Techniques of Integration
When differentiating an expression, we apply basic rules such as the product rule to find its derivative. Integration is a fundamentally more complex task; it requires us to recognize the result of using differentiation rules. The first goal of this course is to develop a set of techniques to help us integrate more complicated expressions than those tackled in the first semester.
The new techniques we will learn include integration by parts, integration by partial fractions, and integration using trigonometric substitutions. We also discuss how to use numerical integration to approximate the value of definite integrals. Finally, we apply integration to solving some important classes of differential equations.
Analytic Geometry
We next discuss the geometry of the conic sections: the parabola, the ellipse, and the hyperbola. These curves have important applications in astronomy, telecommunications, navigation, and many other fields.
The rectangular coordinate system commonly used is sometimes a poor choice for describing the geometry of certain curves. We will develop the polar coordinate system and show how to find tangent lines, volumes, areas, and arc-lengths in polar coordinates.
Infinite Series
A fraction like one-third is really a sum involving infinitely many terms:
0.3333... = 3/10 + 3/100 + 3/1000 + 3/10,000 + ...
which is said to converge to the ratio 1/3. In other examples, such as
1 + 2 + 3 + 4 + 5 + ...
the sum does not converge to any finite number. These are both examples of infinite series. We will develop methods for determining when infinite series converge. We apply the machinery of infinite series to find power series representations of functions. We also discuss the Fourier series representation of periodic functions.
Vectors
We conclude the second semester with an introduction to the geometry of vectors in both two and three-dimensions. This includes the so-called dot¬-product and cross-products of vectors which are the building blocks for modeling many physical phenomena such as work and torque. We also discuss the geometry of lines and planes in space.
UW Colleges Catalog Course Description for MAT 222 Calculus and Analytic Geometry II - 5 credits. A Continuation of MAT 221. Techniques of integration, polar coordinates, conic sections, infinite series and vectors of two and three dimensions. Note: the order of topics covered in MAT 221 and MAT 222 may depend on the text used and the instructor. 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.
Prerequisites: a grade of C or better in MAT 221 or placement based on the department Calculus Proficiency Test or AP exam. MS
After completing this course, the student should be able to:
The course assignments that you will submit during the semester will need to be scanned so that you can submit them to the Dropbox.
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.
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.
Anthony van Groningen