponents mean reciprocals)
a
(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
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 15 / 37
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
ax−y = y (negative exponents mean reciprocals)
a
(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
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 15 / 37
50. Limits of exponential functions
Fact (Limits of exponential y
.
functions) . = (= 2()1/32/3)x
y . 1/ =x( )x
y .
y y y = x . 3x y
. = (. /10)10x= 2x. =
1 . =
y y
If a > 1, then lim ax = ∞
x→∞
and lim ax = 0
x→−∞
If 0 < a < 1, then
lim ax = 0 and y
. =
x→∞
lim a = ∞ x . x
.
x→−∞
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 17 / 37
51. 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
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 18 / 37
52. 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?
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 19 / 37
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?
Answer
$100 + 10% = $110
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 19 / 37
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
$110 + 10% = $110 + $11 = $121
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 19 / 37
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
$100(1.1)t .
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 19 / 37
56. 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?
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 20 / 37
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?
Answer
$100(1.025)4 = $110.38,
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 20 / 37
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, not $100(1.1)4 !
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 20 / 37
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 !
$100(1.025)8 = $121.84
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 20 / 37
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
$100(1.025)4t .
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 20 / 37
61. 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?
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 21 / 37
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?
Answer
$100(1 + 10%/12)12t
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 21 / 37
63. 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?
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 22 / 37
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?
Answer
( r )nt
B(t) = P 1 +
n
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 22 / 37
65. 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?
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 23 / 37
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?
Answer
( ( )
r )nt 1 rnt
B(t) = lim P 1 + = lim P 1 +
n→∞ n n→∞ n
[ ( )n ]rt
1
=P lim 1 +
n→∞ n
independent of P, r, or t
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 23 / 37
67. The magic number
Definition
( )
1 n
e = lim 1 +
n→∞ n
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 24 / 37
68. The magic number
Definition
( )
1 n
e = lim 1 +
n→∞ n
So now continuously-compounded interest can be expressed as
B(t) = Pert .
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 24 / 37
69. Existence of e
See Appendix B
( )
1 n
n 1+
n
1 2
2 2.25
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
70. Existence of e
See Appendix B
( )
1 n
n 1+
n
1 2
2 2.25
3 2.37037
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
71. Existence of e
See Appendix B
( )
1 n
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
72. Existence of e
See Appendix B
( )
1 n
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
73. Existence of e
See Appendix B
( )
1 n
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
1000 2.71692
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
74. Existence of e
See Appendix B
( )
1 n
n 1+
n
1 2
2 2.25
3 2.37037
10 2.59374
100 2.70481
1000 2.71692
106 2.71828
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
75. Existence of e
See Appendix B
( )
1 n
n 1+
We can experimentally n
verify that this number 1 2
exists and is 2 2.25
3 2.37037
e ≈ 2.718281828459045 . . .
10 2.59374
100 2.70481
1000 2.71692
106 2.71828
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
76. Existence of e
See Appendix B
( )
1 n
n 1+
We can experimentally n
verify that this number 1 2
exists and is 2 2.25
3 2.37037
e ≈ 2.718281828459045 . . .
10 2.59374
e is irrational 100 2.70481
1000 2.71692
106 2.71828
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
77. Existence of e
See Appendix B
( )
1 n
n 1+
We can experimentally n
verify that this number 1 2
exists and is 2 2.25
3 2.37037
e ≈ 2.718281828459045 . . .
10 2.59374
e is irrational 100 2.70481
1000 2.71692
e is transcendental
106 2.71828
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
78. 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
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 26 / 37
79. A limit
.
Question
eh − 1
What is lim ?
h→0 h
. . . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 27 / 37
80. A limit
.
Question
eh − 1
What is lim ?
h→0 h
Answer
e = lim → ∞ (1 + 1/n)n = lim (1 + h)1/h . So for a small h, e ≈ (1 + h)1/h .
n h→0
So [ ]h
eh − 1 (1 + h)1/h − 1
≈ =1
h h
. . . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 27 / 37
81. A limit
.
