Course Outline | CalcAPL1x

The full course outline is provided here to help you navigate the course.

Course Updates

1. Introduction Section (Section 0)

1.1 Pre-Course Survey

1.2 Welcome to the Course

1.2.0 How to Use edX and Get Help

1.2.1 Welcome Video (and our First Model)

1.2.2 Overview of Topics

1.3 Goals, Prerequisites and Getting Started

1.3.1 Video: What Should I Already Know to Take This Course?

1.3.2 Pre-requisites and Links to Refresher Resources

1.3.3 Using Desmos in this Course

1.3.4 Exploratory Quiz: Projectile Motion

1.4 Grading, Assignments and Schedule

1.4.1 Course Schedule

1.4.2 Course Structure

1.4.3 Assessment Types and Grade Weights

1.5 Course Policies

1.5.1 Discussion Board and Collaboration Policy

1.5.2 HarvardX Policies

1.6 Introduce yourself!

1.6 Discussion: Introduce Yourself

1.7 Course Community

1.8 Help Forum for Section 0

1.8.1 Using Math Notation in the Discussion Forums

Help Forum for Section 0

Section 1: What Makes a Good Test Question? Mathematical Models to Measure Knowledge and Improve Learning

1.1 What Makes a Good Question? Introduction

1.1.0 Introduction: Item Response Theory

1.1.1 Video: What Makes a Good Question?

1.1.2 Exploratory Quiz: World Geography

1.1.3 Exploratory Quiz: What Makes Some Questions Better Than Others?

1.2 Modeling a Test Question

1.2.1 Video: Parameters of a Question

1.2.2 Ability and Probability

1.2.3 Quiz: Probability as Function of Ability

1.3 Item Response Curves

1.3.1 Video: Item Response Curves

1.3.2 Exploratory Quiz: Features of Item Response Curves and How They Relate to Guessing, Discrimination and Difficulty

1.3.3 (Optional) When is an Item Response Curve NOT a good model

1.4 The Item Response Model and its Parameters

1.4.1 Video: The Item Response Model

1.4.2 What is a parameter?

1.4.3 Exploratory Quiz: What is a parameter?

1.5 Creating and Using Item Response Curves

1.5.0 Creating and Using Item Response Curves

1.5.1 Video: Creating and Using Item Response Curves

1.5.2 Reflecting on "What Makes A Good Test Question?"

1.6 A Dynamic Model for Improving Learning in Education

1.6.1 Video: Which Explanation Should a Student Get?

1.7 How the Parameters of Item Response Model Relate to Difficulty, Discrimination and Guessing

1.7.0 Summary Quiz: How the Parameters of Item Response Model Relate to Difficulty, Discrimination and Guessing

1.7.1 Summary Quiz: Using Calculus to Find End Behavior, Shape and Slope of the Item Response Curve

1.7.2 Summary Quiz: Which parameters represent guessing level, difficulty, and discrimination?

Help Forum for Section 1

Section 2: Economic Applications of Calculus: Price and Demand in a Tale of Two Cities

