DSCI 27A00 - Calculus I

COURSE OVERVIEW

This course offers a rigorous yet accessible entry point into the core ideas of differential and integral calculus at the university level. It is primarily intended for students pursuing programs in science or mathematics, as well as those preparing for graduate-level studies with a strong quantitative component. Beyond foundational mathematical training, the course serves as direct preparation for advanced coursework in economics and the social sciences — including Columbia University's Intermediate Microeconomics and Intermediate Macroeconomics. Teaching is organized into 12 two-hour sessions, each combining theoretical instruction with applied problem-solving.

Who Should Enroll

Students are expected to bring genuine intellectual curiosity and a solid command of high school-level algebra and trigonometry. No prior university-level mathematics is required.

This course is open to students whose prior exposure to calculus remains limited or partial. Specifically, it welcomes those who scored no higher than 4 on the Calculus BC AP exam, no higher than 5 on the Calculus AB AP exam, no higher than 6 on the IB Calculus exam, or no higher than a B in A-Level Further Mathematics. Students holding a French Baccalauréat (spé maths/Math expert) are equally welcome, provided they have not yet achieved a firm grasp of the key concepts this course addresses: functions, limits, derivatives, differentiation techniques, and integration.

Course Structure and Topics

All lecture slides — which constitute the primary study resource for the final exam — will be distributed throughout the semester. While the course textbook is not compulsory, students may find it a valuable companion. The curriculum spans Chapters 1 through 6 of the reference textbook, organized around six central themes:

• Functions and Models: Representing functions in four distinct ways. An overview of essential mathematical models and function families. Building new functions from existing ones. Exponential functions. Inverse functions and logarithms. Trigonometric functions.
• Limits and Derivatives: Intuitive motivation through tangent lines and velocity. Formal definition and computation of limits. Limit laws and their applications. Precise epsilon-delta definition. Continuity and its implications. Asymptotic behavior. Introduction to derivatives as rates of change. The derivative viewed as a function.
• Differentiation Rules: Differentiating polynomial and exponential functions. Product rule and quotient rule. Trigonometric derivatives. Chain rule. Logarithmic differentiation. Applications to rates of change across disciplines. Modeling exponential growth and decay.
• Applications of Differentiation: Finding extreme values of functions. The Mean Value Theorem and its consequences. Using derivatives to analyze and sketch graphs. Handling indeterminate forms via L'Hôpital's Rule. Solving real-world optimization problems. Introduction to antiderivatives.
• Integrals: Estimating areas and distances. Riemann sums and the definite integral. The Fundamental Theorem of Calculus. Indefinite integrals and the Net Change Theorem. Integration by substitution.
• Applications of Integration: Computing areas enclosed between two curves.

Student performance is evaluated through a system of continuous assessment. Over the course of the semester, students will complete two in-class tests covering the material progressively. A comprehensive final exam, encompassing the entirety of the syllabus, completes the evaluation. The final grade reflects all three components.