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 (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.
This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
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
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.
This document contains notes from a Calculus I class at New York University. It discusses related rates problems, which involve taking derivatives of equations relating changing quantities to determine rates of change. The document provides examples of related rates problems involving an oil slick, two people walking towards and away from each other, and electrical resistors. It also outlines strategies for solving related rates problems, such as drawing diagrams, introducing notation, relating quantities with equations, and using the chain rule to solve for unknown rates.
When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
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 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 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.
This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
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
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.
This document contains notes from a Calculus I class at New York University. It discusses related rates problems, which involve taking derivatives of equations relating changing quantities to determine rates of change. The document provides examples of related rates problems involving an oil slick, two people walking towards and away from each other, and electrical resistors. It also outlines strategies for solving related rates problems, such as drawing diagrams, introducing notation, relating quantities with equations, and using the chain rule to solve for unknown rates.
When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
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 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 (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 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 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.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trig functions like arcsin, arccos, and arctan and discusses their domains, ranges, and relationships to the original trig functions. It also provides examples of evaluating inverse trig functions at specific values.
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.
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.
The document discusses exponential growth and decay models in calculus. It covers modeling population growth, radioactive decay using carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided to illustrate exponential growth of bacteria populations using the differential equation y' = ky, and modeling radioactive decay where the relative rate of decay is constant and represented by the differential equation y' = -ky.
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 is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. Examples are provided of finding the antiderivative of power functions like x^3 through identifying the power rule relationship between a function and its derivative.
This document outlines the key rules for differentiation that will be covered in Calculus I class. It introduces the objectives of understanding derivatives of constant functions, the constant multiple rule, sum and difference rules, and derivatives of sine and cosine. It then provides examples of finding the derivatives of squaring and cubing functions using the definition of a derivative. Finally, it discusses properties of the derivatives of these functions.
The document provides an example of using the substitution method to evaluate the indefinite integral ∫(x2 + 3)3 4x dx. It introduces the substitution u = x2 + 3, which allows the integral to be rewritten as ∫u3 2 du and then evaluated as (1/2)u4 = (1/2)(x2 + 3)4. The solution is compared to directly integrating the expanded polynomial. The document outlines the theory and notation of substitution for indefinite integrals.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
The document is about implicit differentiation and contains the following:
- It introduces implicit differentiation using the example of finding the slope of the curve x^2 + y^2 = 1 at the point (3/5, -4/5).
- It shows solving this problem explicitly by isolating y and taking the derivative, as well as implicitly by treating y as a function f(x) and differentiating the equation x^2 + f(x)^2 = 1.
- The objectives, outline, and motivation for implicit differentiation are provided to set up the key concepts covered in the section.
Martin Luther King Jr. fought for civil rights and racial equality in the United States during the 1950s and 1960s. He believed that all people should be treated equally, regardless of race. Through peaceful protests, speeches, and activism, MLK helped change discriminatory laws that had prevented black Americans from having equal rights and access to public services and spaces. His iconic "I Have a Dream" speech was influential in the passage of the Civil Rights Act of 1964. MLK worked to make America a more just and fair society through nonviolent means.
The organization Latinos Progresando was founded in 1998 and is located in Little Village, Chicago, which has a large Latino population. They provide legal services and other programs. The document discusses Deferred Action for Childhood Arrivals, which provides relief from deportation for those who entered the US as children. It outlines the 5 eligibility criteria. The research aims to identify Latino communities that could benefit from information on deferred action from Latinos Progresando. Data was collected from demographic charts, schools, churches and community centers.
This document outlines questions for a website tutorial, including what the client needs from the website, what consumers need, and the intended message for consumers. Specifically, it asks about the client's web design experience and time available, and suggests considering the visual design, structure, and findability of information for consumers.
The document announces birthdays, an upcoming college representative visit, and provides information about Stephen F. Austin State University and the University of Oklahoma. It also includes a writing prompt asking students to describe their ideal college and explain why they would want to attend.
Presented to "Managing the Material: Tackling Visual Arts as Research Data" workshop, organised by Visual Arts Data Service (VADS) in conjunction with the Digital Curation Centre (DCC), through the JISC-funded KAPTUR project. London, 14 September 2012
Disco music of the 1970s had a signature sound with a tempo of 120 BPM. It featured prominent use of drum machines, electric guitars, bass guitars, horn sections, violins, and vocalists. Popular dance songs of this era like "The Hustle" and "Blame it on the Boogie" showcased these instruments and emerging recording technologies that helped define the disco genre.
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.
Lesson 15: Exponential Growth and Decay (Section 021 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.
This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
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 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 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.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trig functions like arcsin, arccos, and arctan and discusses their domains, ranges, and relationships to the original trig functions. It also provides examples of evaluating inverse trig functions at specific values.
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.
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.
The document discusses exponential growth and decay models in calculus. It covers modeling population growth, radioactive decay using carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided to illustrate exponential growth of bacteria populations using the differential equation y' = ky, and modeling radioactive decay where the relative rate of decay is constant and represented by the differential equation y' = -ky.
