Lesson 13: Related Rates Problems

This document discusses heterogeneous agent models without aggregate uncertainty. It introduces a model with a continuum of agents who face idiosyncratic income fluctuations but no aggregate shocks. There is a unique stationary equilibrium with constant interest rates and wages. The document discusses the recursive competitive equilibrium, existence and uniqueness of the stationary equilibrium, transition functions, computation methods, and some qualitative results from calibrating the model.

The document discusses three examples of nonlinear and non-Gaussian DSGE models. The first example features Epstein-Zin preferences to allow for a separation between risk aversion and the intertemporal elasticity of substitution. The second example models volatility shocks using time-varying variances. The third example aims to distinguish between the effects of stochastic volatility ("fortune") versus parameter drifting ("virtue") in explaining time-varying volatility in macroeconomic variables. The document outlines the motivation, structure, and solution methods for these three nonlinear DSGE models.

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.

This document provides an overview of integration by substitution. It begins with announcements about an upcoming review session, evaluations, and final exam. It then discusses the objectives and outline of the section, which are to transform integrals using substitutions, evaluate indefinite integrals using substitutions, and evaluate definite integrals using substitutions. An example is provided to illustrate how to use substitution to evaluate the indefinite integral of x/(x^2 + 1) by letting u = x^2 + 1. The solution uses a new notation of letting u = x^2 + 1 and du = 2x dx to rewrite the integral in terms of u.

This document discusses filtering and likelihood inference. It begins by introducing filtering problems in economics, such as evaluating DSGE models. It then presents the state space representation approach, which models the transition and measurement equations with stochastic shocks. The goal of filtering is to compute the conditional densities of states given observed data over time using tools like the Chapman-Kolmogorov equation and Bayes' theorem. Filtering provides a recursive way to make predictions and updates estimates as new data arrives.

The document discusses the concepts of sampling distributions, including: 1. The sampling distribution of the mean, which follows a normal distribution with mean μ and standard deviation σ/√n when observations are independent and identically distributed. 2. The sampling distribution of the standard deviation, which follows a chi-squared distribution with n-1 degrees of freedom when the population variance is σ2. 3. When the population variance is unknown, the distribution of the sample mean is a t-distribution rather than normal, with heavier tails and degrees of freedom equal to n-1.

The derivative of a composition of functions is the product of the derivatives of those functions. This rule is important because compositions are so powerful.

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.

The document discusses notes for a Calculus I class section on definite integrals. It provides announcements for upcoming quizzes and exams. The objectives are to compute definite integrals using Riemann sums, estimate integrals using techniques like the midpoint rule, and use properties of integrals. The professor outlines the topics to be covered, which include the definition of the definite integral as a limit of Riemann sums and properties of integrals. An example computes the integral of x from 0 to 3.

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 derivative of a sum of functions is the sum of the derivatives of those functions, but the derivative of a product or a quotient of functions is not so simple. We'll derive and use the product and quotient rule for these purposes. It will allow us to find the derivatives of other trigonometric functions, and derivatives of power functions with negative whole number exponents.

Presentation deck used at the Model Schools Conference in Orlando 2012. Presentation on KP Compass and how we use game theory to increase student engagement in our concept driven mastery system. www.kpcompass.com

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This document summarizes a presentation on cluster stability estimation and determining the optimal number of clusters in a dataset. The presentation proposes a method that draws random samples from the dataset and compares the partitions obtained from each sample to estimate cluster stability. It quantifies the consistency between partitions using minimal spanning trees and the Friedman-Rafsky test statistic. Experiments on synthetic and real-world datasets show that the method can accurately determine the true number of clusters by finding the partition that maximizes cluster stability.

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

The derivative of a composition of functions is the product of the derivatives of those functions. This rule is important because compositions are so powerful.

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.

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.

The document provides notes from a Calculus I class at New York University on December 13, 2010. It includes announcements about upcoming review sessions and the final exam. It then outlines the objectives and topics to be covered, which is integration by substitution. Examples are provided to demonstrate how to use u-substitution to evaluate indefinite integrals. The key steps are to let u equal an expression involving x, find du in terms of dx, and then substitute into the integral.

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 is from a Calculus I course at New York University. It outlines the objectives and content for Sections 2.1-2.2 on the derivative and rates of change. The sections will define the derivative, discuss how it models rates of change, and how to find the derivative of various functions graphically and numerically. It will also cover how to find higher-order derivatives and sketch the derivative graph from a function graph.

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.

This document is from a Calculus I course at New York University and covers limits involving infinity. It defines infinite limits, both limits approaching positive and negative infinity, and limits at vertical asymptotes. Examples are provided of known infinite limits, like limits of 1/x as x approaches 0. The document demonstrates finding one-sided limits at points where a function is not continuous using a number line to determine the sign of factors in the denominator.

This document is from a Calculus I class at New York University and covers continuity. It provides announcements about office hours and homework deadlines. It then discusses the objectives of understanding the definition of continuity and applying it to piecewise functions. Examples are provided to demonstrate how to show a function is continuous at a point by evaluating the limit as x approaches the point and showing it equals the function value. The students are asked to determine at which other points the example function is continuous.