Question
eh − 1
What is lim ?
h→0 h
Answer
e = lim → ∞ (1 + 1/n)n = lim (1 + h)1/h . So for a small h, e ≈ (1 + h)1/h .
n h→0
So [ ]h
eh − 1 (1 + h)1/h − 1
≈ =1
h h
eh − 1
It follows that lim = 1.
h→0 h
2h − 1
This can be used to characterize e: lim = 0.693 · · · < 1 and
h→0 h
3h − 1
lim = 1.099 · · · > 1
h→0 h
. . . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 27 / 37
82. 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
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 28 / 37
83. 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 .
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 29 / 37
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 .
Facts
(i) loga (x1 · x2 ) = loga x1 + loga x2
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 29 / 37
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
( )
x1
(ii) loga = loga x1 − loga x2
x2
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 29 / 37
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
( )
x1
(ii) loga = loga x1 − loga x2
x2
(iii) loga (xr ) = r loga x
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 29 / 37
87. Logarithms convert products to sums
Suppose y1 = loga x1 and y2 = loga x2
Then x1 = ay1 and x2 = ay2
So x1 x2 = ay1 ay2 = ay1 +y2
Therefore
loga (x1 · x2 ) = loga x1 + loga x2
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 30 / 37
88. Example
Write as a single logarithm: 2 ln 4 − ln 3.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 31 / 37
89. Example
Write as a single logarithm: 2 ln 4 − ln 3.
Solution
42
2 ln 4 − ln 3 = ln 42 − ln 3 = ln
3
ln 42
not !
ln 3
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 31 / 37
90. Example
Write as a single logarithm: 2 ln 4 − ln 3.
Solution
42
2 ln 4 − ln 3 = ln 42 − ln 3 = ln
3
ln 42
not !
ln 3
Example
3
Write as a single logarithm: ln + 4 ln 2
4
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 31 / 37
91. Example
Write as a single logarithm: 2 ln 4 − ln 3.
Solution
42
2 ln 4 − ln 3 = ln 42 − ln 3 = ln
3
ln 42
not !
ln 3
Example
3
Write as a single logarithm: ln + 4 ln 2
4
Answer
ln 12
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 31 / 37
92. Graphs of logarithmic functions
y
.
. = 2x
y
y
. = log2 x
. . 0, 1)
(
..1, 0) .
( x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 32 / 37
93. Graphs of logarithmic functions
y
.
. = 3x= 2x
y . y
y
. = log2 x
y
. = log3 x
. . 0, 1)
(
..1, 0) .
( x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 32 / 37
94. Graphs of logarithmic functions
y
.
. = .10x 3x= 2x
y y= . y
y
. = log2 x
y
. = log3 x
. . 0, 1)
(
y
. = log10 x
..1, 0) .
( x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 32 / 37
95. Graphs of logarithmic functions
y
.
. = .10=3x= 2x
y xy
y y. = .ex
y
. = log2 x
y
. = ln x
y
. = log3 x
. . 0, 1)
(
y
. = log10 x
..1, 0) .
( x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 32 / 37
96. Change of base formula for exponentials
Fact
If a > 0 and a ̸= 1, then
ln x
loga x =
ln a
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 33 / 37
97. Change of base formula for exponentials
Fact
If a > 0 and a ̸= 1, then
ln x
loga x =
ln a
Proof.
If y = loga x, then x = ay
So ln x = ln(ay ) = y ln a
Therefore
ln x
y = loga x =
ln a
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 33 / 37
98. Example of changing base
Example
Find log2 8 by using log10 only.
Surprised?
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 34 / 37
99. Example of changing base
Example
Find log2 8 by using log10 only.
Solution
log10 8 0.90309
log2 8 = ≈ =3
log10 2 0.30103
Surprised?
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 34 / 37
100. Example of changing base
Example
Find log2 8 by using log10 only.
Solution
log10 8 0.90309
log2 8 = ≈ =3
log10 2 0.30103
Surprised? No, log2 8 = log2 23 = 3 directly.
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 34 / 37
101. Upshot of changing base
The point of the change of base formula
logb x 1
loga x = = · logb x = constant · logb x
logb a logb a
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
. . . . . .
V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 35 / 37