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 is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. Examples are provided of finding the antiderivative of power functions like x^3 through identifying the power rule relationship between a function and its derivative.
This document outlines the key rules for differentiation that will be covered in Calculus I class. It introduces the objectives of understanding derivatives of constant functions, the constant multiple rule, sum and difference rules, and derivatives of sine and cosine. It then provides examples of finding the derivatives of squaring and cubing functions using the definition of a derivative. Finally, it discusses properties of the derivatives of these functions.
The document provides an example of using the substitution method to evaluate the indefinite integral ∫(x2 + 3)3 4x dx. It introduces the substitution u = x2 + 3, which allows the integral to be rewritten as ∫u3 2 du and then evaluated as (1/2)u4 = (1/2)(x2 + 3)4. The solution is compared to directly integrating the expanded polynomial. The document outlines the theory and notation of substitution for indefinite integrals.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
The document is about implicit differentiation and contains the following:
- It introduces implicit differentiation using the example of finding the slope of the curve x^2 + y^2 = 1 at the point (3/5, -4/5).
- It shows solving this problem explicitly by isolating y and taking the derivative, as well as implicitly by treating y as a function f(x) and differentiating the equation x^2 + f(x)^2 = 1.
- The objectives, outline, and motivation for implicit differentiation are provided to set up the key concepts covered in the section.
Martin Luther King Jr. fought for civil rights and racial equality in the United States during the 1950s and 1960s. He believed that all people should be treated equally, regardless of race. Through peaceful protests, speeches, and activism, MLK helped change discriminatory laws that had prevented black Americans from having equal rights and access to public services and spaces. His iconic "I Have a Dream" speech was influential in the passage of the Civil Rights Act of 1964. MLK worked to make America a more just and fair society through nonviolent means.
The organization Latinos Progresando was founded in 1998 and is located in Little Village, Chicago, which has a large Latino population. They provide legal services and other programs. The document discusses Deferred Action for Childhood Arrivals, which provides relief from deportation for those who entered the US as children. It outlines the 5 eligibility criteria. The research aims to identify Latino communities that could benefit from information on deferred action from Latinos Progresando. Data was collected from demographic charts, schools, churches and community centers.
This document outlines questions for a website tutorial, including what the client needs from the website, what consumers need, and the intended message for consumers. Specifically, it asks about the client's web design experience and time available, and suggests considering the visual design, structure, and findability of information for consumers.
The document announces birthdays, an upcoming college representative visit, and provides information about Stephen F. Austin State University and the University of Oklahoma. It also includes a writing prompt asking students to describe their ideal college and explain why they would want to attend.
Presented to "Managing the Material: Tackling Visual Arts as Research Data" workshop, organised by Visual Arts Data Service (VADS) in conjunction with the Digital Curation Centre (DCC), through the JISC-funded KAPTUR project. London, 14 September 2012
Disco music of the 1970s had a signature sound with a tempo of 120 BPM. It featured prominent use of drum machines, electric guitars, bass guitars, horn sections, violins, and vocalists. Popular dance songs of this era like "The Hustle" and "Blame it on the Boogie" showcased these instruments and emerging recording technologies that helped define the disco genre.
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.
Lesson 15: Exponential Growth and Decay (Section 021 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.
This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
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 15: Exponential Growth and Decay (Section 041 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.
Lesson 15: Exponential Growth and Decay (Section 021 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 is a lecture note on derivatives and the shapes of curves. It discusses the mean value theorem and its applications to determining monotonicity and concavity of functions. Specifically, it covers:
- Using the first derivative test to find intervals where a function is increasing or decreasing by determining where the derivative is positive or negative
- Examples of applying this process to functions like x2 - 1 and x2/3(x + 2)
- Definitions of increasing, decreasing, and concavity
- How the second derivative test can determine concavity by examining the sign of the second derivative
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This is the slideshow version from class.
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.
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.
This document is a section from a Calculus I course at New York University that discusses basic differentiation rules. It provides objectives for understanding rules like the derivative of a constant function, the constant multiple rule, sum rule, and derivatives of sine and cosine functions. It then gives examples of finding the derivative of a squaring function using the definition, and introduces the concept of the second derivative.
This document is a section from a Calculus I course at New York University that discusses basic differentiation rules. It covers objectives like differentiating constant, sum, and difference functions. It also reviews derivatives of sine and cosine. Examples are provided, like finding the derivative of a squaring function x^2 using the definition of a derivative. The document outlines the topics and provides explanations and step-by-step solutions.
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.
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 is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. The document provides examples of finding antiderivatives of power functions by using the power rule in reverse.
This document contains lecture notes on differentiation rules from a Calculus I class at New York University. It begins with objectives to understand basic differentiation rules like the derivative of a constant function, the Constant Multiple Rule, the Sum Rule, and derivatives of sine and cosine. It then provides examples of using the definition of the derivative to find the derivatives of squaring and cubing functions. It illustrates the functions and their derivatives on graphs and discusses properties like a function being increasing when its derivative is positive.
