MAT210 One-Variable Calculus with Proofs

Schedule: TTh at 1:30 PM, Friday precept; Fall only

Brief Course Description: MAT 210 will survey the main ideas of calculus in a single variable incorporating an introduction to formal mathematical proofs. The course will place equal emphasis on theory (how to construct formal mathematical definitions and rigorous, logical proofs) and on practice (concrete computational examples involving integration and infinite sequences and series). The course will begin with integrals and computing areas using the technique of mathematical induction. We will continue with basic set theory, functions and properties of the real number system. We will introduce further proof techniques, such as proofs by contradiction and construction of counterexamples. Next we will work with areas and volumes and integration, discussing both the rigorous theoretical definitions and concrete calculations. We will continue with limits and derivatives, focusing on developing the theoretical foundations for the standard definitions and theorems, culminating in the Fundamental Theorem of Calculus, integration by substitution and integration by parts. The course will end with a treatment of sequences and series of real numbers, convergence tests and Taylor’s theorem.

Why take this course?: This course provides a more theoretical foundation in single variable calculus than MAT104, intended to prepare students better for a first course in real analysis (MAT215), but it covers all the computational tools needed to continue to multivariable calculus (MAT201 or MAT203).

Who takes this course?: This course is ideal for prospective mathematics majors who need to start their calculus work at the 104 level. It is suitable for anyone who wants to get an introduction to rigorous mathematical thinking as a preparation for further proof-based courses in the department. So, like 215, it may be of interest to prospective majors in philosophy, physics, computer science or math track economics as well as those interested in mathematics.

Prerequisites: MAT103 or equivalent. We will assume that students have experience with limits, including l’Hôpital’s rule, as well as derivatives, including the standard applications of finding maxima and minima and curve-sketching. We will assume that students have some experience working with definite and indefinite integrals and antiderivatives (for instance, how to integrate a polynomial and how to find areas of regions in the plane bounded by simple curves).