# Waterloo MATH 137 Calculus 1: Topics, Study Guide and Practice

> A study guide for MATH 137 at the University of Waterloo: sequences, limits, continuity, derivatives, Newton's method and the Mean Value Theorem, with practice.

Source: https://practicecalculus.com/courses/university-of-waterloo-math-137
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 137
- **Units:** 0.50
- **Prerequisites:** 4U Calculus and Vectors
- **Antirequisites:** MATH 116, 117, 127 and 147
- **Covers:** Absolute values and inequalities, sequences, limits, continuity, the Intermediate Value Theorem, derivatives, linear approximation, Newton's method, the Mean Value Theorem
- **Next course:** MATH 138 (Calculus 2 for Honours Mathematics)
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- MATH 137 is the first calculus course for Honours Mathematics students at Waterloo, worth 0.50 units.
- It is a differential calculus course that starts with inequalities and sequences before limits of functions.
- It covers Newton's method and uses the Mean Value Theorem to bound errors.
- MATH 147 is the advanced-level alternative, and MATH 138 comes next.

**MATH 137** is Calculus 1 for Honours Mathematics at the University of Waterloo, the calculus course most first-year math students take in the fall alongside MATH 135. It treats differential calculus with more care than a typical first course, starting from inequalities and sequences and ending with error bounds from the Mean Value Theorem. This guide maps every MATH 137 topic to free practice so you can learn by solving problems.

## What Waterloo MATH 137 covers

The Waterloo academic calendar lists the topics as absolute values and inequalities, sequences and their limits, limits of functions and continuity, the Intermediate Value Theorem and approximate solutions of equations, derivatives, linear approximation and Newton's method, applications of derivatives, and the Mean Value Theorem with error bounds and other applications. Some topics are illustrated with computer software.

Compared with a standard **Calculus 1**, MATH 137 covers all of differential calculus but leaves integration for MATH 138. It adds sequences at the start and puts more weight on approximation: Newton's method, linear approximation and bounding the error.

## MATH 137 topic map

| MATH 137 topic | Practice | Read |
| --- | --- | --- |
| Absolute values and inequalities | [Functions and notation](/learn/calculus-1/functions-warmup/function-notation) | [Absolute value function](/functions/absolute-value-function) |
| Sequences and their limits | [Sequences and their limits](/learn/calculus-2/sequences-and-series/sequences) | [Squeeze theorem](/theorems/squeeze-theorem) |
| Limits of functions | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [How to evaluate limits](/blog/how-to-evaluate-limits), [limit laws](/rules/limit-laws) |
| Continuity and the IVT | [Continuity](/learn/calculus-1/limits/continuity) | [Intermediate Value Theorem](/theorems/intermediate-value-theorem), [continuity](/blog/continuity) |
| The derivative | [The limit definition](/learn/calculus-1/the-derivative/limit-definition) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Derivative rules | [Mixed rules practice](/learn/calculus-1/derivative-rules/derivative-rules-practice) | [Chain rule](/rules/chain-rule), [product and quotient rules](/blog/product-and-quotient-rules) |
| Linear approximation and Newton's method | [Tangent lines at work](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Linear approximation](/blog/linear-approximation) |
| The 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) |
| Applications of derivatives | [Increasing and decreasing](/learn/calculus-1/applications-of-derivatives/increasing-and-decreasing) | [Curve sketching](/blog/curve-sketching-with-derivatives), [optimization problems](/blog/optimization-problems) |
## MATH 137 vs MATH 147 vs MATH 127

| Course | Title | Who it is for |
| --- | --- | --- |
| MATH 137 | Calculus 1 for Honours Mathematics | Most incoming Honours Mathematics students |
| MATH 147 | Calculus 1 (Advanced Level) | Students who want a theorem and proof focus, by admission |
| MATH 127 | Calculus 1 for the Sciences | Science students |

These courses are antirequisites, so you take one of them. Waterloo also offers a physics-based section of MATH 137 that covers the same topics with physics examples. After MATH 137, continue to [MATH 138](/courses/university-of-waterloo-math-138), then [MATH 237](/courses/university-of-waterloo-math-237).

## Example problems at the MATH 137 level

**1. An absolute value inequality.** Solve $|2x - 3| < 5$:

$$-5 < 2x - 3 < 5 \quad\Rightarrow\quad -1 < x < 4$$

**2. Newton's method.** Approximate $\sqrt{2}$ as a root of $f(x) = x^2 - 2$, starting at $x_0 = 1$. With $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$:

$$x_1 = 1 - \frac{-1}{2} = 1.5, \qquad x_2 = 1.5 - \frac{0.25}{3} \approx 1.4167$$

which is already close to $\sqrt{2} \approx 1.4142$.

**3. An error bound from the MVT.** Show $|\sin a - \sin b| \le |a - b|$. By the Mean Value Theorem, $\sin a - \sin b = \cos(c)(a - b)$ for some $c$ between $a$ and $b$, and $|\cos c| \le 1$.

## How to do well in MATH 137

- **Take inequalities seriously.** Limits of sequences and error bounds are arguments with inequalities. Practice them early and often.
- **Make derivatives automatic.** Daily short sets in [practice](/practice) on the [power rule](/rules/power-rule) and [chain rule](/rules/chain-rule) free up time for proofs.
- **Know each theorem's hypotheses.** Before using the [Mean Value Theorem](/theorems/mean-value-theorem), check continuity and differentiability.
- **Keep the formulas handy.** The [calculus formula sheet](/formulas) and our list of [common calculus mistakes](/blog/common-calculus-mistakes) prevent the most costly slips.

> **Tip:** when a Newton's method question asks for accuracy, keep iterating until two consecutive values agree to the required decimal places.

## Frequently asked questions

### What is MATH 137 at Waterloo?

MATH 137 is Calculus 1 for Honours Mathematics. It covers sequences and their limits, limits of functions and continuity, derivatives and linear approximation, Newton's method, and the Mean Value Theorem with applications.

### What comes after MATH 137?

MATH 138, Calculus 2 for Honours Mathematics, which needs at least 60% in MATH 137. It covers integration, differential equations, series and Taylor polynomials.

### Is MATH 137 hard?

Waterloo's own advising notes that most incoming students find MATH 135 and MATH 137 challenging enough in their first term. The course goes beyond computing derivatives into why the results hold, so steady weekly practice matters.

### MATH 137 vs MATH 147: which should I take?

MATH 147, Calculus 1 (Advanced Level), focuses on theorems and rigorous proofs and has its own admission requirements. No major requires it. MATH 137 is the standard choice for incoming Honours Mathematics students.

### How should I prepare for MATH 137?

Review 4U Calculus and Vectors, especially derivative rules, then practice working with absolute value inequalities and limits. The free Calculus 1 path on PracticeCalculus covers the computational side step by step.