Similar to Lesson15 -exponential_growth_and_decay_021_slides (20)
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trigonometric functions as the inverses of restricted trigonometric functions, gives their domains and ranges, and discusses their derivatives. The document also provides examples of evaluating inverse trigonometric functions.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
This document summarizes a calculus lecture on linear approximations. It provides examples of using the tangent line to approximate the sine function at different points. Specifically, it estimates sin(61°) by taking linear approximations about 0 and about 60°. The linear approximation about 0 is x, giving a value of 1.06465. The linear approximation about 60° uses the fact that the sine is √3/2 and the derivative is √3/2 at π/3, giving a better approximation than using 0.
This document contains lecture notes on related rates from a Calculus I class at New York University. It begins with announcements about assignments and no class on a holiday. It then outlines the objectives of learning to use derivatives to understand rates of change and model word problems. Examples are provided, including an oil slick problem worked out in detail. Strategies for solving related rates problems are discussed. Further examples on people walking and electrical resistors are presented.
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.
This document is from a Calculus I class lecture on L'Hopital's rule given at New York University. It begins with announcements and objectives for the lecture. It then provides examples of limits that are indeterminate forms to motivate L'Hopital's rule. The document explains the rule and when it can be used to evaluate limits. It also introduces the mathematician L'Hopital and applies the rule to examples introduced earlier.
The document provides an overview of curve sketching in calculus including objectives, rationale, theorems for determining monotonicity and concavity, and a checklist for graphing functions. It then gives examples of graphing cubic and quartic functions step-by-step, demonstrating how to analyze critical points, inflection points, and asymptotic behavior to create the curve. The examples illustrate applying differentiation rules to determine monotonicity from the derivative sign chart and concavity from the second derivative test.
This document is a section from a Calculus I course at New York University covering maximum and minimum values. It begins with announcements about exams and assignments. The objectives are to understand the Extreme Value Theorem and Fermat's Theorem, and use the Closed Interval Method to find extreme values. The document then covers the definitions of extreme points/values and the statements of the Extreme Value Theorem and Fermat's Theorem. It provides examples to illustrate the necessity of the hypotheses in the theorems. The focus is on using calculus concepts like continuity and differentiability to determine maximum and minimum values of functions on closed intervals.
This document is the notes from a Calculus I class at New York University covering Section 4.2 on the Mean Value Theorem. The notes include objectives, an outline, explanations of Rolle's Theorem and the Mean Value Theorem, examples of using the theorems, and a food for thought question. The key points are that Rolle's Theorem states that if a function is continuous on an interval and differentiable inside the interval, and the function values at the endpoints are equal, then there exists a point in the interior where the derivative is 0. The Mean Value Theorem similarly states that if a function is continuous on an interval and differentiable inside, there exists a point where the average rate of change equals the instantaneous rate of
This document discusses the definite integral and its properties. It begins by stating the objectives of computing definite integrals using Riemann sums and limits, estimating integrals using approximations like the midpoint rule, and reasoning about integrals using their properties. The outline then reviews the integral as a limit of Riemann sums and how to estimate integrals. It also discusses properties of the integral and comparison properties. Finally, it restates the theorem that if a function is continuous, the limit of Riemann sums is the same regardless of the choice of sample points.
The document summarizes the steps to solve optimization problems using calculus. It begins with an example of finding the rectangle with maximum area given a fixed perimeter. It works through the solution, identifying the objective function, variables, constraints, and using calculus techniques like taking the derivative to find critical points. The document then outlines Polya's 4-step method for problem solving and provides guidance on setting up optimization problems by understanding the problem, introducing notation, drawing diagrams, and eliminating variables using given constraints. It emphasizes using the Closed Interval Method, evaluating the function at endpoints and critical points to determine maximums and minimums.
The document is about calculating areas and distances using calculus. It discusses approximating areas of curved regions by dividing them into rectangles and letting the number of rectangles approach infinity. It provides examples of calculating areas of basic shapes like rectangles, parallelograms, and triangles. It then discusses Archimedes' work approximating the area under a parabola by inscribing sequences of triangles. The objectives are to compute areas using limits of approximating rectangles and to compute distances as limits of approximating time intervals.
- The document is about evaluating definite integrals and contains sections on: evaluating definite integrals using the evaluation theorem, writing antiderivatives as indefinite integrals, interpreting definite integrals as net change over an interval, and examples of computing definite integrals.
- It discusses properties of definite integrals such as additivity and comparison properties, and provides examples of definite integrals that can be evaluated using known area formulas or by direct computation of antiderivatives.
Here are the key points about g given f:
- g represents the area under the curve of f over successive intervals of the x-axis
- As x increases over an interval, g will increase if f is positive over that interval and decrease if f is negative
- The concavity (convexity or concavity) of g will match the concavity of f over each interval
In summary, the area function g, as defined by the integral of f, will have properties that correspond directly to the sign and concavity of f over successive intervals of integration.
This document provides information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences at NYU. The course will include weekly lectures, recitations, homework assignments, quizzes, a midterm exam, and a final exam. Grades will be determined based on exam, homework, and quiz scores. The required textbook is available in hardcover, looseleaf, or online formats through the campus bookstore or WebAssign. Students are encouraged to contact the professor or TAs with any questions.
