Dalhousie MATH 1000 Calculus I: Syllabus, Topics and Practice

MATH 1000 is Differential and Integral Calculus I at Dalhousie University in Halifax. The calendar calls it a self-contained introduction to calculus, so it starts from functions and limits and does not assume you have seen calculus before. It covers the derivative rules and their applications and ends with a first look at integration. This guide maps each MATH 1000 topic to free practice so you can learn every idea by solving problems.

What Dalhousie MATH 1000 covers

The Dalhousie academic calendar lists the topics as functions, limits, differentiation of polynomial, trigonometric, exponential and logarithmic functions, the product, quotient and chain rules, applications of differentiation, antiderivatives and definite integrals, and integration by substitution. It is a 3 credit hour lecture course. The prerequisite is Nova Scotia Mathematics advanced 11 and 12 or pre-calculus, and the calendar says pre-calculus is highly recommended.

This lines up closely with a standard Calculus 1 course. Integration techniques beyond substitution, such as integration by parts, wait for MATH 1010.

MATH 1000 topic map

MATH 1000 in the Dalhousie sequence

MATH 1000 is equivalent to MATH 1280, and MATH 1215 is an exclusion.

Example problems at the MATH 1000 level

1. A limit by factoring.

2. The quotient rule. For f(x) = \frac{\ln x}{x}, f'(x) = \frac{1 - \ln x}{x^2}, which is zero at x = e.

3. Substitution. With u = x^2 + 1:

How to do well in MATH 1000

• Take the pre-calculus advice seriously. Logs, exponents and trig identities appear in almost every derivative. Our warm-up unit is a quick check.
• Practise rules in mixed sets. Real problems mix the product, quotient and chain rules, so practise them together in mixed rules practice.
• Set up word problems carefully. For optimization and related rates, draw a picture and name each variable before you differentiate.
• Check antiderivatives by differentiating. It is the fastest way to catch a sign or constant error.

Common mistakes in a first calculus course

• Dropping the inner derivative. With the chain rule, the derivative of \sin(3x) is 3\cos(3x), not \cos(3x). Always ask what is inside.
• Forgetting the constant. Every indefinite integral needs + C. On a definite integral, it cancels, so it does not appear.
• Mixing up the quotient rule order. It is the bottom times the derivative of the top, minus the top times the derivative of the bottom, all over the bottom squared.
• Changing variables halfway. With substitution, rewrite everything in terms of u, including dx, before you integrate.

Our list of common calculus mistakes covers more of these, with examples.

Tip: most limit questions that give \frac{0}{0} can be solved by factoring and cancelling, or by multiplying by a conjugate.