Calculus and Analytic Geometry

MAT 221


What is Calculus?

Calculus studies two related questions. In the differential calculus, we study the rates at which quantities change. For example, if we know the position of a moving object, how do we find its velocity? In the integral calculus, we study how rates of change accumulate to arrive at the total change in a quantity. This problem is connected to computing certain geometric areas.

These two concepts, differentiation and integration, are intimately related by the so-called Fundamental Theorem of Calculus. This beautiful result is the central goal of the course.

To make these concepts precise, we first study the mathematical concepts of limit and continuity. The limit is a tool which converts familiar algebraic ideas into calculus ideas and is instrumental in defining the derivative and integral.  Continuity captures the way certain quantities change in a continuous fashion, transitioning from one value to the next, by travelling through all the values in between.

In this course, we study the calculus of logarithmic, exponential, and trigonometric functions. This will require a strong background in pre-calculus topics.

Why Study Calculus?

Calculus is the language of scientists and engineers. It would be impossible to model real-world situations without the ability to accurately describe the way the world changes. As such, this course is a core component of most engineering and science programs. However, the applications of calculus are not limited to the physical sciences but there are numerous applications in Athletics, Biomedical Sciences, Environmental Sciences, Management Science, Business, Economics, and the Social and Behavioral Sciences. Calculus will be one of the most if not the most useful mathematics that anyone in any field of study could take. If this isn’t motivation enough, calculus is one of the great intellectual achievements of history. This alone makes it worthy of study and appreciation.


UW Colleges Catalog Course Description for MAT 221 Calculus and Analytic Geometry I - 5 credits. Analytic geometry, functions, limits and continuity, the derivative, integrals, techniques and applications of differentiation, applications of integration, logarithmic and exponential functions and trigonometric functions. Students may not earn more than six credits by taking both MAT 211 and MAT 221.

Prerequisites: a grade of C or better in MAT 124 or MAT 110 and MAT 113 or equivalent, or placement based on placement test score.

Successful completion of this course will earn five math science (MS) credits toward the Math and Natural Sciences requirement of the Associate of Arts and Science degree.


After completing this course, the student should be able to:

  • Define the concept of limit and successfully evaluate limits and apply the concept to the derivative
  • Determine equations of tangent lines using the derivative and execute the power rule, the sum rule, the constant rule, the product rule, the quotient rule, and the chain rule for computing derivatives of real functions
  • Work with parametric equations and be able to implicitly differentiation and solve related rates and optimization problems
  • Successfully determine local and absolute maxima and minima for real functions and when the function is either increasing or decreasing
  • Determine the concavity of a function and analyze the concavity in terms of the second derivative and accurately sketch the graph of a function using calculus methods
  • Find limits of an indeterminate form and in particular use L’Hopital’s rule
  • Define anti-derivatives and use the basic rules for finding anti-derivatives
  • Use Riemann sums to approximate the area under a curve
  • Define the definite integral as a limit of a sum and use the Fundamental Theorem of Calculus
  • Find an area between curves and the volumes of solids of revolution using integration by circular disk and cylindrical shell methods
  • Accurately employ the applications of integration, including finding arc lengths along curves, determining moments and centers of mass of thin plates in two dimensions
  • Determine the derivative of exponential and logarithmic functions


Technology Requirements

  • Scanner

The course assignments that you will submit during the semester will need to be scanned so that you can submit them to the Dropbox.

  • Graphing calculator

You can use the specific calculator of your choice, but you should choose a calculator with no greater functionality than a TI-86. Please ensure you have a calculator manual as the instructor is not responsible for any technical or operational support for your calculator. In using a calculator, please be clearly aware that all working for problems must be shown and full credit will not be given for answers without supporting processes that demonstrate how the solution was attained.

Software Requirements

  • Microsoft Word (with the equation editor)
  • Microsoft Excel

The most current edition of MS Office (containing MS Word, MS Excel and other valuable programs) is now available to University of Wisconsin students through the Wisconsin Integrated Software Catalog.

  • Adobe Reader
  • Shockwave/Flash
  • RealPlayer
  • QuickTime

MathXL Player and Test Gen plug-ins