# York MATH 1300 Differential Calculus: Topics, Guide and Practice

> A study guide for York University MATH 1300 (Differential Calculus with Applications): limits, continuity, derivatives and proofs, with practice.

Source: https://practicecalculus.com/courses/york-university-math-1300
Updated: 2026-11-08

## At a glance

- **Course code:** SC/MATH 1300 3.00
- **Credits:** 3.00
- **Prerequisites:** SC/MATH 1520 3.00 or SC/MATH 1710 6.00, or 12U Calculus and Vectors (MCV4U) or equivalent
- **Exclusions:** SC/MATH 1013, 1505, 1530, 1550, GL/MATH/MODR 1930, AP/ECON 1530, SC/ISCI 1401, 1410
- **Covers:** Rates of change, limits and continuity, precise definitions and proofs of key results, building up to the Fundamental Theorem of Calculus
- **Next course:** MATH 1310 (Integral Calculus with Applications)
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- MATH 1300 is a 3-credit first calculus course at York University, recommended for math majors and computer science students.
- It is a theoretical course with precise definitions and proofs of some important results.
- It studies rates of change, limits and continuity, and ends with the Fundamental Theorem of Calculus.
- MATH 1013 (Applied Calculus I) is the more applied alternative, and you can only count one of them.

**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 topic | Practice | Read |
| --- | --- | --- |
| Rates of change | [Average rate of change](/learn/calculus-1/the-derivative/average-rate-of-change), [instantaneous rate](/learn/calculus-1/the-derivative/instantaneous-rate-of-change) | [Slope and rate of change](/learn/calculus-1/functions-warmup/slope-and-rate-of-change) |
| Limits | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [What is a limit](/blog/what-is-a-limit), [limit laws](/rules/limit-laws) |
| Squeeze theorem | [Squeeze theorem](/learn/calculus-1/limits/squeeze-theorem) | [Squeeze theorem](/theorems/squeeze-theorem) |
| Continuity | [Continuity](/learn/calculus-1/limits/continuity) | [Continuity guide](/blog/continuity) |
| Intermediate Value Theorem | [Intermediate Value Theorem](/learn/calculus-1/limits/intermediate-value-theorem) | [IVT statement](/theorems/intermediate-value-theorem) |
| The derivative, defined precisely | [The limit definition](/learn/calculus-1/the-derivative/limit-definition), [differentiability](/learn/calculus-1/the-derivative/differentiability) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Derivative rules | [Mixed rules practice](/learn/calculus-1/derivative-rules/derivative-rules-practice) | [Product rule](/rules/product-rule), [chain rule](/rules/chain-rule) |
| Mean Value Theorem | [Tangent lines at work](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Mean Value Theorem](/theorems/mean-value-theorem), [Rolle's theorem](/theorems/rolles-theorem) |
| Extreme values | [Absolute extrema](/learn/calculus-1/applications-of-derivatives/absolute-extrema) | [Extreme Value Theorem](/theorems/extreme-value-theorem), [Fermat's theorem](/theorems/fermats-theorem) |
| Area and the Fundamental Theorem | [Accumulation functions](/learn/calculus-1/integrals/accumulation-functions), [FTC part 1](/learn/calculus-1/integrals/fundamental-theorem-part-1) | [FTC part 1](/theorems/fundamental-theorem-of-calculus-part-1) |

## MATH 1300 vs MATH 1013

| Course | Title | Style |
| --- | --- | --- |
| MATH 1300 | Differential Calculus with Applications | Theoretical, precise definitions and proofs |
| MATH 1013 | Applied Calculus I | Theory and applications of differential calculus, intro to integrals |

They exclude each other for credit. After MATH 1300, continue to [MATH 1310](/courses/york-university-math-1310). Engineering and physical science students who later need multivariable calculus often take [MATH 2015](/courses/york-university-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.**

$$\frac{d}{dx}x^2 = \lim_{h\to 0}\frac{(x + h)^2 - x^2}{h} = \lim_{h\to 0}(2x + h) = 2x$$

**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](/theorems/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.

## Frequently asked questions

### What is MATH 1300 at York?

MATH 1300 is Differential Calculus with Applications. York's Department of Mathematics and Statistics describes it as the study of rates of change of real valued functions, with precise definitions, a close study of limits and continuity, and proofs of important results, culminating in a theorem about the rate of change of the area under a graph (the Fundamental Theorem of Calculus).

### MATH 1300 vs MATH 1013?

Both are first calculus courses and they are credit exclusions of each other. MATH 1300 is theoretical, with definitions and proofs, and is recommended for math majors and computer science students. MATH 1013, Applied Calculus I, focuses on the theory and applications of differential calculus and introduces integral calculus.

### What do I need for MATH 1300?

SC/MATH 1520 3.00 or SC/MATH 1710 6.00, or 12U Calculus and Vectors (MCV4U) or equivalent.

### What comes after MATH 1300?

MATH 1310, Integral Calculus with Applications. MATH 1300 is also a prerequisite for MATH 1014, MATH 2001 and MATH 2200.