MA102: Single-Variable Calculus II

Course Syllabus for "MA102: Single-Variable Calculus II"

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This course is the second installment of Single-Variable Calculus. In Part I (MA101), we studied limits, derivatives, and basic integrals as a means to understand the behavior of functions. In this course (Part II), we will extend our differentiation and integration abilities and apply the techniques we have learned. Additional integration techniques, in particular, are a major part of the course. In Part I, we learned how to integrate by various formulas and by reversing the chain rule through the technique of substitution. In Part II, we will learn some clever uses of substitution, how to reverse the product rule for differentiation through a technique called integration by parts, and how to rewrite trigonometric and rational integrands that look impossible into simpler forms. Series, while a major topic in their own right, also serve to extend our integration reach: they culminate in an application that lets you integrate almost any function you’d like. Integration allows us to calculate physical quantities for complicated objects: the length of a squiggly line, the volume of clay used to make a decorative vase, or the center of mass of a tray with variable thickness. The techniques and applications in this course also set the stage for more complicated physics concepts related to flow, whether of liquid or energy, addressed in Multivariable Calculus (MA103). Part I covered several applications of differentiation, including related rates. In Part II, we introduce differential equations, wherein various rates of change have a relationship to each other given by an equation. Unlike with related rates, the rates of change in a differential equation are various-degree derivatives of a function, including the function itself. For example, acceleration is the derivative of velocity, but the effect of air resistance on acceleration is a function of velocity: the faster you move, the more the air pushes back to slow you down. That relationship is a differential equation.

Learning Outcomes

Upon successful completion of this course, you will be able to:

• define and describe the indefinite integral;
• compute elementary definite and indefinite integrals;
• explain the relationship between the area problem and the indefinite integral;
• use the midpoint, trapezoidal, and Simpson’s rule to approximate the area under a curve;
• state the fundamental theorem of calculus;
• use change of variables to compute more complicated integrals;
• integrate transcendental, logarithmic, hyperbolic, and trigonometric functions;
• find the area between two curves;
• find the volumes of solids using ideas from geometry;
• find the volumes of solids of revolution using disks, washers, and shells;
• find the surface area of a solid of revolution;
• compute the average value of a function;
• use integrals to compute displacement, total distance traveled, moments, centers of mass, and work;
• use integration by parts to compute definite and indefinite integrals;
• use trigonometric substitution to compute definite and indefinite integrals;
• use the natural logarithm in substitutions to compute integrals;
• integrate rational functions using the method of partial fractions;
• compute improper integrals of both types;
• graph and differentiate parametric equations;
• convert between Cartesian and polar coordinates;
• graph and differentiate equations in polar coordinates;
• write and interpret a parameterization for a curve;
• find the length of a curve described in Cartesian coordinates, described in polar coordinates, or described by a parameterization;
• compute areas under curves described by polar coordinates;
• define convergence and limits in the context of sequences and series;
• find the limits of sequences and series;
• discuss the convergence of the geometric and binomial series;
• show the convergence of positive series using the comparison, integral, limit comparison, ratio, and root tests;
• show the divergence of a positive series using the divergence test;
• show the convergence of alternating series;
• define absolute and conditional convergence;
• show the absolute convergence of a series using the comparison, integral, limit comparison, ratio, and root tests;
• manipulate power series algebraically;
• differentiate and integrate power series;
• compute Taylor and MacLaurin series;
• recognize a first order differential equation;
• recognize an initial value problem;
• solve a first order ODE/IVP using separation of variables;
• draw a slope field given an ODE;
• use Euler’s method to approximate solutions to basic ODE; and
• apply basic solution techniques for linear, first order ODE to problems involving exponential growth and decay, logistic growth, radioactive decay, compound interest, epidemiology, and Newton’s Law of Cooling.

Course Requirements

In order to take this course, you must:

√    Have continuous broadband Internet access.

√    Have the ability/permission to install plug-ins or software (e.g. Adobe Reader or Flash).

√    Have the ability to download and save files and documents to a computer.

√    Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.).

√    Be competent in the English language.

√    Have read the Saylor Student Handbook.

√    Have completed MA101:Single-Variable Calculus I.

Course Information

Welcome to MA102 Single-Variable Calculus II. Below, please find general information on this course and its requirements. Picking up where MA101 left off, this course looks at integration in more depth, before moving on to cover a variety of topics, including parametric equations, sequences and series, and the basics of ordinary differential equations. Calculus II can often feel like an ill-assorted grab-bag, but each of the disparate topics it introduces either extends the reach of our integration and differentiation techniques or addresses applications of those techniques.

Course Designer: Clare Wickman

Primary Resources: This course is comprised of a range of different free, online materials. However, the course makes primary use of the following materials:

Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials.  You will also need to complete:

• Unit 1 Assessments
• Unit 2 Assessments
• Unit 3 Assessments
• Unit 4 Assessments
• Unit 5 Assessments
• Unit 6 Assessments
• The Final Exam

Note that you will only receive an official grade on your Final Exam. However, in order to adequately prepare for this exam, you will need to work through all of the resources and assessments in each unit.

In order to “pass” this course, you will need to earn a 70% or higher on the Final Exam.  Your score on the exam will be tabulated as soon as you complete it. If you do not pass the exam, you may take it again.

Time Commitment: This course should take you approximately 125 hours to complete. At the beginning of each unit, there is a detailed list of time advisories for each subunit. These estimates factor in the time required to watch each lecture, work through each reading thoughtfully, and complete each assessment. However, these should be seen as guidelines, not goals; each learner is different, and you may find that your pace changes throughout the course. Mastery of the material, rather than strict adherence to the time estimates, is the measure of success in this course. It may be useful to take a look at these time advisories, to determine how much time you have over the next few weeks to complete each unit, and then to set goals for yourself. For example, Unit 1 should take approximately 20.25 hours. Perhaps you can sit down with your calendar and decide to complete sub-subunit 1.1.1 (a total of 2 hours) on Monday night; sub-subunits 1.1.2 and 1.1.3 (a total of 3.5 hours) on Tuesday night; sub-subunit 1.1.4 (a total of 4 hours) on Wednesday night; etc.

Tips/Suggestions: If a lecture stops making sense to you, pause it – this is a luxury you only have in a course of this nature! – and return to the readings to get up-to-speed on the material. Remember to note down the time at which you paused the lecture, in case your browser times out. As noted in the “Course Requirements,” Single-variable Calculus Part I (MA101) is a pre-requisite for this course. If you are struggling with the mathematics as you progress through this course, consider taking a break to revisit MA101.