1.1 A Tale of Two Cities: Public Transit Fares in New York and Boston

1.1.0 Introduction: Economic Applications of Calculus

1.1.1 Video: Public Transit Fares in New York City and Boston

1.1.2 Exploratory Quiz: Why the Fare Decrease in Boston?

1.2 Price and Demand

1.2.1 Video: Introduction to Demand Functions

1.2.2 Exploratory Quiz: Linear Functions to Model Price and Demand

1.2.3 Quiz: Absolute vs. Percent Change in Demand

1.3 Measuring What Matters: Price Elasticity of Demand

1.3.1 Video: Price Elasticity of Demand

1.3.2 Quiz: Compute and Interpret Elasticity Values

1.4 Price Elasticity of Demand: A Calculus Viewpoint

1.4.1 Video: Price Elasticity of Demand in the Tale of Two Cities

1.4.2 Exploratory Quiz: on PPED and PED formulas

1.4.3 Video: Price Elasticity of Demand: A Calculus Viewpoint

1.4.4 Quiz: Point Price Elasticity Computations

1.5 A Tale of Two Cities, Revisited

1.5.1 Video: The Tale of Two Cities, Revisited

1.6 Elasticity and Maximizing Revenue

1.6.1 Exploratory Quiz: Elastic versus Inelastic

1.6.2 Quiz: Maximizing Revenue in the Boston Example

1.6.3 Elasticity and Maximizing Revenue

1.7 Summary Quiz - A little more about math and economics from Kiran Gajwani

1.7.1 Video: Elasticity in Developing Countries

1.7.2 Summary Quiz: Elasticity in Action

1.7.3 Summary and Additional Resources

1.8 Help Forum for Section 2

Help Forum for Section 2

Section 3: From X-Rays to CT-Scans: Mathematics and Medical Imaging

1.1 Medical Imaging in a Real Patient's Case

1.1.0 Introduction: Medical Imaging

1.1.1 Video: Medical Imaging in a Real Patient's Case

1.1.2 Exploratory Quiz: What is an X-ray?

1.2 What is an x-ray?

1.2.1 Video: What is an x-ray?

1.2.2 X-rays and Projection Graphs

1.2.3 Exploratory Quiz: Interpreting Projection Graphs

1.2.4 Quiz: Interpreting Projection Graphs

1.3 A Mathematical Model for an X-Ray

1.3.1 Video: A Mathematical Model for an X-Ray

1.3.2 Quiz: Making Sense of the Lambert-Beer Model

1.4 X-Rays through a Non-Uniform Material

1.4.1 Video: X-Rays through a Non-Uniform Material

1.4.2 X-Rays through a Non-Uniform Material

1.4.3 Quiz: The formula for an x-ray projection

1.4.4 Exploratory Quiz: Thinking about the Attenuation Function, $\mu (x)$

1.5 Finding the Attenuation Function from Many Views

1.5.1 Video: Finding the Attenuation Function from Many Views

1.5.2 Optional Exploratory Quiz: An Example of Iterative Reconstruction

1.6 CT Scans: What They Can Reveal in a Real Patient's Case

1.7 Summary Quiz: From X-rays to CT-scans

1.7.1 Summary Quiz Part 1

1.7.2 Summary Quiz Part 2

1.8 Additional Resources (Optional)

1.9 Help Forum for Section 3

Section 4: What is Middle Income? Thinking about Income Distributions with Statistics and Calculus