The document is a section from a Calculus I course at NYU from June 22, 2010. It discusses using the substitution method to evaluate indefinite integrals. Specifically, it provides an example of using the substitution u=x^2+3 to evaluate the integral of (x^2+3)^3 4x dx. The solution transforms the integral into an integral of u^3 du and evaluates it to be 1/4 u^4 + C.
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This document contains lecture notes from a Calculus I class at New York University on September 14, 2010. The notes cover announcements, guidelines for written homework, a rubric for grading homework, examples of good and bad homework, and objectives for the concept of limits. The bulk of the document discusses the heuristic definition of a limit using an error-tolerance game approach, provides examples to illustrate the game, and outlines the path to a precise definition of a limit.
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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
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Lesson15 -exponential_growth_and_decay_021_slides
1. Section 3.4
Exponential Growth and Decay
V63.0121.021, Calculus I
New York University
October 28, 2010
Announcements
Quiz 3 next week in recitation on 2.6, 2.8, 3.1, 3.2
. . . . . .
2. Announcements
Quiz 3 next week in
recitation on 2.6, 2.8, 3.1,
3.2
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 2 / 40
3. Objectives
Solve the ordinary
differential equation
y′ (t) = ky(t), y(0) = y0
Solve problems involving
exponential growth and
decay
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 3 / 40
4. Outline
Recall
The differential equation y′ = ky
Modeling simple population growth
Modeling radioactive decay
Carbon-14 Dating
Newton’s Law of Cooling
Continuously Compounded Interest
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 4 / 40
5. Derivatives of exponential and logarithmic functions
y y′
ex ex
ax (ln a) · ax
1
ln x
x
1 1
loga x ·
ln a x
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 5 / 40
6. Outline
Recall
The differential equation y′ = ky
Modeling simple population growth
Modeling radioactive decay
Carbon-14 Dating
Newton’s Law of Cooling
Continuously Compounded Interest
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 6 / 40
7. What is a differential equation?
Definition
A differential equation is an equation for an unknown function which
includes the function and its derivatives.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 7 / 40
8. What is a differential equation?
Definition
A differential equation is an equation for an unknown function which
includes the function and its derivatives.
Example
Newton’s Second Law F = ma is a differential equation, where
a(t) = x′′ (t).
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 7 / 40
9. What is a differential equation?
Definition
A differential equation is an equation for an unknown function which
includes the function and its derivatives.
Example
Newton’s Second Law F = ma is a differential equation, where
a(t) = x′′ (t).
In a spring, F(x) = −kx, where x is displacement from equilibrium
and k is a constant. So
k
−kx(t) = mx′′ (t) =⇒ x′′ (t) + x(t) = 0.
m
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 7 / 40
10. What is a differential equation?
Definition
A differential equation is an equation for an unknown function which
includes the function and its derivatives.
Example
Newton’s Second Law F = ma is a differential equation, where
a(t) = x′′ (t).
In a spring, F(x) = −kx, where x is displacement from equilibrium
and k is a constant. So
k
−kx(t) = mx′′ (t) =⇒ x′′ (t) + x(t) = 0.
m
The √ general solution is x(t) = A sin ωt + B cos ωt, where
most
ω = k/m.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 7 / 40
11. Showing a function is a solution
Example (Continued)
Show that x(t) = A sin ωt + B cos ωt satisfies the differential equation
k √
x′′ + x = 0, where ω = k/m.
m
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 8 / 40
12. Showing a function is a solution
Example (Continued)
Show that x(t) = A sin ωt + B cos ωt satisfies the differential equation
k √
x′′ + x = 0, where ω = k/m.
m
Solution
We have
x(t) = A sin ωt + B cos ωt
x′ (t) = Aω cos ωt − Bω sin ωt
x′′ (t) = −Aω 2 sin ωt − Bω 2 cos ωt
Since ω 2 = k/m, the last line plus k/m times the first line result in zero.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 8 / 40
13. The Equation y′ = 2
Example
Find a solution to y′ (t) = 2.
Find the most general solution to y′ (t) = 2.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 9 / 40
14. The Equation y′ = 2
Example
Find a solution to y′ (t) = 2.
Find the most general solution to y′ (t) = 2.
Solution
A solution is y(t) = 2t.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 9 / 40
15. The Equation y′ = 2
Example
Find a solution to y′ (t) = 2.
Find the most general solution to y′ (t) = 2.
Solution
A solution is y(t) = 2t.
The general solution is y = 2t + C.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 9 / 40
16. The Equation y′ = 2
Example
Find a solution to y′ (t) = 2.
Find the most general solution to y′ (t) = 2.
Solution
A solution is y(t) = 2t.
The general solution is y = 2t + C.
Remark
If a function has a constant rate of growth, it’s linear.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 9 / 40
17. The Equation y′ = 2t
Example
Find a solution to y′ (t) = 2t.
Find the most general solution to y′ (t) = 2t.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 10 / 40
18. The Equation y′ = 2t
Example
Find a solution to y′ (t) = 2t.
Find the most general solution to y′ (t) = 2t.
