Single-Variable Calculus II

Purpose of Course  showclose

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.  While this end goal remains the same, we will now focus on adapting what we have learned to applications.  By the end of this course, you should have a solid understanding of functions and how they behave.  You should also be able to apply the concepts we have learned in both Parts I and II of Single-Variable Calculus to a variety of situations.

We will begin by revisiting and building upon what we know about the integral.  We will then explore the mathematical applications of integration before delving into the second major topic of this course: series.  The course will conclude with an introduction to differential equations.

Learning Outcomes  showclose

Upon successful completion of this course, students 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 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.
  • 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  showclose

In order to take this course you must:
√    Have access to a computer.
√    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.

Unit Outline show close