1.1.0 Introduction: What is Middle Income? (Statistics)

1.1.1 Video: What is Middle Income? Looking at Income Distributions

1.1.2 Histograms of Data

1.1.3 Exploratory Quiz: How To Make A Histogram

1.2 Looking at Income Distributions: Frequency and Relative Frequency Histograms

1.2.0 Video: A Better View of the Data: Relative Frequency Histograms

1.2.1 Reading Histograms

1.2.2 Quiz: Reading Histograms

1.3 What is Middle Income? Mean, Median and Beyond

1.3.1 Video: Defining Middle Income

1.3.1.0 Summary: Mean, Median and Middle Income

1.3.2 Quiz: Mean, Median and Middle Income

1.4 From Histogram of Data to Continuous Model: Probability Density Functions

1.4.1 Video: From Histogram to Probability Density Functions

1.4.2 From Histogram to Probability Density Functions

1.4.3 Quiz: Probability Density Functions to Model Data

1.5 Using Integration to Estimate Households in Middle Income Range

1.5.0 Video: How Many Households are in the Middle Income Range?

1.5.1 More About Gamma Distributions

1.5.2 Exploratory Quiz: Probability Density Functions in Action

1.6 What is Middle Income? Mean and Median for the Continuous Model

1.6.0 Introduction: Mean and Median for the Continuous Model

1.6.1 Video: Mean and Median for the Continuous Model

1.6.2 Expected Value, Mean and Riemann Sums

1.6.3 Video: How Good is our Model for Predicting Billionaires?

1.7 Optional Video: A Little More from Nina about Statistics in her work

1.8 Summary Quiz

1.9 Help Forum for Section 4

Section: 5 Population Dynamics Part I: the Evolution of Population Models

1.1 Introduction to Population Models

1.1.1 Introduction and Learning Goals for Section

1.1.2 Video: Our Dynamic Earth and Malthus' Population Model

1.1.3 Exploratory Quiz: Malthus' Three Principles

1.2 Limits on Growth: Verhulst's Model

1.2.1 Video: Limits on Growth: Verhulst's Model

1.2.2 Quiz: Logistic Growth Model

1.2.3 Qualitative Analysis of Differential Equations

1.2.4 Quiz: Interpreting Models

1.3 Species Interact: D'Ancona's Puzzle and Volterra's Predator-Prey Model

1.3.1 Video: D'Ancona's Puzzle and Volterra's Predator-Prey Model

1.3.2 Exploratory Quiz: D'Ancona's Observation

1.4 Volterra's Model in Action: Marlin and Sardines

1.4.1 Video: Volterra's Model in Action: Marlin and Sardines

1.4.2 Explanation of Lotka-Volterra Predator Prey Model Terms

1.4.3 Qualitative Analysis of a System of Differential Equations via Phase Planes

1.4.4 Optional Video: Phase Plane Analysis: A Crash Course

1.4.5 Quiz: Creating a Phase Plane for Marlin and Sardine

1.4.5 Quiz Part 2: Creating a Phase Plane for Marlin and Sardine

1.4.5 Quiz Part 3: Creating a Phase Plane for Marlin and Sardine

1.4.5 Quiz Part 4: Creating a Phase Plane for Marlin and Sardine

1.4.5 Quiz Part 5: Creating a Phase Plane for Marlin and Sardine

1.5 How Populations Change in a Predator-Prey System

1.5.1 Video: How Populations Change

1.5.2 Quiz: Populations Oscillate

1.6 Summary Quiz: Population Models

1.6.1 Quiz: A General Predator-Prey Phase Plane

1.6.2 Quiz: Other Models of Species Interaction

1.7 Help Forum for Section 5-6

Section 6: Population Dynamics II: A Biological Puzzle OR How Fishing Affects a Predator-Prey System

2.1 Introduction to the Biological Puzzle

2.1.1 Introduction and Learning Goals for Section

2.2 The Average Value of Populations in a Predator-Prey System

2.2.1 Computing Average Value

2.2.2 Exploratory Quiz: Computing Average Value

2.2.3 Video: Computing Average Value, Part I

2.2.4 Exploratory Quiz: Computing Average Value, Part II

2.2.5 Video: Computing Average Value, Part II

2.2.6 Quiz: Average Value of Sardines and Marlin

2.3 The Effect of Fishing on the Predator-Prey System

2.3.1 Video: The Effect of Fishing on the Predator-Prey System

2.3.2 Exploratory Quiz: What happens when there is fishing?

2.4 D'Ancona's Puzzle Again and Implications for Fishing Regulations

2.4.1 Video: D'Ancona's Puzzle Explained

2.5 Making a Good Model: Issues and Limitations (Optional)

2.5.1 Video: Making a Good Model

2.5.2 No Perfect Model?

2.5.3 Exploratory Quiz: Rates of Predation and Synthesis

2.6 Summary Quiz

2.6.1 Quiz: Effect of Fishing on Predator and Prey Fish

2.6.2 Quiz: Paradox of the Pesticides

2.7 Help Forum for Section 6

Section 7: Extinction, Chaos and other Bifurcation Behavior

1.1 What is a Bifurcation? Introduction

1.1.0 Introduction

1.1.1 Video: Change and Bifurcations

1.1.2 Change and Bifurcations

1.1.3 Exploratory Quiz: Parameters and Variables

1.1.4 Video: More Examples of Bifurcations: Fishing and Weather

1.1.5 Quiz: Parameters, Variables and Bifurcations

1.2 Modeling Fish Populations

1.2.1 Video: Modeling Fish Populations and Harvesting

1.2.2 Making Sense of the Graph of dP/dt versus P

1.2.2 Equilibrium Points and Stability

1.2.3 Quiz: Making Sense of the Graph of dP/dt versus P

1.2.4 Exploratory Quiz: What Effect does Fishing have on Population?

1.3 Survival or Extinction: The Effect of Harvesting on Fish Population

1.3.1 Exploratory Quiz: What Effect does Fishing have on Population?

1.3.2 Video: How Does Fishing Affect the Population?

1.3.3 The Meaning of Critical Values and Bifurcations

1.3.4 (Optional) Qualitative Analysis of Differential Equations: Visualizing the Long-Term Behavior