Solution
A solution is y(t) = t2 .
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 10 / 40
19. The Equation y′ = 2t
Example
Find a solution to y′ (t) = 2t.
Find the most general solution to y′ (t) = 2t.
Solution
A solution is y(t) = t2 .
The general solution is y = t2 + C.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 10 / 40
20. The Equation y′ = y
Example
Find a solution to y′ (t) = y(t).
Find the most general solution to y′ (t) = y(t).
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 11 / 40
21. The Equation y′ = y
Example
Find a solution to y′ (t) = y(t).
Find the most general solution to y′ (t) = y(t).
Solution
A solution is y(t) = et .
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 11 / 40
22. The Equation y′ = y
Example
Find a solution to y′ (t) = y(t).
Find the most general solution to y′ (t) = y(t).
Solution
A solution is y(t) = et .
The general solution is y = Cet , not y = et + C.
(check this)
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 11 / 40
23. Kick it up a notch: y′ = 2y
Example
Find a solution to y′ = 2y.
Find the general solution to y′ = 2y.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 12 / 40
24. Kick it up a notch: y′ = 2y
Example
Find a solution to y′ = 2y.
Find the general solution to y′ = 2y.
Solution
y = e2t
y = Ce2t
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 12 / 40
25. In general: y′ = ky
Example
Find a solution to y′ = ky.
Find the general solution to y′ = ky.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 13 / 40
26. In general: y′ = ky
Example
Find a solution to y′ = ky.
Find the general solution to y′ = ky.
Solution
y = ekt
y = Cekt
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 13 / 40
27. In general: y′ = ky
Example
Find a solution to y′ = ky.
Find the general solution to y′ = ky.
Solution
y = ekt
y = Cekt
Remark
What is C? Plug in t = 0:
y(0) = Cek·0 = C · 1 = C,
so y(0) = y0 , the initial value of y. . . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 13 / 40
28. Constant Relative Growth =⇒ Exponential Growth
Theorem
A function with constant relative growth rate k is an exponential
function with parameter k. Explicitly, the solution to the equation
y′ (t) = ky(t) y(0) = y0
is
y(t) = y0 ekt
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 14 / 40
29. Exponential Growth is everywhere
Lots of situations have growth rates proportional to the current
value
This is the same as saying the relative growth rate is constant.
Examples: Natural population growth, compounded interest,
social networks
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 15 / 40
30. Outline
Recall
The differential equation y′ = ky
Modeling simple population growth
Modeling radioactive decay
Carbon-14 Dating
Newton’s Law of Cooling
Continuously Compounded Interest
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 16 / 40
31. Bacteria
Since you need bacteria to
make bacteria, the amount
of new bacteria at any
moment is proportional to
the total amount of
bacteria.
This means bacteria
populations grow
exponentially.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 17 / 40
32. Bacteria Example
Example
A colony of bacteria is grown under ideal conditions in a laboratory. At
the end of 3 hours there are 10,000 bacteria. At the end of 5 hours
there are 40,000. How many bacteria were present initially?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 18 / 40
33. Bacteria Example
Example
A colony of bacteria is grown under ideal conditions in a laboratory. At
the end of 3 hours there are 10,000 bacteria. At the end of 5 hours
there are 40,000. How many bacteria were present initially?
Solution
Since y′ = ky for bacteria, we have y = y0 ekt . We have
10, 000 = y0 ek·3 40, 000 = y0 ek·5
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 18 / 40
34. Bacteria Example
Example
A colony of bacteria is grown under ideal conditions in a laboratory. At
the end of 3 hours there are 10,000 bacteria. At the end of 5 hours
there are 40,000. How many bacteria were present initially?
Solution
Since y′ = ky for bacteria, we have y = y0 ekt . We have
10, 000 = y0 ek·3 40, 000 = y0 ek·5
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 18 / 40
35. Bacteria Example
Example
A colony of bacteria is grown under ideal conditions in a laboratory. At
the end of 3 hours there are 10,000 bacteria. At the end of 5 hours
there are 40,000. How many bacteria were present initially?
Solution
Since y′ = ky for bacteria, we have y = y0 ekt . We have
10, 000 = y0 ek·3 40, 000 = y0 ek·5
Dividing the first into the second gives
4 = e2k =⇒ 2k = ln 4 =⇒ k = ln 2. Now we have
10, 000 = y0 eln 2·3 = y0 · 8
10, 000
So y0 = = 1250.
8 . . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 18 / 40
36. Could you do that again please?
We have
10, 000 = y0 ek·3
40, 000 = y0 ek·5
Dividing the first into the second gives
40, 000 y e5k
= 0 3k
10, 000 y0 e
=⇒ 4 = e2k
=⇒ ln 4 = ln(e2k ) = 2k
ln 4 ln 22 2 ln 2
=⇒ k = = = = ln 2
2 2 2
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 19 / 40
37. Outline
Recall
The differential equation y′ = ky
Modeling simple population growth
Modeling radioactive decay
Carbon-14 Dating
Newton’s Law of Cooling
Continuously Compounded Interest
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 20 / 40
38. Modeling radioactive decay
Radioactive decay occurs because many large atoms spontaneously
give off particles.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 21 / 40
39. Modeling radioactive decay
Radioactive decay occurs because many large atoms spontaneously
give off particles.
