York MATH 1300 Differential Calculus: Topics, Guide and Practice

MATH 1300 is Differential Calculus with Applications at York University in Toronto. York's mathematics department recommends it for math majors and computer science students, and it is more theoretical than many first calculus courses: it defines limits and derivatives precisely and proves some of the key results. This guide maps the MATH 1300 topics to free practice so you can build the computational skills that support the theory.

What York MATH 1300 covers

York's Department of Mathematics and Statistics describes differential calculus as the study of rates of change of real valued functions. In MATH 1300 this notion is precisely defined and key facts about it are proved. That requires a close study of limits and continuity, and the course culminates in the proof of a theorem about the rate of change of the area bounded by the graph of a function, which is the Fundamental Theorem of Calculus. The department calls it a theoretical course emphasizing precise definitions and featuring proofs of some important results.

The description does not give a week-by-week list, so the table below follows the usual order of a proof-based Calculus 1 course. Check your section's outline for the exact topics.

MATH 1300 topic map

MATH 1300 vs MATH 1013

They exclude each other for credit. After MATH 1300, continue to MATH 1310. Engineering and physical science students who later need multivariable calculus often take MATH 2015.

Example problems at the MATH 1300 level

1. A limit by definition. Show \lim_{x\to 2}(3x - 1) = 5. Given \varepsilon > 0, take \delta = \frac{\varepsilon}{3}. If |x - 2| < \delta, then |(3x - 1) - 5| = 3|x - 2| < \varepsilon.

2. The squeeze theorem. Since -x^2 \le x^2\sin\frac{1}{x} \le x^2, and both bounds go to 0, \lim_{x\to 0} x^2\sin\frac{1}{x} = 0.

3. A derivative from the definition.

4. The IVT. f(x) = x^3 + x - 1 is continuous, f(0) = -1 and f(1) = 1, so it has a root between 0 and 1.

How to do well in MATH 1300

• Learn definitions word for word. In a proof-based course, a precise definition is the starting point of every argument.
• Check every hypothesis. Before using the Mean Value Theorem, say why the function is continuous and differentiable.
• Keep computing too. Proofs get the attention, but exams still ask you to find limits and derivatives quickly. Short daily sets help.
• Write proofs in full sentences. Explain each step, and say which result you are using.

Tip: for an epsilon-delta proof, work backwards on scrap paper to find \delta, then write the proof forwards.