When we think of the material covered by a course, it’s common to think about the topics we cover: functions, rates of change, derivatives, etc. But doing math is more than this, and our focus in this course is on more than the individual topics in mathematics that appear in our textbook and video resources. Our course learning goals reflect this: by the end of the course, we expect that:
- Students will be able to appropriately choose and apply various problem solving strategies in the context of mathematical problems, such as identifying relevant data and appropriate mathematical tools, considering a simpler or special case of a given problem, testing specific parameter values, or considering an alternate representation of the scenario.
- Students will be able to effectively communicate mathematical solutions and arguments from calculus, choosing graphical, numerical, algebraic, and/or verbal points of view as appropriate, both with mathematical language, vocabulary, and precision, as well as with non-technical, context-appropriate language.
- And, of course, students will have mastered the mathematical content learning objectives; broadly, course topics include functions and graphs, derivatives and their applications to real-life problems in various fields, and an introduction to integration.
Mathematical Content
This is a first course in single-variable Calculus, and it covers that content. This means that we learn about functions and rates of change, and how change can accumulate. More formally, we cover the following topics (the section and chapter numbers in parentheses are those from our textbook, Hughes Hallett, Calculus):
- Functions and Limits (sections 1.1–1.8)
- The Derivative (sections 2.1–2.6)
- Short-Cuts to Differentiation (sections 3.1–3.4, 3.6, 3.7, 3.9, 3.10)
- Applications of the Derivative (sections 4.1–4.6)
- The Definite Integral (sections 5.1–5.4)
- Antiderivatives and Constructing Antiderivatives (sections 6.1–6.2)
There are many aspects to these very large topics that are fundamental to the course we teach, not least of which are the connections between them that build deep understanding. Throughout Math 115 we seek to build conceptual understanding, emphasizing the meaning of the derivative as a rate of change; the relationships between functions and their derivatives in different representations; application of derivatives and differentiation to real-world situations; modeling; and the accumulation of change (the Fundamental Theorem of Calculus) in applied contexts that extend beyond velocity and acceleration.