This means that in a sample of
a bunch of atoms, we can
assume a certain percentage of
them will “go off” at any point.
(For instance, if all atom of a
certain radioactive element
have a 20% chance of decaying
at any point, then we can
expect in a sample of 100 that
20 of them will be decaying.)
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 21 / 40
40. Radioactive decay as a differential equation
The relative rate of decay is constant:
y′
=k
y
where k is negative.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 22 / 40
41. Radioactive decay as a differential equation
The relative rate of decay is constant:
y′
=k
y
where k is negative. So
y′ = ky =⇒ y = y0 ekt
again!
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 22 / 40
42. Radioactive decay as a differential equation
The relative rate of decay is constant:
y′
=k
y
where k is negative. So
y′ = ky =⇒ y = y0 ekt
again!
It’s customary to express the relative rate of decay in the units of
half-life: the amount of time it takes a pure sample to decay to one
which is only half pure.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 22 / 40
43. Computing the amount remaining of a decaying
sample
Example
The half-life of polonium-210 is about 138 days. How much of a 100 g
sample remains after t years?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 23 / 40
44. Computing the amount remaining of a decaying
sample
Example
The half-life of polonium-210 is about 138 days. How much of a 100 g
sample remains after t years?
Solution
We have y = y0 ekt , where y0 = y(0) = 100 grams. Then
365 · ln 2
50 = 100ek·138/365 =⇒ k = − .
138
Therefore
= 100 · 2−365t/138
365·ln 2
y(t) = 100e− 138
t
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 23 / 40
45. Computing the amount remaining of a decaying
sample
Example
The half-life of polonium-210 is about 138 days. How much of a 100 g
sample remains after t years?
Solution
We have y = y0 ekt , where y0 = y(0) = 100 grams. Then
365 · ln 2
50 = 100ek·138/365 =⇒ k = − .
138
Therefore
= 100 · 2−365t/138
365·ln 2
y(t) = 100e− 138
t
Notice y(t) = y0 · 2−t/t1/2 , where t1/2 is the half-life.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 23 / 40
46. Carbon-14 Dating
The ratio of carbon-14 to
carbon-12 in an organism
decays exponentially:
p(t) = p0 e−kt .
The half-life of carbon-14 is
about 5700 years. So the
equation for p(t) is
ln2
p(t) = p0 e− 5700 t
Another way to write this would
be
p(t) = p0 2−t/5700
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 24 / 40
47. Computing age with Carbon-14 content
Example
Suppose a fossil is found where the ratio of carbon-14 to carbon-12 is
10% of that in a living organism. How old is the fossil?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 25 / 40
48. Computing age with Carbon-14 content
Example
Suppose a fossil is found where the ratio of carbon-14 to carbon-12 is
10% of that in a living organism. How old is the fossil?
Solution
p(t)
We are looking for the value of t for which = 0.1
p0
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 25 / 40
49. Computing age with Carbon-14 content
Example
Suppose a fossil is found where the ratio of carbon-14 to carbon-12 is
10% of that in a living organism. How old is the fossil?
Solution
p(t)
We are looking for the value of t for which = 0.1 From the
p0
equation we have
2−t/5700 = 0.1
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 25 / 40
50. Computing age with Carbon-14 content
Example
Suppose a fossil is found where the ratio of carbon-14 to carbon-12 is
10% of that in a living organism. How old is the fossil?
Solution
p(t)
We are looking for the value of t for which = 0.1 From the
p0
equation we have
t
2−t/5700 = 0.1 =⇒ − ln 2 = ln 0.1
5700
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 25 / 40
51. Computing age with Carbon-14 content
Example
Suppose a fossil is found where the ratio of carbon-14 to carbon-12 is
10% of that in a living organism. How old is the fossil?
Solution
p(t)
We are looking for the value of t for which = 0.1 From the
p0
equation we have
t ln 0.1
2−t/5700 = 0.1 =⇒ − ln 2 = ln 0.1 =⇒ t = · 5700
5700 ln 2
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 25 / 40
52. Computing age with Carbon-14 content
Example
Suppose a fossil is found where the ratio of carbon-14 to carbon-12 is
10% of that in a living organism. How old is the fossil?
Solution
p(t)
We are looking for the value of t for which = 0.1 From the
p0
equation we have
t ln 0.1
2−t/5700 = 0.1 =⇒ − ln 2 = ln 0.1 =⇒ t = · 5700 ≈ 18, 940
5700 ln 2
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 25 / 40
53. Computing age with Carbon-14 content
Example
Suppose a fossil is found where the ratio of carbon-14 to carbon-12 is
10% of that in a living organism. How old is the fossil?
