Topics include applications of the integral, techniques of integration, numerical integration, improper integrals, solutions of separable differential equations, infinite series, polynomial approximations of functions, power series, Taylor and MacLaurin series, and parametric and polar equations. Utilizes numerical, graphical, and algebraic approaches whenever possible. Includes applications throughout the course. Requires a graphing calculator, with the TI-84 Plus series recommended, and access to an online homework management system.

### Goals, Topics, and Objectives

- To study the nature and significance of calculus for students of science, technology, engineering, and mathematics disciplines.
- To demonstrate various applications of calculus to problems from the social sciences, physical sciences, and engineering.
- To present an exposition of calculus that incorporates graphical, numerical, and algebraic analysis, without undue emphasis on theoretical abstraction or routine mechanical manipulation.
- To use technology to illustrate calculus concepts and verify calculus solutions to application problems.
- To provide students with an exposure to the logical reasoning of mathematics.

- Applications of Integration
- Compute areas of regions between two curves.
- Compute the volumes of solids of revolution formed by revolving a region about horizontal or vertical lines using the Disk and the Washer Methods.
- Compute the arc length of a curve.
- Solve physical science problems involving mass, work, and force.

- Integration Techniques
- Evaluate definite and indefinite integrals using substitution, parts, trigonometric identities, trigonometric substitution, partial fractions, and completing the square.
- Evaluate definite and indefinite integrals using tables.
- Approximate definite integrals using the midpoint rule, the trapezoidal rule, and Simpson's Rule, with and without the use of a graphing utility.

- Improper Integrals
- Evaluate limits using L'Hopital's Rule.
- Determine whether an improper integral over infinite intervals converges or diverges and, if it converges, find its value.
- Determine whether an improper integral with an unbounded integrand converges or diverges and, if it converges, find its value.
- Use improper integrals to solve applications involving volumes of solids of revolution, areas, arc length, and other net change problems.

- Introduction to Differential Equations
- Determine the order of a differential equation and whether it is linear or nonlinear.
- Verify that a function is a solution to a given differential equation.
- Solve separable differential equations.

- Sequences
- Find the general term of a sequence given the initial terms of the sequence.
- Determine if a sequence converges or diverges and, if it converges, find its limit.
- Determine if a sequence is bounded, monotonic, or neither.
- Determine if a sequence is geometric.
- Draw a graph that represents a sequence using a graphing utility.

- Infinite Series
- Determine if an infinite series is geometric and whether it converges or diverges and, if it converges, find the value to which it converges.
- Determine if an infinite series is telescoping and whether it converges or diverges and, if it converges, find the value to which it converges.
- Determine if an infinite series converges or diverges using the Divergence Test, the Integral Test, and the p-series Convergence Test.
- Estimate what a convergent infinite series converges to by using the Remainder Estimate for the Integral Test and a graphing utility.
- Determine if an infinite series converges or diverges using the Ratio, Root, Comparison, and Limit Comparison Tests.
- Determine if an infinite alternating series converges or diverges using the Alternating Series Test.
- Estimate what a convergent alternating series converges to by using the Remainder Theorem for Alternating Series and a graphing utility.
- Determine if an infinite series converges absolutely or conditionally.
- Solve application problems involving geometric series.

- Power Series
- Find the nth-order Taylor polynomial of a given function centered at "a" and use it to approximate the function at a given value.
- Compute the absolute error in the remainder of a Taylor Polynomial of a given function centered at "a" using Taylor's Theorem along with a graphing utility.
- Find the interval of convergence and radius of convergence of power series by using the convergence and divergence tests of infinite series.
- Find a power series of a function by algebraically combining other power series and find the new interval of convergence.
- Find the derivative and the integral of a power series, the function represented by these processes, and the new interval of convergence for the resulting series.
- Find a Taylor and a MacLaurin Series for a given function and find the interval of convergence.
- Find derivatives and integrals using Taylor series, the functions represented by these processes, and the new intervals of convergence.
- Find limits of functions using Taylor Series.

- Parametric and Polar Equations
- Graph parametric equations with and without the use of a graphing utility, making sure to indicate the orientation of the curve.
- Eliminate the parameter (if possible) in parametric equations to obtain Cartesian equations.
- Find parametric equations corresponding to Cartesian equations.
- Find dy/dx when y and x are defined in terms of a parameter t.
- Find the slope of the tangent line to a parametric curve at a specific point.
- Convert between Cartesian and polar coordinates.
- Graph basic polar curves with and without the use of a graphing utility.
- Find the slope of the tangent line to a polar curve at a specific point.
- Find the points at which a polar curve has a horizontal or a vertical tangent line.
- Find analytically all points of intersection of given polar curves.
- Find the areas of regions bounded by polar curves.
- Find the arc length of a polar curve.

### Assessment and Requirements

- All students will be required to complete a comprehensive final examination that assesses the learning of all course objectives.
- This final exam must be weighted in a manner so that this exam score is worth a minimum of fifteen percent (15%) of the final course grade. In selected semesters this final exam may be a common final exam administered to all sections of Math 183.
- All students will be required to complete online homework. This online homework must be weighted in such a manner so that it is worth between six percent (6%) and twelve percent (12%) of the final course grade.
- Additional assessment of student achievement may include assignments, quizzes, and exams.
- Application problems must not only be included on chapter exams but also on the final exam.
- Some exam problems should require the use of a graphing calculator.

- Application problems must be covered in all mathematics courses. Every section in any course outline that includes application problems must be covered.
- A graphing calculator is required of each student and the Mathematics Department recommends and uses the TI-84 Plus series.
- Access to an online homework management system is also required.

### Outcomes

- Mathematics

- Quantitative Literacy

- Category 3: Mathematics (College Algebra Track)

### Credit for Prior College-Level Learning

A student may receive credit for MATH-183 by earning one of the minimum scores from the following exams:

- At least 3 on the Advanced Placement (AP) Calculus BC test, or
- At least 5 on the International Baccalaureate-Higher Level (IB-HL) Further Mathematics Exam.

A student may receive credit for Math 183 by earning a minimum score of 3 on the Advanced Placement Calculus BC test.