1.3.5 Quiz: (Optional) Qualitative Analysis of the Fishing Model

1.3.6 Quiz: How does fishing level affect what can happen to fish populations in the long term?

1.4 Summary: Bifurcations and Fishing

1.4.1 Video: Some Last Thoughts on Bifurcations

1.4.2 Summary Quiz: Survival or Extinction: The Effect of a Proportional Fishing Rate

1.5 Mathematics and Biology: A Symbiotic Relationship

1.5.1 Video: Mathematics and Biology: A Symbiotic Relationship

Help Forum for Section 7

Section 8: Outbreak! Budworm Populations and Bifurcations

1.1 Introduction to the Budworm Model

1.1.0 Introduction

1.1.1 Video: A Population Model for Budworms

1.1.2 A Population Model for Budworms

1.1.3 Exploratory Quiz: Can the Logistic Model Capture Outbreaks?

1.2 Modifying the Budworm Model: Predators!

1.2.1 Video: A Budworm Population Model with Predation

1.2.1.1 A Budworm Population Model with Predation

1.2.2 Quiz: Equilibria of the Budworm Model with Predation

1.3 Equilibrium Solutions and Carrying Capacity

1.3.1 Video: Equilibrium Solutions and Carrying Capacity

1.3.2 Exploratory Quiz: Increasing the Carrying Capacity $q$

1.4 Increasing the Carrying Capacity Further

1.4.1 Video: Increasing the Carrying Capacity Further

1.4.2 Quiz: What Can Happen to the Budworm Population?

1.5 Summary Quiz: Outbreaks and Other Bifurcation Behavior

1.6 The Effect of Carrying Capacity and Reproduction Rate: A Summary

Help Forum for Section 8

Section 9: Species in Competition: Coexistence or Exclusion

1.1 A Model of Species in Competition

1.1.0 Introduction

1.1.1 Video: Species in Competition

1.1.2 A System of Differential Equations for Species in Equal Competition

1.1.3 Exploratory Quiz: Species in Competition and Phase Plane Analysis

1.2 How Weak Competition Affects the System

1.2.1 Video: How Weak Competition Affects the System

1.2.2 How Can We Determine Solution Trajectories?

1.2.3 Quiz: From Weak to Strong Competition

1.3 How Strong Competition Affects the System

1.3.1 Video: How Strong Competition Affects the System

1.3.2 Exploratory Quiz: What is the bifurcation for this system?

1.4 The Bifurcation Value and Competition in Economic Systems

1.4.1 Video: The Bifurcation Value and Competition in Economic Systems

1.5 Summary Quiz

1.5.1 Summary Quiz: Species in Competition and Bifurcation Values

Help Forum for Section 9

Section 10: E = mc²: Taylor Approximation and the Energy Equation (OPTIONAL)

1.1 Introducing the Energy-Mass Equation

1.1.1 Video: Introducing the Energy-Mass Equation

1.1.2 The Energy-Mass Equation and Reference Frames

1.1.3 Exploratory Quiz: The Energy-Mass Equation and Taylor Approximation

1.2 Taylor Approximation for the Energy-Mass Equation

1.2.1 Video: Taylor Approximation for the Energy-Mass Equation

1.2.2 Quiz: Computing the Taylor Approximation

1.3 Making Sense of the First Few Terms of the Approximation

1.3.1 Video: Making Sense of the First Few Terms of the Approximation

1.3.2 Quiz: Making Sense of the First Few Terms of the Approximation

1.4 Higher Terms and Faster Speeds

1.4.1 Video: Higher Terms and Faster Speeds

1.4.1.1 Mboyo's Board Work

1.4.2 Exploratory Quiz: Catching an Error in Computation

1.5 String Theory, the Energy Equation and a Physicist's Dream

1.6 Summary Quiz: E = mc²: Taylor Approximation and the Energy Equation

1.6.1 Summary Quiz: Taylor Approximation and Connecting Old and New Theories

1.6.2 Summary Quiz: Part 2: Spectral Radiance of Electromagnetic Radiation

1.7 Additional Resources (Optional)

Help Forum for Section 10

Section 11: Final Assessment

Video: In Closing... and final assessment

Final Assessment Instructions and Part I

Final Assessment Instructions

Final Assessment Part I: Find an example of calculus applied

Final Assessment Part II: Self-Review Practice and Instructions

Help Forum for Final Assessments

Final Assessment Part II; Option I: Physics Pondering a Pendulum (CHOOSE ONE)

1.0 Introduction and Intuition

1.2 Exploratory Quiz

2.0 Model of Pendulum Motion

2.1 Quiz: Pendulum Model

2.2 Quiz: Pendulum Model Part 2

3.0 Qualitative Analysis of the Linearized Model

3.1 From Second-Order Differential Equation to a System of Differential Equations

3.2 Quiz: Phase Plane for the Simplified Pendulum Model

4.0 Solving the Simplified Pendulum Model

4.1 Quiz: Solving the Simplified Pendulum Model

5.0 How does gravity or pendulum weight and length affect the period?

5.1 Quiz: How does gravity or pendulum weight and length affect the period?

6.0 Summary Quiz

6.1 Summing Up

6.1.1 Summary Quiz Part I

6.2 Summary Quiz Part II: Self Assessment

6.3 Summary Quiz Part III: What Happens in the Original System for the Pendulum?