Solution
p(t)
We are looking for the value of t for which = 0.1 From the
p0
equation we have
t ln 0.1
2−t/5700 = 0.1 =⇒ − ln 2 = ln 0.1 =⇒ t = · 5700 ≈ 18, 940
5700 ln 2
So the fossil is almost 19,000 years old.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 25 / 40
54. Outline
Recall
The differential equation y′ = ky
Modeling simple population growth
Modeling radioactive decay
Carbon-14 Dating
Newton’s Law of Cooling
Continuously Compounded Interest
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 26 / 40
55. Newton's Law of Cooling
Newton’s Law of Cooling
states that the rate of
cooling of an object is
proportional to the
temperature difference
between the object and its
surroundings.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 27 / 40
56. Newton's Law of Cooling
Newton’s Law of Cooling
states that the rate of
cooling of an object is
proportional to the
temperature difference
between the object and its
surroundings.
This gives us a differential
equation of the form
dT
= k(T − Ts )
dt
(where k < 0 again).
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 27 / 40
57. General Solution to NLC problems
To solve this, change the variable y(t) = T(t) − Ts . Then y′ = T′ and
k(T − Ts ) = ky. The equation now looks like
dT dy
= k(T − Ts ) ⇐⇒ = ky
dt dt
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 28 / 40
58. General Solution to NLC problems
To solve this, change the variable y(t) = T(t) − Ts . Then y′ = T′ and
k(T − Ts ) = ky. The equation now looks like
dT dy
= k(T − Ts ) ⇐⇒ = ky
dt dt
Now we can solve!
y′ = ky
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 28 / 40
59. General Solution to NLC problems
To solve this, change the variable y(t) = T(t) − Ts . Then y′ = T′ and
k(T − Ts ) = ky. The equation now looks like
dT dy
= k(T − Ts ) ⇐⇒ = ky
dt dt
Now we can solve!
y′ = ky =⇒ y = Cekt
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 28 / 40
60. General Solution to NLC problems
To solve this, change the variable y(t) = T(t) − Ts . Then y′ = T′ and
k(T − Ts ) = ky. The equation now looks like
dT dy
= k(T − Ts ) ⇐⇒ = ky
dt dt
Now we can solve!
y′ = ky =⇒ y = Cekt =⇒ T − Ts = Cekt
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 28 / 40
61. General Solution to NLC problems
To solve this, change the variable y(t) = T(t) − Ts . Then y′ = T′ and
k(T − Ts ) = ky. The equation now looks like
dT dy
= k(T − Ts ) ⇐⇒ = ky
dt dt
Now we can solve!
y′ = ky =⇒ y = Cekt =⇒ T − Ts = Cekt =⇒ T = Cekt + Ts
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 28 / 40
62. General Solution to NLC problems
To solve this, change the variable y(t) = T(t) − Ts . Then y′ = T′ and
k(T − Ts ) = ky. The equation now looks like
dT dy
= k(T − Ts ) ⇐⇒ = ky
dt dt
Now we can solve!
y′ = ky =⇒ y = Cekt =⇒ T − Ts = Cekt =⇒ T = Cekt + Ts
Plugging in t = 0, we see C = y0 = T0 − Ts . So
Theorem
The solution to the equation T′ (t) = k(T(t) − Ts ), T(0) = T0 is
T(t) = (T0 − Ts )ekt + Ts
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 28 / 40
63. Computing cooling time with NLC
Example
A hard-boiled egg at 98 ◦ C is put in a sink of 18 ◦ C water. After 5
minutes, the egg’s temperature is 38 ◦ C. Assuming the water has not
warmed appreciably, how much longer will it take the egg to reach
20 ◦ C?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 29 / 40
64. Computing cooling time with NLC
Example
A hard-boiled egg at 98 ◦ C is put in a sink of 18 ◦ C water. After 5
minutes, the egg’s temperature is 38 ◦ C. Assuming the water has not
warmed appreciably, how much longer will it take the egg to reach
20 ◦ C?
Solution
We know that the temperature function takes the form
T(t) = (T0 − Ts )ekt + Ts = 80ekt + 18
To find k, plug in t = 5:
38 = T(5) = 80e5k + 18
and solve for k.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 29 / 40
65. Finding k
Solution (Continued)
38 = T(5) = 80e5k + 18
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 30 / 40
66. Finding k
Solution (Continued)
38 = T(5) = 80e5k + 18
20 = 80e5k
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 30 / 40
70. Finding k
Solution (Continued)
38 = T(5) = 80e5k + 18
20 = 80e5k
1
= e5k
( )4
1
ln = 5k
4
1
=⇒ k = − ln 4.
5
Now we need to solve for t:
t
20 = T(t) = 80e− 5 ln 4 + 18
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 30 / 40
71. Finding t
Solution (Continued)
t
20 = 80e− 5 ln 4 + 18
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 31 / 40
72. Finding t
Solution (Continued)
t
20 = 80e− 5 ln 4 + 18
t
2 = 80e− 5 ln 4
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 31 / 40
73. Finding t
Solution (Continued)
t
20 = 80e− 5 ln 4 + 18
t
2 = 80e− 5 ln 4
1 t
= e− 5 ln 4
40
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 31 / 40
74. Finding t
Solution (Continued)
t
20 = 80e− 5 ln 4 + 18
t
2 = 80e− 5 ln 4
1 t
= e− 5 ln 4
40
t
− ln 40 = − ln 4
5
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 31 / 40
75. Finding t
Solution (Continued)
t
20 = 80e− 5 ln 4 + 18
t
2 = 80e− 5 ln 4
1 t
= e− 5 ln 4
40
t
− ln 40 = − ln 4
5
ln 40 5 ln 40
=⇒ t = 1
= ≈ 13 min
5 ln 4 ln 4
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 31 / 40
76. Computing time of death with NLC
Example
A murder victim is discovered at
midnight and the temperature of
the body is recorded as 31 ◦ C.
