# Queen's MATH 121 Differential and Integral Calculus: Study Guide

> A study guide for Queen's University MATH 121: differentiation, integration, differential equations and multivariable calculus, with free practice.

Source: https://practicecalculus.com/courses/queens-university-math-121
Updated: 2026-11-08

## At a glance

- **Course code:** MATH 121
- **Units:** 6.00 (full year)
- **Learning hours:** 240 (72 lecture, 24 tutorial, 144 private study)
- **Prerequisites:** None. Recommended: MHF4U and MCV4U or equivalent, or 4U AFIC, or permission of the Department
- **Exclusions:** Maximum of 6.0 units from MATH 120, 121, 123, 124, 126, 127, 128; maximum of one course from MATH 121 and MATH 130
- **Covers:** Differentiation and integration with applications to biology, physics, chemistry, economics and social sciences, differential equations, multivariable differential calculus
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- MATH 121 is a 6.00-unit, full-year calculus course at Queen's for students majoring outside Mathematics and Statistics.
- It covers differentiation and integration with applications in the sciences, economics and social sciences.
- It also covers differential equations and multivariable differential calculus.
- MATH 123 and MATH 124 split the same material into two 3.00-unit terms.

**MATH 121** is Differential and Integral Calculus at Queen's University in Kingston. It is a full-year, 6.00-unit course, the main calculus course for students in life sciences, health sciences, economics and other programs outside Mathematics and Statistics. In one year it covers both derivatives and integrals, then adds differential equations and a first look at calculus in several variables. This guide maps every MATH 121 topic to free practice.

## What Queen's MATH 121 covers

The Queen's Arts and Science calendar describes MATH 121 as differentiation and integration with applications to biology, physics, chemistry, economics and social sciences; differential equations; and multivariable differential calculus. Learning hours are listed as 240: 72 lecture, 24 tutorial and 144 private study. There is no formal prerequisite, but MHF4U and MCV4U (or equivalent), 4U AFIC, or permission of the Department are recommended.

The calendar also lists learning outcomes: applying differential and integral calculus to technical problems, extending one-variable analysis to multivariable functions, working with standard models, and communicating results in proper notation. Together that is a standard **Calculus 1** and most of **Calculus 2**, with some **Calculus 3** at the end. The MATH 124 entry notes that its topics are the same as the winter term of MATH 121: techniques of integration, differential equations and multivariable differential calculus.

## MATH 121 topic map

| MATH 121 topic | Practice | Read |
| --- | --- | --- |
| Limits and rates of change | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit), [instantaneous rate](/learn/calculus-1/the-derivative/instantaneous-rate-of-change) | [How to evaluate limits](/blog/how-to-evaluate-limits) |
| Differentiation rules | [Mixed rules practice](/learn/calculus-1/derivative-rules/derivative-rules-practice) | [Chain rule](/rules/chain-rule), [product rule](/rules/product-rule) |
| Exponential and log models | [Exponential and log derivatives](/learn/calculus-1/derivative-rules/exponential-and-log-derivatives) | [Exponential functions](/functions/exponential-functions) |
| Applications of derivatives | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization), [related rates](/learn/calculus-1/applications-of-derivatives/related-rates) | [Optimization problems](/blog/optimization-problems) |
| Integration and the FTC | [Riemann sums](/learn/calculus-1/integrals/riemann-sums), [FTC part 2](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [FTC guide](/blog/fundamental-theorem-of-calculus) |
| Techniques of integration | [u-substitution](/learn/calculus-1/integrals/u-substitution), [integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts guide](/blog/integration-by-parts) |
| Applications of integration | [Area between curves](/learn/calculus-1/applications-of-integrals/area-between-curves), [average value](/learn/calculus-1/applications-of-integrals/average-value) | [Area between curves guide](/blog/area-between-curves) |
| Differential equations | [Separable equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations), [logistic growth](/learn/calculus-2/improper-integrals-and-applications/logistic-growth) | [Slope fields](/learn/calculus-2/improper-integrals-and-applications/slope-fields) |
| Functions of several variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives guide](/blog/partial-derivatives) |
| Gradient and optimization | [Gradient](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives), [second derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Lagrange multipliers](/blog/lagrange-multipliers) |

## First-year calculus options at Queen's

| Course | Units | Who it is for |
| --- | --- | --- |
| [MATH 120](/courses/queens-university-math-120) | 6.00 | Majors or Joint Honours in Mathematics, Statistics or Physics |
| MATH 121 | 6.00 | Majors outside Mathematics or Statistics |
| MATH 123 and MATH 124 | 3.00 each | The same material split into two terms |
| MATH 126 (or 127 and 128) | 6.00 | Mainly BAH students, social science applications |

You can count at most 6.0 units from MATH 120, 121, 123, 124, 126, 127 and 128. After MATH 121, a common next calculus course is [MATH 221](/courses/queens-university-math-221).

## Example problems at the MATH 121 level

**1. A biology model.** A population grows as $P(t) = 200e^{0.05t}$. Its growth rate is $P'(t) = 10e^{0.05t}$, which is $10$ per unit time at $t = 0$.

**2. Integration by parts.**

$$\int_0^1 x e^{2x}\,dx = \left[\tfrac{x}{2}e^{2x} - \tfrac{1}{4}e^{2x}\right]_0^1 = \frac{e^2 + 1}{4}$$

**3. Newton's law of cooling.** $\frac{dT}{dt} = -k(T - 20)$ has solution $T = 20 + (T_0 - 20)e^{-kt}$.

**4. A partial derivative in economics.** For $Q = 10K^{0.5}L^{0.5}$, the marginal product of labour is $\frac{\partial Q}{\partial L} = 5K^{0.5}L^{-0.5}$.

## How to do well in MATH 121

- **Use the full year well.** A 6.00-unit course rewards steady work. Do a short practice set every week rather than cramming before exams.
- **Go to tutorials.** The calendar lists 24 tutorial hours. They are the best place to ask about the first step.
- **Connect to your field.** The course uses biology, physics, chemistry and economics applications, so translate each word problem into a function first.
- **Review over the winter break.** Integration techniques in the second term build directly on derivative rules from the first.

> **Tip:** for any separable differential equation, check your solution by substituting it back into the equation.

## Frequently asked questions

### What is MATH 121 at Queen's?

MATH 121 is Differential and Integral Calculus, a 6.00-unit course. The Queen's Arts and Science calendar lists differentiation and integration with applications to biology, physics, chemistry, economics and social sciences; differential equations; and multivariable differential calculus. It is intended for students who wish to major or specialize in a subject other than Mathematics or Statistics.

### Do I need calculus before MATH 121?

There is no formal prerequisite. The calendar recommends MHF4U and MCV4U (or equivalent), or 4U AFIC, or permission of the Department.

### MATH 121 vs MATH 120?

MATH 120 is the 6.00-unit calculus course for students planning a Major or Joint Honours Plan in Mathematics, Statistics or Physics, and it includes sequences and series. MATH 121 is for other majors. You can count at most 6.0 units from the first-year calculus courses.

### What can I take after MATH 121?

MATH 121 is a prerequisite for second-year courses such as MATH 221 (Vector Calculus) and, with linear algebra, MATH 280 (Advanced Calculus) and MATH 225 or 231 (differential equations).