MA 115 - Calculus I
This two-semester introduction to calculus, MA 115 and MA 116, prepares students for sophomore-level topics in mathematical analysis (differential equations and vector calculus), and calculus-based subjects in science and engineering. Upon completion of the course, students will have a working knowledge of the fundamental definitions and theorems of elementary calculus, be able to complete routine derivations associated with calculus, recognize elementary applications of differential and integral calculus, and be literate in the language and notation of calculus.
Term I (MA115) develops the differential and integral calculus for functions of one variable.
Term II (MA116) is an introduction to differential and integral calculus for parametric curves, multi-variable functions, and power series representations.
Upon completing this course, it is expected that a student will be able to do the following:
1. Mathematical Foundations
- Limits of Indeterminate Forms: Explain the concept of a limit and evaluate elementary examples of indeterminate forms.
- Continuity: Demonstrate a working knowledge of continuity for functions of one variable.
- Derivative - First Principles: State and apply the fundamental definition of the derivative, understand its relationship to the tangent line, and recognize when a function is not differentiable.
- Evaluating Derivatives: Correctly evaluate the derivative of any function constructed via composition, multiplication, division, and addition of elementary functions.
- Implicit Functions: Distinguish between implicitly- and explicitly-defined functions and be able to determine derivative information for implicit functions.
- Definite Integral - First Principles: Describe the definite integral as the signed area under the curve, y = g(x), and state the definition as the limit of Riemann sums that approximate this area.
- Integration Techniques: Successfully apply the Substitution Method and Integration by Parts to express antiderivatives in terms of elementary functions.
- Fundamental Theorem: Evaluate definite integrals and demonstrate a working knowledge of the inverse relationship between differentiation and integration.
2. Applications of Mathematics:
- Curve Sketching: Use information from the first and second derivatives to understand the behavior of a function and to sketch its graph.
- Optimization: Solve elementary optimization problems and characterize the critical points of functions of one variable.
- Initial Value Problems: Explain what is meant by the most general anti-derivative and solve elementary initial value problems of the form x''(t) = g(t).
- Application of Integration: Describe the area of a planar region as a definite integral and recognize elementary applications for which the definite integral is the appropriate tool.
Stewart, James, Calculus: Concepts and Contexts, 4th Edition, Brooks/Cole Pub., 2008.