One hour later, the temperature
of the body is 29 ◦ C. Assume
that the surrounding air
temperature remains constant
at 21 ◦ C. Calculate the victim’s
time of death. (The “normal”
temperature of a living human
being is approximately 37 ◦ C.)
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 32 / 40
77. Solution
Let time 0 be midnight. We know T0 = 31, Ts = 21, and
T(1) = 29. We want to know the t for which T(t) = 37.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 33 / 40
78. Solution
Let time 0 be midnight. We know T0 = 31, Ts = 21, and
T(1) = 29. We want to know the t for which T(t) = 37.
To find k:
29 = 10ek·1 + 21 =⇒ k = ln 0.8
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 33 / 40
79. Solution
Let time 0 be midnight. We know T0 = 31, Ts = 21, and
T(1) = 29. We want to know the t for which T(t) = 37.
To find k:
29 = 10ek·1 + 21 =⇒ k = ln 0.8
To find t:
37 = 10et·ln(0.8) + 21
1.6 = et·ln(0.8)
ln(1.6)
t= ≈ −2.10 hr
ln(0.8)
So the time of death was just before 10:00pm.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 33 / 40
80. Outline
Recall
The differential equation y′ = ky
Modeling simple population growth
Modeling radioactive decay
Carbon-14 Dating
Newton’s Law of Cooling
Continuously Compounded Interest
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 34 / 40
81. Interest
If an account has an compound interest rate of r per year
compounded n times, then an initial deposit of A0 dollars becomes
( r )nt
A0 1 +
n
after t years.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 35 / 40
82. Interest
If an account has an compound interest rate of r per year
compounded n times, then an initial deposit of A0 dollars becomes
( r )nt
A0 1 +
n
after t years.
For different amounts of compounding, this will change. As
n → ∞, we get continously compounded interest
( r )nt
A(t) = lim A0 1 + = A0 ert .
n→∞ n
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 35 / 40
83. Interest
If an account has an compound interest rate of r per year
compounded n times, then an initial deposit of A0 dollars becomes
( r )nt
A0 1 +
n
after t years.
For different amounts of compounding, this will change. As
n → ∞, we get continously compounded interest
( r )nt
A(t) = lim A0 1 + = A0 ert .
n→∞ n
Thus dollars are like bacteria.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 35 / 40
84. Continuous vs. Discrete Compounding of interest
Example
Consider two bank accounts: one with 10% annual interested
compounded quarterly and one with annual interest rate r compunded
continuously. If they produce the same balance after every year, what
is r?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 36 / 40
85. Continuous vs. Discrete Compounding of interest
Example
Consider two bank accounts: one with 10% annual interested
compounded quarterly and one with annual interest rate r compunded
continuously. If they produce the same balance after every year, what
is r?
Solution
The balance for the 10% compounded quarterly account after t years is
A1 (t) = A0 (1.025)4t = P((1.025)4 )t
The balance for the interest rate r compounded continuously account
after t years is
A2 (t) = A0 ert
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 36 / 40
86. Solving
Solution (Continued)
A1 (t) = A0 ((1.025)4 )t
A2 (t) = A0 (er )t
For those to be the same, er = (1.025)4 , so
r = ln((1.025)4 ) = 4 ln 1.025 ≈ 0.0988
So 10% annual interest compounded quarterly is basically equivalent
to 9.88% compounded continuously.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 37 / 40
87. Computing doubling time with exponential growth
Example
How long does it take an initial deposit of $100, compounded
continuously, to double?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 38 / 40
88. Computing doubling time with exponential growth
Example
How long does it take an initial deposit of $100, compounded
continuously, to double?
Solution
We need t such that A(t) = 200. In other words
ln 2
200 = 100ert =⇒ 2 = ert =⇒ ln 2 = rt =⇒ t = .
r
For instance, if r = 6% = 0.06, we have
ln 2 0.69 69
t= ≈ = = 11.5 years.
0.06 0.06 6
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 38 / 40
89. I-banking interview tip of the day
ln 2
The fraction can also
r
be approximated as either
70 or 72 divided by the
percentage rate (as a
number between 0 and
100, not a fraction between
0 and 1.)
This is sometimes called
the rule of 70 or rule of 72.
72 has lots of factors so it’s
used more often.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 39 / 40
90. Summary
When something grows or decays at a constant relative rate, the
growth or decay is exponential.
Equations with unknowns in an exponent can be solved with
logarithms.
Your friend list is like culture of bacteria (no offense).
. . . . . .
V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 40 / 40