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

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 vs MATH 147 vs MATH 127

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, then MATH 237.

Example problems at the MATH 137 level

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

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)}:

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 on the power rule and chain rule free up time for proofs.
• Know each theorem's hypotheses. Before using the Mean Value Theorem, check continuity and differentiability.
• Keep the formulas handy. The calculus formula sheet and our list of 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.